Statistics

🗂️ THE 10 SYLLABUS TOPICS — how this file is divided

The FAA syllabus lists 10 statistics topics, and the 2024 paper asked exactly one question from each, in order. This file is being divided to match. Each topic starts with a blue TOPIC banner and ends with a grey END banner, so you always know which topic you are reading.

#	Syllabus topic	Divided?
(i)	Primary and secondary data	✅
(ii)	Methods of collecting primary & secondary data	✅
(iii)	Preparation of questionnaires	✅
(iv)	Tabulation and compilation of data	✅
(v)	Measures of central tendency	✅
(vi)	Theory of probability	✅ written in full — was entirely missing
(vii)	Theory of attributes	✅
(viii)	Theory of index numbers	✅
(ix)	Demography — Census	✅
(x)	Vital statistics	✅

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⬜ FOUNDATION — General Concepts

Background only — meaning, characteristics, limitations and basic vocabulary. Not one of the 10 syllabus topics. Topic (i) begins after this.

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1. Five Basics / Fundamentals of Statistics

1.1 Variable

An attribute or characteristic of an item being analysed.

1.2 Population / Universe

All the members, or the entire group of units, which is the focus of study. Example: all the citizens of a country; the residents of a particular geographical location.

1.3 Parameter

A characteristic used to define a given population — a numeric measure. Example: average income of every individual in a country.

1.4 Sample

A set of observations drawn from a population — a subset of the population selected for analysis. Example: a sample of a medicine.

1.5 Statistic

A characteristic that defines the given sample — a numerical measure of the sample. Example: number of RBCs in a blood sample.

The pairing to remember: a parameter describes a population; a statistic describes a sample.

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2. Origin and Definition

Derivation of the word "Statistics":

graph TD
    A[Word: Statistics] --> B["==Status==<br>(Latin word)"]
    A --> C["==Statista==<br>(Italian word)"]
    B --> D["Political State<br>(Political status)"]
    C --> E["(Statesman)"]

Definition — Statistics is the study of:

graph TD
    A["==Collection=="] --> B["==Classification / organisation=="]
    B --> C["==Analysis=="]
    C --> D["==Presentation / Interpretation=="]
    D -.-> E[of Data]

Key people

What statistics is, and what it deals with

Statistics is a branch of Science / Mathematics / Mathematical Science.

graph TD
    A[Data] --> B["Qualitative data<br>(Attributes / characteristics)"]
    A --> C["Quantitative data<br>(Number / Quantity)"]

Data + Statistical Techniques = Statistics

Characteristics of statistics

• Data is numerically expressed
• A systematic manner is followed
• Expresses relations within the data (comparison)
• Provides metrical results
• Complexities in data are simplified

Fields of application

Economics · Commerce · Banking & Insurance · Public Opinion (exit polls, etc.)

Limitations — what statistics cannot do

• ❌ Study qualitative phenomena
• ❌ Reveal the entire background
• ❌ Be precise and exact (it works on averages)
• ❌ Resist misuse — the full context is always needed

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3. Other Important Terms

Skewness and normal distribution

      Height of                             Income
      Students
          . ˙ .                           . ˙ .       . ˙ .
        ˙       ˙    Normal     Bihar   ˙       ˙   ˙       ˙   J&K
      ˙           ˙  Distribution     ˙           ˙           ˙
  +---+---+---+---+---+---+---+    +---+---+---+---+---+---+---+---+
 0.8 1.0 1.2 1.4 1.6 1.8 2.0      5K  10K 15K 20K 25K 30K 40K
                                  ^ Left Tail            Right Tail ^
                                  (Negative)             (Positive)

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🔷 TOPIC (i) — PRIMARY AND SECONDARY DATA

Syllabus topic (i) of 10 · runs until Topic (ii)

Covers: data collection · primary data · secondary data · primary vs secondary · types of data · census vs sample and sampling error 🎯 Asked: 2024 · Q71 — secondary data is primarily sourced from?

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1. Data Collection

The step after the research problem is identified and the research design is made is data collection.

Based on the approach to information gathering, data is categorised as:

graph TD
    A[Data] --> B[Primary Data]
    A --> C[Secondary Data]

2. Types of Data

➕ Added — not in the original notes.

2.1 By nature

2.2 Quantitative data splits again

2.3 By time reference

The quick test: count it in whole numbers → discrete · measure it to decimals → continuous · only label it → qualitative.

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3. Primary Data

Data collected for the first time by the researcher himself, for a specific purpose. Example: a survey on job satisfaction of employees; health needs of a community.

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4. Secondary Data

Data that is already collected, produced or published by others. Example: use of hospital records or census records.

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5. Primary vs Secondary Data

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6. Census vs Sample, and Errors in Collection

➕ Added — not in the original notes.

6.1 Census (complete enumeration) vs Sample survey

Testing that destroys the unit (bulbs, matchsticks, blood) must use sampling — a census would destroy the whole population.

6.2 Sampling error vs Non-sampling error

6.3 Essentials of a good sample

Representative · Adequate (large enough) · Independent · Homogeneous · Free from bias

6.4 Sampling methods

In probability sampling every unit has a known, non-zero chance of selection — which is what makes the sampling error calculable.

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🔷 TOPIC (ii) — METHODS OF COLLECTING PRIMARY AND SECONDARY DATA

Syllabus topic (ii) of 10 · runs until Topic (iv)

Covers: primary methods — experiment, interview, observation, questionnaire, schedules, local correspondents, projective techniques · secondary sources · precautions 🎯 Asked: 2024 · Q72 — example of primary data collection · 2022 · Q35 — method of collecting secondary data · 2022 · Q31 — sampling error

⚠️ Topic (iii) — Preparation of Questionnaires — is nested inside this topic, because the questionnaire is one of the primary collection methods. It is bracketed separately below.

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PART A — PRIMARY DATA COLLECTION METHODS

Note: the original notes number these 1, 2 … then jump to 6–9 for the "other important methods" and 10 for projective techniques. That numbering is kept as written.

1. Experiment Method

Manipulating one variable to determine whether changes in it cause changes in another.

Three kinds of variables [x + y = z]

• Independent variable
• Dependent variable
• Controlled variable / controlled environment

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2. Interview Method

Based on oral or verbal stimuli.

graph TD
    A[Interview Method] --> B[Direct Interview]
    A --> C[Indirect Interview]

2.1 Direct Interview

The investigator directly contacts the respondent.

2.2 Indirect Interview

Also called Indirect Personal Interview or Indirect Oral Interview. The investigator collects information from someone else who has the required information about the subject.

2.3 Telephonic Interview

The dominant method. Questions are asked over the phone.

2.4 Other important interview methods

2.5 Types of interview questions

⚠️ Leading and loading questions must be avoided in questionnaire design — see Topic (iii).

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3. Observation Method

Data from the field is collected by the observer through observation.

3.1 Classification of the observation method

graph TD
    Root[Classification] --> Plan[Plan]
    Root --> Conditions[Conditions]
    Root --> Participation[Participation]

    Plan --> Structured
    Plan --> Unstructured

    Conditions --> Controlled["Controlled<br>(Controlled conditions)"]
    Conditions --> Uncontrolled["Uncontrolled<br>(Naturalistic conditions)"]

    Participation --> C_Obs["==Complete observer=="]
    Participation --> Obs_Part["==Observer as participant=="]
    Participation --> Part_Obs["==Participant as observer=="]
    Participation --> C_Part["==Complete participant=="]

    C_Part -.-> Note["(Participant observation & Non-participant obs)"]

Disguised observation — the observer's identity is unknown to the subject. Example: "mystery shopping".

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🔷 TOPIC (iii) — PREPARATION OF QUESTIONNAIRES

Syllabus topic (iii) of 10 · nested inside Topic (ii), because the questionnaire is a primary collection method

Covers: what a questionnaire is · merits and demerits · principles of preparation · construction steps (including the pre-test / pilot survey) · distribution · questionnaire vs schedule 🎯 Asked: 2024 · Q73 — which statements on questionnaire preparation are correct?

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1. The Questionnaire

The heart of the survey operation.

A set of printed or written questions with a choice of answers, used for surveys or statistical studies — usually mailed.

Used to obtain: information · feelings · beliefs · perceptions about the research participant. Quantitative, qualitative and mixed data can all be collected through a questionnaire.

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2. Principles for Preparing a Questionnaire

1. Should be research oriented — the objective must be fulfilled
2. The research participant should be kept in view
3. Simple and familiar language should be used
4. Questions should be clear, easy to understand and free of ambiguity
5. Avoid leading and loading questions; avoid personal questions
6. Avoid double-barrelled questions
7. Avoid double-negative questions
8. Use mutually exclusive and exhaustive response categories for closed-ended questions
9. The questionnaire should be properly organised and easy to use
10. Proper instructions for filling it must be provided

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3. Construction of a Questionnaire

graph TD
    S1["==Determine Research objective=="] --> S2["==Type of questionnaire to use=="]
    S2 --> S3["==Determine the content of Individual=="]
    S3 --> S4["==Type of Questions to use=="]
    S4 --> S5["==Determine the wording of questions=="]
    S5 --> S6["==Determining the sequence of Questions=="]
    S6 --> S7["==Determining the length of Questionnaire=="]
    S7 --> S8["==Layout=="]
    S8 --> S9["==Check the Questions=="]
    S9 --> S10["==Pre-test (Pilot survey)=="]
    S10 --> S11["==Revision & Final draft=="]

⭐ Step 10 — the Pre-test (Pilot survey) — is the step the exam asks about. It comes after checking the questions and before the final draft.

4. Questionnaire Distribution

graph LR
    A[Questionnaire distribution] --> B[Printed]
    B --> B1[mailed]
    B --> B2[Fax]

    A --> C[Electronic]
    C --> C1[email]

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PART B — TOPIC (ii) RESUMED: REMAINING PRIMARY METHODS

Schedules, local correspondents and projective techniques are collection methods, so they belong to Topic (ii). The "Questionnaires vs Schedules" comparison below also relates to Topic (iii).

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4. Schedules

A questionnaire filled in by enumerators. The investigator or enumerator personally visits informants with the questionnaire, asks the questions and notes the responses.

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5. Data from Local Correspondents

The investigator appoints a local correspondent or agent who collects information on the investigator's behalf.

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6. Questionnaires vs Schedules

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7–9. Other Important Methods

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10. Projective Techniques

Also known as Indirect Interviewing.

• Used to reach underlying motives and intentions
• Applied where the respondent resists revealing them
• Requires intensive training

10.1 Pictorial techniques

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PART C — SECONDARY DATA COLLECTION

Secondary data can be published or unpublished.

1. Published sources

2. Unpublished data

Some research institutes and universities do not publish their data, but it can still be used as a secondary source.

3. Precautions When Using Secondary Data

Demerits of secondary data:

• Proper data-collection procedure may not have been used
• Out-dated data
• Bias
• Does not fulfil the accuracy needs of the research
• Not suitable for the period of investigation

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4. One-Shot Recapitulation

Examples

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🔷 TOPIC (iv) — TABULATION AND COMPILATION OF DATA

Syllabus topic (iv) of 10 · runs until Topic (v)

Covers: Classification of data & its principles · statistical series · time / geographical / magnitude / condition classification · arrangement of data (individual, discrete, continuous) · tabulation & parts of a table · types of tables · graphical representation — histogram, bar diagram, frequency polygon, frequency curve, ogives · true, non-true & open-end classes 🎯 Asked in the papers: 2024 · Q74 — which statement about tabulation is FALSE · 2022 · Q38 — shape of a frequency polygon as classes increase

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Classification of Data

Process of arranging data into different groups or classes based on common characteristics.

Characteristics of classification of Data:-

• It should be unambiguous
• Flexible to adjustments
• It should perform homogeneous grouping
• Its basis should be clearly defined and adhered to.

Objectives:- $\rightarrow$ To simplify the huge data $\rightarrow$ To facilitate comparison. $\rightarrow$ To provide basis for tabulation $\rightarrow$ To make data understandable. $\rightarrow$ Clearly specify similarities and dissimilarities.

Representation of Data

graph LR
    A[Representation of Data] --- B[Text]
    A --- C[Tabular]
    A --- D[Graphical]

Principles of classification:-

1. Exhaustive :-

• Every item should be classified.
• NO Residual or miscellaneous class.

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2. Mutually Exclusive :- Every single item should fall in only one class.

3. Stability :- Common principle should be maintained / followed for whole classification.

4. Suitability :- Classification should be suitable as per objective of research.

5. Flexibility :- Classification should be adjustable to new situation / new adjustments.

6. Homogeneity :- Data items of a class should be similar in characteristics.

Statistical Series or Statistical classification:

Arrangement of data in a logical order.

graph TD
    Root["==Statistical Series=="]
    
    Root --> Char["On the basis of<br>characteristics"]
    Root --> Const["On the basis of<br>construction"]
    
    Char --> C1["Time-Series"]
    Char --> C2["Spatial<br>series"]
    Char --> C3["Condition<br>series"]
    Char --> C4["Magnitude<br>series"]
    
    C2 -.-> OR["(Or)"]
    OR -.-> Simple["Simple<br>(one attribute)"]
    OR -.-> Manifold["Manifold<br>(more than<br>one attributes)"]
    
    Const --> S1["Individual<br>series"]
    Const --> S2["Discrete<br>series"]
    Const --> S3["Continuous<br>series"]

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Important Classification:-

1. Time-Series / Chronological / Temporal classification:-

Based on time of occurence. e.g population of India from 2010 - 2015

2. Geographical / spatial classification:-

Classification on the basis of Area/Region/Location. e.g:- sales report of company

3. Magnitude / Numerical / Quantitative classification:-

Based on quantity

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4. Condition / Seasonal classification:-

When classification is done on the basis of seasonal variations/situations. e.g sales of ice cream / cold drinks in summers and winters.

Simple classification :- Classification on the basis of only one attribute. e.g:- On the basis of Gender

graph TD
    A[Gender] --> B[Male]
    A --> C[Female]

Manifold classification :- Classification based on more than one attributes.

graph TD
    Pop[Population] --> Lit[Literate]
    Pop --> Illit[Illiterate]
    
    Lit --> Emp1[Employed]
    Lit --> Unemp1[Unemployed]
    
    Illit --> Emp2[Employed]
    Illit --> Unemp2[Unemployed]
    
    Emp1 --> M1[Married]
    Emp1 --> UM1[Unmarried]
    
    Unemp1 --> M2[M]
    Unemp1 --> UM2[UM]
    
    Emp2 --> M3[M]
    Emp2 --> UM3[UM]
    
    Unemp2 --> M4[M]
    Unemp2 --> UM4[UM]

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Arrangement of Data

Data can be arranged in two ways:

1. Serial (or) Alphabetic order
2. Ascending (or) Descending order.

e.g. $\rightarrow$ Primary, secondary, Data, Questionnaire (ungrouped data) / Raw data $\downarrow$ Data, Primary, Questionnaire, Secondary.

e.g:- 13, 19, 12, 10, 25, 32, 29, 37 (ungrouped / Raw data) $\downarrow$ Arrayed Data $\rightarrow$ 10, 12, 13, 19, 25, 29, 32, 37 $\rightarrow$ Ascending / Increasing order $\hookrightarrow$ 37, 32, 29, 25, 19, 13, 12, 10 $\rightarrow$ Descending / Decreasing order

Individual series:-

Each item is given separate value. e.g

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2. Discrete Series | Discrete Frequency Distribution :-

Each individual value is presented with frequency.

e.g:-

3. Continuous series | continuous frequency distribution :-

Shows range of values of different items.

e.g:-

• 0-5 $\rightarrow$ class interval = $5 - 0 = \text{\textcircled{5}}$ (Range)
  • $0 \rightarrow$ lower limit
  • $5 \rightarrow$ upper limit
• class mark = mid value = $\frac{\text{lower limit} + \text{upper limit}}{2}$ $= \frac{0+5}{2} = \text{\textcircled{2.5}}$

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Tabulation of Data

Systematic and Logical representation of numeric data in rows ($\rightarrow$ horizontal) and columns ($\downarrow$ vertical).

Objectives:- (1) To simplify the complex data. (2) To bringout important features. (3) To facilitate comparison. (4) To facilitate statistical analysis. (5) To save space and time.

Limitations:- (1) Lack of description (2) Incapable of presenting individual items. (3) Needs ample knowledge & understanding.

General Format:-

Table No: <Title> <Head note> (if any)

• S. Note: -
• Foot note / Note

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Main parts of table

(1) Table NO:-

• First item mentioned on top of table
• Identification and References.

(2) Title:- $\rightarrow$

• Second item, just above the table / by right side of T.NO.

(3) Head-note / Prefatory:-

• 3rd item, just above the table.
• Information about unit of data. e.g.: currency ₹ (or) $ Quintals or tonnes
• Generally given in brackets.

(4) Caption / Col-Headings / characteristic:-

• Top of each column.
• Explains data in column.

(5) Stub / Row Heading:-

• Title of horizontal rows.

(6) Body:-

• Numeric data.
• In cols and rows.

(7) Foot Note:-

• To explain non-explanatory things.
• Exceptions if any.
• Circumstances affecting data.

(8) Source Note:-

• Statement indicating source of data.

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Table NO. 2.1:- Records of Graduation students faculty wise H.Note: (Includes all registered students)

Source note: Admission dept. ABC college. Footnote: Late registrations are not included.

Characteristics of Good table:-

• Title compatible with objective.
• Comparable.
• Ideal size.
• Col-total, Row total, total included.
• Stubs.
• Headings.
• Simple, Economical & Attractive.
• No Abbreviations.
• No Over crowding.

Types of Tables

graph TD
    Root[Types of Tables]
    
    Root --> Obj[objectivity / purpose]
    Root --> Nat[Nature / originality]
    Root --> Char[characteristic / constructio]
    
    Obj --> O1["==General Purpose=="]
    Obj --> O2["==Special Purpose=="]
    
    Nat --> N1["==Primary / original=="]
    Nat --> N2["==Secondary / Derived=="]
    
    Char --> C1["==Simple (1-way)=="]
    Char --> C2["==Complex=="]
    
    C2 --> C2_1["==Double / Two-way=="]
    C2 --> C2_2["==Treble / 3-way=="]
    C2 --> C2_3["==Manifold=="]

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Classification of Tables

• On the basis of "Purpose" ① General Purpose / Master Tables

  • General use.
  • Not meant for special purpose. e.g.: $\rightarrow$ Census

  ② Special purpose table / Summary Table.

  • Derived from general table.
  • Serves special purpose.
  • Useful for calculation of analytical statistics like ratio, percentage etc. e.g.: $\rightarrow$ Calculating Ratio of Male:Female

• On the basis of "Originality" (1) Original Table / Primary Table

  • Data is presented in the form in which it is collected.

  (2) Derived Table / Secondary Table

  • Converted into any form as per requirement. e.g.: Limited columns as required.

• As per construction / characteristics (1) Simple Table

  • "One-way" table.
  • Data presented based on the "one" characteristic only.

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Table 1.1 Faculty-wise Number of students

Source note. One-way Table Foot note.

(2) Complex tables

• More than one attribute presented simultaneously.

  (i) Double / Two-way Table

  • Data is tabulated on the basis of two inter-related characteristics.

Table 1.2 Faculty - wise number of Male and Female students

S. note. F. note.

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• Three-way / Trebles
  • Data is tabulated on the basis of 3-inter-related characteristics.

Table 1.3 Faculty wise, (Gender & semester) list of students

S. note. 3-way Table F. note.

• Manifold (Higher order Table)
  • Data is tabulated on the basis of large no. of interrelated characteristics.

Table 1.4 Faculty wise (UG / PG) Gender based students in each sem.

S. note F. note

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Graphical Representation of Data

Presentation of statistical data on graph paper in the form of lines or curves.

Merits:- (i) Simplifies complex data. (ii) Helps in forecasting. (iii) Variations in the values of variables.

Demerits:- (i) Precise values are not shown. (ii) May give wrong predictions. (iii) Difficult to interpret.

Procedure to construct a Graph:-

               y-axis (ordinate)
                      ^
                      |
         O₂           |           O₁
                      |
<---------------------+---------------------> x-axis
                      |                   (abscissa)
         O₃           |           O₄
                      |
                      v

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(i) Scale:- Appropriate scale. (ii) Heading:- Suitable and precise heading. (iii) Proper Indications:- More than one curve, properly differentiated. (iv) False Base line:- y-axis. (v) Index:- lines & colors. (vi) Data Table:- full knowledge of data. (vii) Drawing line or curve:- Joining the points. (viii) Paper size:- Appropriate size of graph paper. (ix) From L to R & B to Top:- Left to right & Bottom to top.

Frequency Diagram:-

Frequency distribution is represented using graphs.

Histogram $\rightarrow$ Two dimensional diagram, set of rectangles with base as the intervals between class boundaries. ** Histogram is not drawn for discrete data.

Equal classes:-

xychart-beta
title "Histogram (Student marks)"
x-axis ["0-5", "5-10", "10-15", "15-20", "20-25"]
y-axis "Students" 0 --> 50
bar [20, 30, 10, 40, 50]

Unequal classes:-

xychart-beta
title "Histogram (Student marks - Unequal classes)"
x-axis ["0-5", "5-15", "15-20", "20-30", "30-35"]
y-axis "Students" 0 --> 40
bar [20, 30, 10, 40, 10]

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Bar Diagram :-

• Bars with arbitrary width are drawn to represent the data.
• Drawn for both discrete and continuous data.
• Some space has to be left between consecutive bars.

e.g.: Production of Rice in J&K from 1997 - 2004

xychart-beta
title "Bar diagram: Production of Rice"
x-axis ["1997", "1998", "1999", "2000", "2001", "2002", "2003", "2004"]
y-axis "Production (tonnes)" 0 --> 40
bar [10, 20, 30, 15, 20, 25, 30, 35]

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Frequency Polygons:-

Frequency polygon / Frequency polygon curve is constructed by joining the mid points of the frequencies presented using rectangles (histograms). ** constructed for both discrete as well as continuous series.

e.g.: Construct frequency polygon for continuous series:-

xychart-beta
title "Frequency Polygon for Age VS No. of students"
x-axis ["5-10", "10-15", "15-20", "20-25", "25-30"]
y-axis "No. of students" 0 --> 25
bar [5, 10, 15, 20, 5]
line [5, 10, 15, 20, 5]

e.g.:- construct Frequency polygon for discrete series

xychart-beta
title "Frequency polygon Age vs No. of students (discrete series)"
x-axis ["1", "2", "3", "4", "5", "6", "7", "8", "9", "10"]
y-axis "No. of students" 0 --> 25
line [0, 0, 0, 10, 15, 20, 10, 5, 4, 0]

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Frequency curve :-

Obtained by drawing smooth freehand curve passing through the closely associated points. ** Can be drawn for both discrete as well as continuous series.

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Histogram → Frequency Polygon → Frequency Curve is one continuous progression:

1. Histogram — bars over each class
2. Frequency polygon — join the mid-points of the tops of those bars
3. ⭐ Frequency curve — now make the class intervals SMALLER, so the number of classes INCREASES. Each corner of the polygon gets shallower, and the line becomes increasingly SMOOTH. In the limit (infinitely many, infinitely narrow classes) the polygon becomes the frequency curve.

[!pyq] 🎯 2022 · Q38 — "What happens to the shape of a frequency polygon as the number of classes increases?" → ⭐ It becomes increasingly smooth

Also worth knowing: the area under a frequency polygon equals the area of its histogram · a polygon can compare two or more distributions on one graph (a histogram cannot).

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Types of curves on the basis of shape:-

1. Normal curve: $\rightarrow$ symmetric curve / Bell curve / Gaussian distribution
2. U-shaped curve
3. Positive skewed curve
4. Negative skewed
5. Bi-modal curve
6. J-shaped curve
7. Reverse J-shaped curve
8. Mixed:- Multi-modal curve

Cummulative frequency curve / ogive curve :-

Less than (LTB) cumulative frequency $\rightarrow$ Less than ogive curve:

Greater than (GTB) cumulative frequency $\rightarrow$ Greater than ogive curve:

** intersection of Less than ogive & greater than ogive = Median

True class intervals

There is no gap between the successive classes. Upper limit of each class is equal to lower limit of the succeeding class. e.g:-

$\rightarrow$ Class Boundaries

Non-True classes:-

Upper limit of each class is not equal to lower limit of successive class. e.g

$\rightarrow$ class limits

Open-end classes and closed-end classes.

When a class limit is missing either at lower end (first class) or at upper end (last class) or limits are not specified at both ends — open end classes. When limits are specified at both ends — closed end classes.

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examples of open-end classes:-

closed-end example:-

Some important points about compilation & Tabulation of Data:-

1. Repository: Tables to show data in orderly manner.
2. Marginal Frequency: $\frac{\text{Row total}}{\text{Grand total}} = \frac{\text{column total}}{\text{Grand total}}$
3. Relative Frequency = $\frac{\text{Individual frequency}}{\text{Total no. of observations}}$
4. Univariate Frequency Distribution: One variable
5. Bivariate Frequency Distribution: Two variables

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🔷 TOPIC (v) — MEASURES OF CENTRAL TENDENCY

Syllabus topic (v) of 10 · runs until Topic (vi)

Covers: Construction of frequency distributions · inclusive vs exclusive method · Mean (individual, discrete, continuous; combined mean, missing values, correction, weighted mean) · Median · Mode · ➕ added: empirical relation, comparison of the three averages, GM & HM, partition values 🎯 Asked in the papers: 2024 · Q75 — arrange mean / median / mode / n in decreasing order

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Measures of Central Tendency

Construction of frequency distribution:-

Unarranged data can be distributed into classes and corresponding frequencies be assigned. Steps to follow:-

1. Sorting of Data.
2. Calculate the Range.
3. Decide number of classes.
4. Calculate class width.
5. Tally & count the observations.

e.g:- 0, 1, 4, 3, 12, 13, 19, 15, 20, 23, 27, 29, 37, 31, 41, 46, 50, 48 $\rightarrow$ 18 observations.

Step 1:- Sorting the data $\underline{0}, 1, 3, 4, 12, 13, 15, 19, 20, 23, 27, 29, 31, 37, 41, 46, 48, \underline{50}$

Step 2:- calculate the range { 0 - 50 } = 50

Step 3:- Decide the number of classes. Let it be '5'

Step 4:- calculate class width. $\text{width} = \frac{\text{upper Range} - \text{Lowest Range}}{\text{No. of classes}}$

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$\text{width} = \frac{50 - 0}{5} = \frac{50}{5} = \text{==10==}$

Steps:- count observations & tally. No. of observations = frequency.

Inclusive and Exclusive Method of distribution / series.

$\rightarrow$ Inclusive Method / series (Both limits are included)

$\downarrow$ conversion to Exclusive Method distribution

$\text{adjustment } (h) = \frac{\text{Lower limit } (i_2) - \text{Upper limit } (i_1)}{2}$ $h = \frac{LL(i_2) - UL(i_1)}{2}$ $h = \frac{30 - 29}{2} = \text{==0.5==}$

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Mean: Average or the most common value in the collection of numbers.

• Simple Arithmetic Mean
• Weighted Mean.

Simple Arithmetic Mean:- Average of the collection of numbers. Formulas for calculating Simple Arithmetic Mean:

Individual Series

Q,, Find the A. Mean of the given series: $10, 12, 15, 20, 30, 40$

Sol:- $\text{Mean}_{\text{Individual Series}} = \frac{\sum x}{N} = \frac{10 + 12 + 15 + 20 + 30 + 40}{6}$ $\bar{x} = \frac{127}{6} = \text{==21.16==}$

$\bar{x} = \frac{\sum x}{N} = \frac{75}{5} = \text{==15==}$

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Discrete series :-

Note: N = $\sum f = 17$

Now $\bar{x} = \frac{\sum fx}{N} = \frac{88}{17} = 5.17$

Continuous series :-

Now $\bar{x} = \frac{\sum fx}{\sum f} = \frac{780}{22} = \text{==35.45==}$

Combined Mean:-

$\bar{x}_{1,2,3...} = \frac{\bar{x}_1 f_1 + \bar{x}_2 f_2 + \bar{x}_3 f_3 + \dots}{f_1 + f_2 + f_3 + \dots}$

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$\bar{x}_{1,2,3,4} = \frac{\bar{x}_1 f_1 + \bar{x}_2 f_2 + \bar{x}_3 f_3 + \bar{x}_4 f_4}{f_1 + f_2 + f_3 + f_4}$ $= \frac{50 \times 30 + 60 \times 40 + 70 \times 20 + 100 \times 15}{30 + 40 + 20 + 15}$ $\Rightarrow \bar{x}_{1,2,3,4} = \frac{1500 + 2400 + 1400 + 1500}{105}$ $= \frac{6800}{105} = \text{==64.76==}$

Find the missing values:-

Q,, The mean temperature for four days noted is 120°c. If the temperature for day 1, day 2 & day 4 is 30, 35 and 40 respectively. Find the temperature of day 3?

Sol:- $\bar{x} = 120^\circ\text{c}$ $\Rightarrow \frac{\text{day } 1 + \text{day } 2 + \text{day } 3 + \text{day } 4}{4} = 120$ $\Rightarrow \frac{30 + 35 + \text{day } 3 + 40}{4} = 120$ $\Rightarrow \frac{105 + \text{day } 3}{4} = 120$ $\Rightarrow 105 + \text{day } 3 = 480$ $\Rightarrow \text{day } 3 = 480 - 105 = \text{==375}^\circ\text{==}$

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Q,, Find the missing value if the mean value = 10.

$\bar{x} = \frac{\sum fx}{\sum f} = \frac{250 + 20x_1}{75}$ $\Rightarrow 10 = \frac{250 + 20x_1}{75}$ $\Rightarrow 750 = 250 + 20x_1$ $\Rightarrow 500 = 20x_1$ $\Rightarrow \frac{500}{20} = x_1$ $x_1 = \text{==25==}$

Correction in the mean value:-

Q,, In a class, average marks of 30 students is 40. If the correct marks for one student is 46 which was misread as 42. calculate the new correct mean.

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Sol:- $\bar{x} = \frac{\sum x}{N}$ $40 = \frac{\sum x}{30} \Rightarrow \sum x = 40 \times 30 = 1200$

$\sum x =$ Sum of all the marks.

• Add correct marks
• Subtract wrong marks. $= \sum x + 46 - 42$ $= 1200 + 46 - 42 = \text{==1204==}$

New $\sum x = 1204$. $\therefore$ New correct mean $\bar{x} = \frac{\sum x}{N} = \frac{1204}{30} = \text{==40.13==}$

Weighted Arithmetic Mean:-

When the items of a series are not of equal importance / weightage. $\bar{x}_w = \frac{\sum wx}{\sum w}$ where $\bar{x}_w =$ weighted Mean. $\sum wx =$ sum of product of weights and items. $\sum w =$ sum of weights.

Q,, student scores 20 marks in statistics, 35 in english, 40 in maths and 45 in Geography. calculate weighted mean, if the marks are weighted as 2, 1, 3, 4 respectively.

Sol:-

$\bar{x}_w = \frac{\sum wx}{\sum w} = \frac{375}{10} = \text{==37.5==}$

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Median: Median is the middle value which separates the higher half and lower half of data sample/population.

Individual Series :-

• Sort the data in ascending or descending order
• Calculate Median by formula:- $M = \text{value of } \left( \frac{n+1}{2} \right)^{\text{th}} \text{ item}$

e.g 1:- $2, 7, 9, 3, 6, 8, 12$ Sol:- Sort: $2, 3, 6, 7, 8, 9, 12$ $\rightarrow$ Trick: (odd) Middle value = Median. here $n = 7 = \text{no. of observations / items}$ $\therefore M = \text{value of } \left( \frac{n+1}{2} \right)^{\text{th}} \text{ item}$ $= \text{value of } \left( \frac{7+1}{2} \right)^{\text{th}} \text{ item}$ $= \text{value of (4)}^{\text{th}} \text{ item}$ $\Rightarrow \text{==} M = 7 \text{==}$

e.g 2:- $10, 15, 20, 30, 35, 40, 50, 60$ Sol:- Already sorted $\rightarrow$ Trick: $\frac{\text{Sum of two middle value}}{2} = M$ $M = \text{value of } \left( \frac{n+1}{2} \right)^{\text{th}} \text{ item}$ $= \text{value of } \left( \frac{8+1}{2} \right)^{\text{th}} \text{ item}$ $= \text{value of } (4.5)^{\text{th}} \text{ item}$ $= \text{value of } 4^{\text{th}} + 0.5(\text{difference between } 4^{\text{th}} \text{ \& } 5^{\text{th}} \text{ item})$ $= 30 + 0.5(5)$ $M = 30 + 2.5 = \text{==32.5==}$

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e.g 3:- $\rightarrow$ sort values of X.

$M = \text{value of } \left( \frac{n+1}{2} \right)^{\text{th}} \text{ term}$ $= \text{v. of } \left( \frac{7+1}{2} \right)^{\text{th}} \text{ term}$ $= \text{v. of } 4^{\text{th}} \text{ term}$ $M = \text{==20==}$

Discrete Series:-

(X) (frequencies) *1. Sort values in Ascending or decending order. *2. Cummulate frequencies *3. $M = \text{value of } \left( \frac{n+1}{2} \right)^{\text{th}} \text{ item}$

e.g:-

$n = 63$

$M = \left( \frac{n+1}{2} \right)^{\text{th}} \text{ value}$ $= \left( \frac{63+1}{2} \right)^{\text{th}} \text{ value} = \text{==32==} \rightarrow$ check class and corresponding X-value $\therefore M = 50$

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Continuous series:-

(1) Sort groups in ascending or decending order. (2) change inclusive series into exclusive series. (3) cummulate frequencies (4) Calculate $m = \text{value of } \left( \frac{n}{2} \right)^{\text{th}} \text{ item}$

$M = L_1 + \frac{i}{f} (m - c_f)$

e.g:-

$n = 27$ $m = \left( \frac{n}{2} \right)^{\text{th}} \text{ value} = \left( \frac{27}{2} \right)^{\text{th}} \text{ value} = \text{==13.5==} \rightarrow \text{Falls in 30-40 class.}$

Now $M = L_1 + \frac{i}{f} (m - c_f)$ $= 30 + \frac{10}{7} (13.5 - 10)$ $= 30 + \frac{10}{7} (3.5)$ $= 30 + \frac{10}{2}$ $= 30 + 5 = \text{==35==}$

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Mode: (Z): Value in the data set that appears most frequently. (or) value that is repeated maximum number of times.

Individual series:-

observation that is repeated maximum times e.g:- 1, 2, 3, 4, $\underline{3}$, 6, 5, 9, 10, 6, $\underline{3}$, 1 Mode = 3 (repeated maximum times)

e.g 2:- 1, 2, 3, 6, 7, 9 All Mode (no observation repeated).

e.g 3:- 4, $\underline{3}$, 4, 2, $\underline{3}$, 5, 6, 8. Mode = 3 & 4 (Bimodal).

• If there are more than two modes in a data set, it is called Multi-modal data.

[!fix] ❗ MISSING FORMULA — the empirical relation These notes cover Mean, Median and Mode thoroughly but never state the relation that links them:

⭐ Mode = 3 Median − 2 Mean

Equivalently Mean − Mode = 3 (Mean − Median). For a symmetric distribution Mean = Median = Mode. This is the standard one-mark question of this topic and is needed for 2024·Q75-type items where all three must be ordered.

Discrete series:-

Observation that is corresponding to the maximum frequency.

$\therefore Z = \text{==20==}$

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Continuous Series:-

Sol:-

• check if for exclusiveness
• select the maximum/highest frequency
• its corresponding class becomes modal class. {40-50}

Now $Z = l + \left| \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right| \times h$

$l \rightarrow \text{lower limit of modal class}$ $f_1 \rightarrow \text{frequency of modal class}$ $f_0 \rightarrow \text{frequency pre-modal class}$ $f_2 \rightarrow \text{frequency post-modal class}$ $h = \text{upper - lower limit of modal class.}$

$Z = 40 + \left| \frac{12 - 10}{2 \times 12 - 10 - 6} \right| \times 10$ $= 40 + \left| \frac{2}{8} \right| \times 10 = 40 + \frac{1}{4} \times 10$ $= 40 + \frac{5}{2} = \text{==42.5==}$

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$X: 4, 6, 8, 10, 12$ If in the above example mean is increased by 2, what will happen to the individual observation if all are equally affected.

Sol:- $\bar{x} = \frac{\sum x}{N} = \frac{4+6+8+10+12}{5} = \frac{40}{5} = 8$

$\therefore \sum x = 40$ $\bar{x} = 8$ If mean is increased by 2 then New mean = $8 + 2 = 10$

$\therefore 10 = \frac{\sum x'}{5} \Rightarrow \sum x' = 50$

Now $\sum x' - \sum x = 50 - 40 = \text{==10==}$

This 10 needs to be equally distributed. each observation will get $\frac{10}{5} = 2$ increment.

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🔴 ⭐ THE EMPIRICAL RELATION (the single most-tested formula of this topic)

$\textbf{Mode} = 3\,\textbf{Median} - 2\,\textbf{Mean}$

Rearranged forms you may need: $\text{Median} = \frac{\text{Mode} + 2\,\text{Mean}}{3} \qquad \text{Mean} = \frac{3\,\text{Median} - \text{Mode}}{2}$

• Also written Mean − Mode = 3 (Mean − Median)
• ⭐ For a SYMMETRIC distribution: Mean = Median = Mode
• Positively skewed: Mean > Median > Mode · Negatively skewed: Mean < Median < Mode

e.g. Mean = 25, Median = 24 → Mode = 3(24) − 2(25) = 72 − 50 = 22

⭐ Comparison of the three averages

⭐ Σ(x − x̄)² is MINIMUM when taken about the MEAN.

Types of averages

Mathematical: Arithmetic Mean (AM) · Geometric Mean (GM) · Harmonic Mean (HM) Positional: Median · Mode

$GM = \sqrt[n]{x_1 \times x_2 \times \dots \times x_n} \qquad\qquad HM = \frac{n}{\sum \frac{1}{x}}$

⭐ AM ≥ GM ≥ HM (always; equal only when all values are identical) · ⭐ GM² = AM × HM Use GM for rates of growth / ratios / index numbers · Use HM for speeds and rates per unit.

➕ Partition Values — Quartiles, Deciles, Percentiles

Values that divide an ordered data set into equal parts.

⭐ Also: Q₁ = P₂₅ · Q₃ = P₇₅ · D₁ = P₁₀

Individual & discrete series (N = number of items): $Q_i = \text{size of } \left(\frac{i(N+1)}{4}\right)^{th} \text{ item} \quad D_i = \left(\frac{i(N+1)}{10}\right)^{th} \quad P_i = \left(\frac{i(N+1)}{100}\right)^{th}$

Continuous series (interpolation, cf = cumulative frequency of the preceding class): $Q_i = l + \frac{\frac{iN}{4} - cf}{f} \times h$

Related: Quartile Deviation (Semi-Interquartile Range) $= \dfrac{Q_3 - Q_1}{2}$ · Interquartile Range $= Q_3 - Q_1$

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🔷 TOPIC (vi) — THEORY OF PROBABILITY

Syllabus topic (vi) of 10 · runs until Topic (vii)

Covers: basic terms · types of events · the three definitions of probability · addition theorem · multiplication theorem · conditional probability · odds · Bayes theorem 🎯 Asked in the papers: 2024 · Q76 — which probability statements are correct · 2022 · Q34 — P(Maths or Physics)

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Basic terms

Types of events

Definitions of probability

(1) Classical / Mathematical (a priori): $P(A) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} = \frac{m}{n}$ Requires outcomes to be equally likely, mutually exclusive and exhaustive.

(2) Empirical / Statistical (a posteriori): based on actual repeated trials — $P(A) = \lim_{n \to \infty} \frac{\text{Number of times A occurred}}{n}$

(3) Axiomatic (Kolmogorov): ⭐ $0 \le P(A) \le 1$ · $P(S) = 1$ · for mutually exclusive events $P(A \cup B) = P(A) + P(B)$

Addition theorem — "OR" / union

General (works always): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

⭐ If A and B are mutually exclusive, $P(A \cap B) = 0$, so: $P(A \cup B) = P(A) + P(B)$

Three events: $P(A \cup B \cup C) = P(A)+P(B)+P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$

Multiplication theorem — "AND" / intersection

⭐ If A and B are INDEPENDENT: $P(A \cap B) = P(A) \times P(B)$

If DEPENDENT: $P(A \cap B) = P(A) \times P(B \mid A)$

Conditional probability

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(B) \ne 0$ ⭐ If A and B are independent, $P(A \mid B) = P(A)$ — knowing B tells you nothing about A.

Odds

• Odds in favour of A = $m : (n-m)$ = favourable : unfavourable
• Odds against A = $(n-m) : m$
• If odds in favour are $a : b$ then $P(A) = \dfrac{a}{a+b}$

Bayes theorem (know the shape, rarely computed)

$P(A_i \mid B) = \frac{P(A_i) \cdot P(B \mid A_i)}{\sum P(A_j) \cdot P(B \mid A_j)}$ Used to revise a prior probability after new evidence arrives.

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Worked examples

Q1. (the 2022 · Q34 type) In a class, P(passing Maths) = 2/5, P(passing Physics) = 3/10, P(passing both) = 1/5. Find P(passing Maths or Physics).

Sol:- Use the general addition theorem — $P(M \cup P) = P(M) + P(P) - P(M \cap P) = \frac{2}{5} + \frac{3}{10} - \frac{1}{5}$ $= \frac{4}{10} + \frac{3}{10} - \frac{2}{10} = \frac{5}{10} = \mathbf{\frac{1}{2}}$

Q2. A die is thrown once. Find the probability of getting an even number or a number greater than 4.

Sol:- $A = \{2,4,6\} \Rightarrow P(A) = \frac{3}{6}$ ; $B = \{5,6\} \Rightarrow P(B) = \frac{2}{6}$ ; $A \cap B = \{6\} \Rightarrow P(A \cap B) = \frac{1}{6}$ $P(A \cup B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \mathbf{\frac{2}{3}}$

Q3. Two coins are tossed. Find P(at least one head).

Sol:- $S = \{HH, HT, TH, TT\}$, $n(S) = 4$. Easier by complement — $P(\text{at least one head}) = 1 - P(\text{no head}) = 1 - \frac{1}{4} = \mathbf{\frac{3}{4}}$ ⭐ "At least one" is almost always fastest via the complement.

Q4. A bag has 5 red and 3 black balls. Two are drawn with replacement. P(both red)?

Sol:- With replacement → independent — $P = \frac{5}{8} \times \frac{5}{8} = \mathbf{\frac{25}{64}}$ Without replacement (dependent): $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$

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🔷 TOPIC (vii) — THEORY OF ATTRIBUTES

Syllabus topic (vii) of 10 · runs until Topic (viii)

Covers: Attributes & notation · number of classes · order of frequencies · algebraic expressions · contingency tables · consistency of data · independence & association — proportion method and (AB) = (A)(B)/N · Yule's coefficient of association 🎯 Asked in the papers: 2024 · Q77 — class of 100 students, match the boys/girls figures · 2022 · Q39 — ultimate class frequency for independent attributes

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Theory of Attributes

Deals with the qualitative characteristics calculated using quantitative measurements. e.g:- honesty, habit of smoking etc.

Attributes:- "Qualitative characteristics of an individual."

Dicotomony / Dichotomous classification:- Attribute divides class into two at each level.

graph TD
    A[Population] --> B[Male]
    A --> C[Female]
    B --> D[Literate]
    B --> E[Illiterate]
    C --> F[Literate]
    C --> G[Illiterate]

Types of Attributes

graph TD
    A[Types of Attributes] --> B["Positive Attributes<br>(Presence)<br>Attribute"]
    A --> C["Negative Attributes<br>(Absence)<br>Attribute"]

Symbols / Notations / Mnemonics:- Presence of Attribute:- Capital Letters; $A, B, C \text{ etc.}$ Absence of Attribute:- Greek Letters; $\alpha, \beta, \gamma \text{ etc.}$

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e.g:- $A \rightarrow \text{Presence of honesty}$ $\alpha \rightarrow \text{Absence of honesty}$ $B \rightarrow \text{Literate}$ $\beta \rightarrow \text{Illiterate (Absence of Literacy)}$ $C \rightarrow \text{Employed}$ $\gamma \rightarrow \text{Unemployed (Absence of employment)}$

• class:- Homogeneous & Mutually exclusive group.
• class Frequency:- Number of items in each class.
  • denoted by brackets over class symbols. e.g: $\rightarrow (A), (AB), (BC), (Ac) \rightarrow \text{==+ve frequencies==}$ $\rightarrow (\alpha), (\beta), (\gamma), (\alpha\beta) \text{ etc } \rightarrow \text{==-ve frequencies==}$ $\rightarrow (AB\gamma), (\alpha B\gamma), (A\beta) \text{ etc } \rightarrow \text{==Contrarary frequencies==}$

Number of classes:-

1 - Attribute — 3 classes — $A, \alpha, N$ 2 - Attribute — 9 classes $\rightarrow N, A, B, \alpha, \beta, AB, A\beta, \alpha B, \alpha\beta$. 3 - Attributes — 27 classes. n - Attributes = $3^{\text{No. of Attributes}}$

e.g No. of Attributes = 7. $\therefore \text{No. of classes} = 3^7 = \text{==2,187 classes==}$

Order of frequencies:-

Order - 0 : $N$ Order - 1 : $(A) (\alpha) : (B) (\beta) : (C) (\gamma)$ Order - 2 : $(AB) (A\beta), (\alpha B), (\alpha\beta) : (AC), (A\gamma), (\alpha C) (\alpha\gamma)$ Order - 3 : $(ABC), (\alpha BC) / (ABC), (AB\gamma) \dots$

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$N = \text{==}(A), (\alpha)\text{==} : \text{==}(B), (\beta)\text{==} : \text{==}(C), (\gamma)\text{==}$

$A = (AB) + (A\beta)$ $B = (AB) + (\alpha B)$

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Some Algebraic Expressions:-

$(A) = (AB) + (A\beta)$ $(B) = (AB) + (\alpha B)$ $(\alpha) = (\alpha B) + (\alpha\beta)$ $(\beta) = (A\beta) + (\alpha\beta)$

$(ABC) + (AB\gamma) + (\alpha B)$ $= (AB) + (\alpha B) = (B)$

Formula for any order frequency classes:-

No. of classes of $n^{\text{th}}$ order frequency $n! = n \times n-1 \times n-2 \times \dots \times 1$ $1! = 1$ $0! = 1$ $^nC_n = 1$ $^nC_r = \frac{n!}{(n-r)! r!}$

$^nC_r \times 2^r \text{ classes}$

• $n \rightarrow$ No. of attributes
• $r \rightarrow$ order

Q,, Find the number of Order-2 classes if the number of attributes is 2. Sol:- Number of 2 order class = $^nC_r \times 2^r$ $= ^2C_2 \times 2^2$ $= 1 \times 2^2 = 4$

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Q,, Calculate the 1st-order classes if the number of attributes = 2. Sol:- No. of Order-1 classes = $^nC_r \times 2^r$ $= ^2C_1 \times 2^1$ $= \frac{2!}{(2-1)!(1!)} \times 2^1$ $= \frac{2 \times 1}{(1!)(1!)} \times 2$ $= \frac{2}{1} \times 2 = \text{==4==}$

Q,, calculate No. of order-2 classes if the number of attributes = 3 Sol:- No. of order-2 classes = $^nC_r \times 2^r$ $= ^3C_2 \times 2^2$ $= \frac{3!}{(3-2)! (2!)} \times 4$ $= \frac{3 \times 2 \times 1}{(1!) (2 \times 1)} \times 4$ $= \frac{6}{2} \times 4 = \text{==12==}$

Contingency table:

Note: Ultimate frequencies $\rightarrow$ Highest order Frequencies $2^n$

• Contingency matrix shows the relation between two variables / attributes.
• Used by Karl Pearson for first time — $\rightarrow$ theory of contingency $\rightarrow$ Relation to Association & Normal correlation

Contingency table / Cross-Tabulation / Cross-Tab.

graph TD
    A[Contingency table] --> B["==2-Attributes=="]
    A --> C[3-Attributes]
    
    B -.-> B1[* 9 square table]
    B -.-> B2[* 2x2 table]

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Practice Questions on Contingency matrix

Q1. If (A) = 40, (AB) = 50, B = 80 & N = 160. Find out the remaining values. Sol:-

Q2:- If the values (AB) = 60, (A$\beta$) = 40, ($\alpha$B) = 30, ($\alpha\beta$) = 20. Find out the rest of values. Sol:-

H/W:- Q3:$\rightarrow$ If (A)=130, (B)=100, ($\beta$)=120 and ($\alpha$)=110 find the other values.

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Applications of theory of Attributes:-

Consistency of data:-

Rule:-

1. No class frequency can be negative. Frequency of every class $\ge 0$ (OR)
2. No class frequency can be greater than N. (Each class frequency $\le N$)

Q,, Determine consistency in the given data: (AB) = 70, ($\alpha$B) = 40, ($\alpha\beta$) = 60, B = 100.

Conclusion: No frequency is negative, nor any frequency is greater than N. $\therefore$ Data is consistent

Q,, Find out if the data is consistent or not. values given are N = 300, (A) = 200, B = 180, (AB) = 170.

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$\rightarrow$ Data is consistent

Q:- N = 400, (A) = 300, (B) = 280, (AB) = 170. Determine consistency. Sol:-

$\rightarrow$ Data Inconsistent $\because \text{==} (\alpha\beta) = -10 \text{==}$

H/W Q,, (A) = 200, (B) = 100, ($\alpha$B) = 80, N = 300. Determine consistency.

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Trick :- To check consistency of data. $\downarrow$ Check for the ultimate frequencies, If any ultimate frequency is "negative", the data is inconsistent otherwise not.

Q,, If (AB)=100, ($\alpha$B)=40, (A$\beta$)=30 & ($\alpha\beta$)=60. Find if Data is consistent or not. Sol:- No ultimate frequency is negative. $\therefore$ Data is consistent.

H/W:- Q:- Determine the consistency of Data. (A$\beta$)=220, ($\alpha$B)=130, ($\alpha\beta$)=110, (A)=180. Find consistency of Data.

2. Independence And Association:-

• Attributes are said to be Independent if there does not exist any relation between them. e.g:- Gender and success, Beauty and Intelligence.

$\rightarrow$ Two attributes are said to be associated if they are related in one way or other. e.g:-

• Positive Association: Present or Absent together. e.g.: unemployment & poverty.
• Negative Association: one is present & another absent. e.g.: Education & Ignorance.

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Methodology to check Association and Independence of Attributes:-

(1) Proportion Method :-

(i) $\frac{(AB)}{(B)} = \frac{(A\beta)}{(\beta)}$ $\rightarrow$ Independent (ii) $\frac{(AB)}{(B)} > \frac{(A\beta)}{(\beta)}$ $\rightarrow$ Positive Association (iii) \frac{(AB)}{(B)} &lt; \frac{(A\beta)}{(\beta)} $\rightarrow$ Negative Association

Note: ($\alpha$,$\beta$), (A,$\beta$), ($\alpha$,B) are also Independent.

Q,, (AB) = 100, (B) = 10, (A$\beta$) = 150 & ($\beta$) = 15. Find out how A & B are Associated? Sol:- $\frac{(AB)}{(B)} = \frac{100}{10} = 10$ $\frac{(A\beta)}{(\beta)} = \frac{150}{15} = 10$ $\therefore \frac{(AB)}{(B)} = \frac{(A\beta)}{(\beta)} \rightarrow$ Independent.

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Q,, (AB) = 200, (B)= 10, (A$\beta$) = 250 and ($\beta$) = 25 Sol:- $\frac{(AB)}{(B)} = \frac{200}{10} = 20$ $\frac{(A\beta)}{(\beta)} = \frac{250}{25} = 10$ $\therefore \frac{(AB)}{(B)} > \frac{(A\beta)}{(\beta)} \rightarrow$ Positively Associated.

H/W Q// (AB) = 210, (B) = 10, (A$\beta$) = 310, ($\beta$) = 10. Find association?

(2) Comparison Method :-

(i) $(AB) = \frac{(A) \times (B)}{N} \rightarrow$ Independent (ii) $(AB) > \frac{(A) \times (B)}{N} \rightarrow$ Positively Associated (iii) (AB) &lt; \frac{(A) \times (B)}{N} \rightarrow Negatively Associated

Q,, Type of Association in the given data N = 106, (A) = 70, (B) = 36, (AB) = 20. Sol:- $(AB) = 20$ $\frac{(A) \times (B)}{N} = \frac{70 \times 36}{106} = 23.77$

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\therefore (AB) &lt; \frac{(A) \times (B)}{N} \rightarrow Negatively associated.

3:- Yule's coefficient of Association method:-

$Q_{AB} = \frac{(AB)(\alpha\beta) - (A\beta)(\alpha B)}{(AB)(\alpha\beta) + (A\beta)(\alpha B)}$

Representation of values of Q. (i) If Q lies between 0 to 1 $\rightarrow$ Positively Associated (ii) Q lies between -1 to 0 $\rightarrow$ Negatively Associated (iii) Q = 0 $\rightarrow$ Independent (iv) Q = 1 $\rightarrow$ completely (Perfectly) Associated (v) Q = -1 $\rightarrow$ completely (Perfectly) Disassociated.

  Completely
Disassociated      Negatively Associated         Positively Associated         Completely
      |----------------------|-----------------------------|----------------------| Associated
     -1                   -0.5             0              0.5                     1
      ^                      ^             ^               ^                      ^
   strong                  weak       Independent         weak                  strong

(AB) or (αβ)                                                                 (Aβ) or (αB)
    = 0                                                                           = 0

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Q:- If (AB) = 300, (B) = 15, (A$\beta$) = 350 and $\beta$ = 25. Find the Association between A & B. Sol:- Since the values given are (AB), (B), (A$\beta$) & ($\beta$), we can use proportion method. $\frac{(AB)}{(B)} = \frac{300}{15} = 20$ $\frac{(A\beta)}{(\beta)} = \frac{350}{25} = 14$ $\therefore \frac{(AB)}{(B)} > \frac{(A\beta)}{(\beta)} \rightarrow$ Positive Association b/w A & B.

Q,, If N = 100, (A) = 50, (B) = 30 & (AB) = 40. Find Association between A & B. Sol:- Since the values given are (AB), (A), (B) and N $\therefore$ Comparison method will be time saving. $(AB) = 40$ $\frac{(A) \times (B)}{N} = \frac{50 \times 30}{100} = \frac{1500}{100} = 15$ Here $(AB) > \frac{(A) \times (B)}{N} \rightarrow$ Positive Association between A and B.

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Question Asked in PAA (JKSSB) - Year 2020

Q:: If N = 200, which of the options match the ultimate class frequencies, given that there are two independent attributes A = 100, B = 140? (A) 60 (B) 70 (C) 80 (D) 90 Sol:- Given values are N, (A), (B) It is also given in question that Attributes A & B are independent. $\therefore$ As per the given values, comparison method is to be used. $(AB)$? $(AB) = \frac{(A) \times (B)}{N} \rightarrow$ Condition for independent Attributes in Comparison Method. $(AB) = \frac{100 \times 140}{200} = \frac{140}{2} = \text{==70== (b)}$

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3:- Yule's coefficient of Association method:-

(Repeated Page/Content)

$Q_{AB} = \frac{(AB)(\alpha\beta) - (A\beta)(\alpha B)}{(AB)(\alpha\beta) + (A\beta)(\alpha B)}$

Representation of values of Q. (i) If Q lies between 0 to 1 $\rightarrow$ Positively Associated (ii) Q lies between -1 to 0 $\rightarrow$ Negatively Associated (iii) Q = 0 $\rightarrow$ Independent (iv) Q = 1 $\rightarrow$ completely (Perfectly) Associated (v) Q = -1 $\rightarrow$ completely (Perfectly) Disassociated.

  Completely
Disassociated      Negatively Associated         Positively Associated         Completely
      |----------------------|-----------------------------|----------------------| Associated
     -1                   -0.5             0              0.5                     1
      ^                      ^             ^               ^                      ^
   strong                  weak       Independent         weak                  strong

(AB) or (αβ)                                                                 (Aβ) or (αB)
    = 0                                                                           = 0

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Q:- (AB) = 60, ($\alpha\beta$) = 20, A$\beta$ = 80, ($\alpha$B) = 30. Find the Association. Sol:- Since the given values are (AB), ($\alpha\beta$), (A$\beta$) and ($\alpha$B) $\therefore$ Yule's coefficient method will be used. $Q = \frac{(AB)(\alpha\beta) - (A\beta)(\alpha B)}{(AB)(\alpha\beta) + (A\beta)(\alpha B)}$ $= \frac{60 \times 20 - 80 \times 30}{60 \times 20 + 80 \times 30}$ $= \frac{1200 - 2400}{1200 + 2400} = \frac{-1200}{3600} = \frac{-12}{36}$ $Q = \text{==-0.33==} \rightarrow \text{weakly negatively associated.}$

Q2:- (AB) = 70, ($\alpha\beta$) = 30, (A$\beta$) = 0, ($\alpha$B) = 60. Find Association. Sol:- $Q = \frac{(AB)(\alpha\beta) - (A\beta)(\alpha B)}{(AB)(\alpha\beta) + (A\beta)(\alpha B)}$ $= \frac{70 \times 30 - 0 \times 60}{70 \times 30 + 0 \times 60}$ $= \frac{2100}{2100} = \text{==1==} \rightarrow \text{Perfectly / completely Associated.}$

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Q:- Find the Association between Literate Husband and Literate wife. Literate husband with Literate wife = 80 Literate husband with illiterate wife = 30 Illiterate husband with Literate wife = 100 Illiterate husband with illiterate wife = 60.

Sol:- A : Literate Husband B : Literate Wife $\alpha$ : Illiterate Husband $\beta$ : Illiterate wife

(AB) = 80, (A$\beta$) = 30, ($\alpha$B) = 100, ($\alpha\beta$) = 60 $Q = \frac{(AB)(\alpha\beta) - (A\beta)(\alpha B)}{(AB)(\alpha\beta) + (A\beta)(\alpha B)}$ $= \frac{80 \times 60 - 30 \times 100}{80 \times 60 + 30 \times 100} = \frac{4800 - 3000}{4800 + 3000}$ $= \frac{1800}{7800} = \frac{18}{78} = \text{==0.23==} \rightarrow \text{weak positively Associated}$

H/W Q!- (AB) = 90, ($\alpha\beta$) = 40, $\alpha$B = 60 and A$\beta$ = 35. Find association?

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🔷 TOPIC (viii) — THEORY OF INDEX NUMBERS

Syllabus topic (viii) of 10 · runs until Topic (ix)

Covers: Characteristics · problems in construction · types · simple (unweighted) index · quantity index · value index · weighted aggregative — Laspeyres, Paasche, ⭐ Fisher's Ideal · weighted average of price relatives · CPI · WPI · CPI vs WPI · ⭐ Tests of Adequacy (Unit, Time Reversal, Factor Reversal, Circular) 🎯 Asked in the papers: 2024 · Q78 — Fisher Ideal Value Index — the hardest question on the 2024 paper

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Index Numbers

Statistical measure that shows changes in variables with respect to time, geography or other characteristics.

• First time calculated by Italian statistician "Giovanni Rinaldo carli" $\downarrow$ Calculated ratio of prices for (grain, wine and oil). 1500 and 1750
• Also known as "Economic Barometer"

Characteristics:-

(1) Expressed in percentage form. (2) Relative or comparative measurement of a group of commodities. (3) Represent the specialised averages. (4) e.g; consumer Price Index, cost of living index, Industrial production Index.

Problems in construction of Index Numbers:-

• Purpose of Index number should be pre-defined, every index number has its specific use.
• Selection of Base year:-
  • Base year should not be too near or too far.
  • Should be calamity free.

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• Selection of commodities.
• Choosing the source of data.
• Choice of average.
• Choice of calculation method.

Limitations of Index Numbers:-

• Index numbers give approximate values.
• Based on samples only.
• Quality of the products is not taken in consideration.
• For every purpose, different index numbers are to be constructed.

Types of Index Numbers:-

1. Price Index numbers
2. Quantity Index Numbers.
3. Value Index numbers

• Price Index measures change in price b/w Base year and current year. $\hookrightarrow$ Whole sale Price Index Number $\hookrightarrow$ Retail Price Index Number
• Quantity Index numbers show average changes in quantities, produced, consumed or sold. e.g Imports, Exports, production in Industries etc.
• Value Index Number represent the product of commodity and the quantity.

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Methods of constructing Index Numbers

graph TD
    A[Index Numbers] --> B["==Unweighted / Simple Index Numbers=="]
    A --> C["==Weighted Index Numbers=="]
    
    B --> D["==Simple Aggregative Method=="]
    B --> E["==Simple Average of Price Relatives=="]
    
    C --> F["==Weighted Aggregative Method=="]
    C --> G["==Weighted Average of Price Relatives=="]

Simple Index Numbers:-

Each item has got the same weight therefore no individual weights are assigned. (i) Simple Aggregative method (ii) Simple Average of Price Relatives.

(i) Simple Aggregative Method. Procedure:- (i) Add all the current year prices of various commodities. (ii) Add all the base year prices of various commodities.

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Formula:- Price of "1" on "0" $\rightarrow$ $P_{01} = \frac{\sum P_1}{\sum P_0} \times 100$

• $P_{01} \rightarrow$ Index number of current year
• $\sum P_1 \rightarrow$ total (sum) of the current year prices.
• $\sum P_0 \rightarrow$ total (sum) of the Base year prices.

Q,,

Sol:- $P_{01} = \frac{\sum P_1}{\sum P_0} \times 100 = \frac{20+30+35+40+45}{10+20+25+30+35} \times 100$ $P_{01} = \frac{170}{120} \times 100 = \frac{1700}{12} = \text{==141.66==}$ Price index of current year is 141.66 (or) Price of current year has increased by 41.66%.

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Quantity Index

Formula:- $q_{01} = \frac{\sum q_1}{\sum q_0} \times 100$

• $q_{01} \rightarrow$ quantity index no. of current year.
• $\sum q_1 \rightarrow$ sum total of current year quantities.
• $\sum q_0 \rightarrow$ sum total of Base year quantities.

Q,, Calculate the quantity index number for 2019 when the base year is 2011.

Sol:- $q_{01} = \frac{\sum q_1}{\sum q_0} \times 100 = \frac{95}{45} \times 100 = \frac{9500}{45}$ $= 211.11$ $\therefore$ Quantity index for current year (2019) = 211.11 (or) Quantity of current year has increased by 111.11.

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(ii) Simple Average of Price Relatives Method

Procedure:- (i) calculate relative price of current year (current year price by Base year price) $\left( \frac{P_1}{P_0} \times 100 \right)$ (ii) obtain the sum of Relative prices / Price Relative $\sum \frac{P_1}{P_0} \times 100$ (iii) Divide the sum of Relative prices by total number of commodities. $P_{01} = \frac{\sum \left( \frac{P_1}{P_0} \times 100 \right)}{N}$

Q:- Using Price Relative Method, for the year 2020 find out index values from the given data.

Sol:-

N = 5 | $\sum \left( \frac{P_1}{P_0} \times 100 \right) =$ 806.67

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$P_{01} = \frac{\sum \left( \frac{P_1}{P_0} \times 100 \right)}{N} = \frac{806.67}{5} = \text{==161.33==}$

The Price index for the year 2020 is 161.33 or there is increase in the price of 2020 by 61.33% compared to 2010.

H/W:- calculate the Quantity Index Number for the Previous question Assuming prices in (Rs.) as quantity in quintals.

Formula:- $q_{01} = \frac{\sum \left( \frac{q_1}{q_0} \times 100 \right)}{N}$

$P_{01} = \frac{\sum \left( \frac{P_1}{P_0} \times 100 \right)}{N} = \frac{806.67}{5} = \text{==161.33==}$

The Price index for the year 2020 is 161.33 or there is increase in the price of 2020 by 61.33% compared to 2010.

H/W:- calculate the Quantity Index Number for the Previous question Assuming prices in (Rs.) as quantity in quintals.

Formula:- $q_{01} = \frac{\sum \left( \frac{q_1}{q_0} \times 100 \right)}{N}$

Ans:- Same as above question.

Value Index Number :

$V_{01} = \frac{\sum P_1 q_1}{\sum P_0 q_0} \times 100$

$\Rightarrow V_{01} = \frac{39}{36} \times 100 = 108.33$ (or) Prices increase by 8.33% from base year.

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Weighted Index Numbers :-

Appropriate weights are assigned to various commodities to show their relative importance.

1. Weighted Aggregative Method :-

There are lot of methods to calculate weighted index numbers viz; Laspeyre's, Paasche's, Fisher's, Kelley's, Bowley's, Dorbish and few more methods.

Laspeyre's Method :-

It is used to compare the expenditure of basket of commodities of Base year and current year.

Price Index $P_{01} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$ * Quantities of base year are taken as weights.

Quantity Index $q_{01} = \frac{\sum q_1 p_0}{\sum q_0 p_0} \times 100$ * Prices of Base year as weights.

Paasche's Method :-

Used to know the cost of basket of commodities in current year when the same basket cost ₹100 in Base year.

Price Index $P_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$

Quantity Index $q_{01} = \frac{\sum q_1 p_1}{\sum q_0 p_1} \times 100$

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Quantities of current year and Prices of current year are used as weights.

Fisher's Method :-

Also called as Ideal method.

Price Index:- $P_{01} = \sqrt{L \times P}$ $P_{01} = \sqrt{ \frac{\sum P_1 q_0}{\sum P_0 q_0} \times \frac{\sum P_1 q_1}{\sum P_0 q_1} } \times 100$

• It is the geometric mean of Laspeyre's and Paasche's method.
• Both Base and current year Quantities are used as weights.

Quantity Index :- $q_{01} = \sqrt{L \times P}$ $q_{01} = \sqrt{ \frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 p_1}{\sum q_0 p_1} } \times 100$

• Both Base year and current year Prices are used as weights.

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Q,, Calculate Price and quantity Index numbers using Paasche's, Lapeyere's and Fisher's method.

Laspeyre's method:

Price Index $\rightarrow$ $P_{01} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$ $= \frac{46}{41} \times 100$ $= 112.19$ | $12.19\% \uparrow$

Quantity Index $\rightarrow$ $q_{01} = \frac{\sum q_1 p_0}{\sum q_0 p_0} \times 100$ $= \frac{63}{41} \times 100$ $= 153.65$ | $53.65\% \uparrow$

Paasche's Method :-

Price Index :- $P_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100$ $= \frac{58}{63} \times 100$ $= 92.06$ / $7.94\% \downarrow$ Fallen

Quantity Index $q_{01} = \frac{\sum q_1 p_1}{\sum q_0 p_1} \times 100$ (Correction: formula written in notes uses $p_1$ for weights) $= \frac{\sum q_1 p_1}{\sum q_0 p_1} \times 100$ $= \frac{58}{46} \times 100$ $= 126.08$ / $26.08\% \uparrow$

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Fisher's Method :

Price Index $P_{01} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100$ $= \sqrt{\frac{46}{41} \times \frac{58}{63}} \times 100$ $= \sqrt{\frac{2668}{2583}} \times 100$ $= \sqrt{1.032} \times 100$ $= 1.015 \times 100$ $= 101.5$ or $1.5\% (\uparrow)$

Quantity Index $q_{01} = \sqrt{\frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 p_1}{\sum q_0 p_1}} \times 100$ $= \sqrt{\frac{63}{41} \times \frac{58}{46}} \times 100$ $= \sqrt{\frac{3654}{1886}} \times 100$ $= \sqrt{1.937} \times 100$ $= 1.391 \times 100$ $= 139.1$ or $39.1\% (\uparrow)$

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(2) Weighted Average of Price Relatives Method:

Procedure (i) Calculate Price Relatives of current year $R = \frac{p_1}{p_0} \times 100$

(ii) calculate the value weights $V = (p_0 q_0)$

Price Index $P_{01} = \frac{\sum RV}{\sum V}$

Q,, calculate the weighted Average Price Relatives Index for given data.

$P_{01} = \frac{\sum RV}{\sum V} = \frac{35,199.8}{264} = \text{==133.33==}$

(or) $\frac{133.33 - 100}{100} \times 100 = 33.33\%$ Increase in prices of 2018 on 2011.

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Consumer Price Index (CPI) :- (weighted)

Also known as:-

• Real Price Index Number.
• Cost of Living Index number.
• Price of Living Index Number.
• Retail Price Index Number.

CPI is used to measure the price of basket of goods and services of a particular class or region at particular point of time in comparison to Base year.

Applications of CPI:

(1) To know increase or decrease in cost of living. (2) Used by government to make salary and (D.A). (3) Used to find purchasing power of money and real income/wages.

• $\text{Purchasing power of money} = \frac{1}{\text{Cost of living Index}}$
• $\text{Real wages} = \frac{\text{Money wages}}{\text{Cost of Living Index}} \times 100$

(4) Used by government in framing price policy and income policy.

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Types of CPI:

• (CPI - IW) - Industrial workers (1982 B.Y) $\rightarrow$ 2001 $\rightarrow$ 2016 [Ministry of Labour, Labour Bureau (Shimla)]
• (CPI - AL) - Agricultural Labours (1986-87 B.Y)
• (CPI - RL) - Rural Labours (1986-87 B.Y)
• (CPI - UNME) - Urban Non-Manual employees (1984-85 B.Y) $\rightarrow$ (CSO) $\downarrow$ Now (NSO) [MOSPI - Ministry of Statistics & Program Implementation]

2011 - CPI (R), CPI (U), CPI (Combined)

Problems in construction of CPI:-

1. Difference in price of goods and services.
2. Difference in the standard of living.
3. Choice of Base year.
4. Difference in proportion of expenditure.

Methods of construction of CPI:-

• Aggregate Expenditure Method
• Family Budget Method

(1) Aggregate Expenditure method / Weighted Aggregate Method :- Quantities of base year are taken as weights $CPI = \frac{\text{Expenditure in current year}}{\text{Expenditure in Base year}} \times 100$ $CPI = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$ ** Laspeyere's method

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(2) Family Budget Method / Weighted Average of Price Relatives :- Expenditure in the base period are taken as weights. $P = \frac{p_1}{p_0} \times 100$ Weights "$W$" = $p_0 q_0$ $CPI = \frac{\sum WP}{\sum W}$

Q,, Find the cost of living index in the given Data.

Sol:- Cost of Living = $\frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 = \frac{72}{36} \times 100 = \text{==200==}$

** Based on Lapeyre's Price Index.

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Q,, Data about the middle class family is as follows.. Expense on food 30%, Rent 15%, clothing 20%, Fuel 10%, others 25% on base year. Price (₹) 2001: 100, 20, 70, 20, 40 Price (₹) 2011: 90, 20, 140, 15, 60 Find Cost of living.

Sol:-

$\text{Cost of Living} = \frac{12700}{100} = \text{==127==}$

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Whole sale Price Index:-

It measures the general changes in the whole sale price of goods in the country.

• Based on the commodities produced and distributed (first stage of transaction).
• Rise or fall in prices at wholesale level spills over to the retail level after lag.
• Published by Economic Advisor, Ministry of Commerce and Industry.
• First time published 10th January 1942 (1939 B.Y).
• 7th revision is with Base year (2011-2012) $\downarrow$ chaired by Dr. Sumitra Chaudhari. $\hookrightarrow$ Education, Health, etc not included. * taxes are also not included.
• WPI Food Index (CSO) separately presented.
• 697 commodities included in WPI.

WPI Basket

graph TD
    A[WPI Basket] --> B["Primary Articles<br>(22.60%)<br>117 items<br>Rice, wheat, fruits etc."]
    A --> C["Fuel & Power Articles<br>(13.20%)<br>16 items<br>Power, coal petroleum etc."]
    A --> D["Manufactured Articles<br>(64.20%)<br>564 items<br>oil (edible), sugar, chemicals etc."]

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Uses of WPI:-

(1) Estimation of Inflation: $\frac{X_n - X_{n-1}}{X_{n-1}} \times 100$ $X_n = \text{WPI for } n^{\text{th}} \text{ week.}$ $X_{n-1} = \text{WPI for } (n-1)^{\text{th}} \text{ week.}$

$\text{Yearly inflation rate} = \left( \frac{\text{Current year}_{WPI}}{\text{Previous year}_{WPI}} \times 100 \right) - 100$

(2) Estimation of Monetary value and Real value. (3) Used for estimating GDP by CSO. (4) Used by Business contractors (Demand & supply). (5) By Global investors for investment decisions.

Method for Calculation:

Stage 1: Elementary price Indices using Jevon's index (Geometric mean for price). Stage 2: Elementary aggregated using Laspeyre's index formula.

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WPI (Vs) CPI

Tests of Adequacy $\rightarrow$ Index Numbers

(1) Unit Test (2) Time Reversal Test (3) Factor Reversal Test (4) Circular Test (extention of TRT)

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(1) Unit tests Index number formulae should be independent of the units in which prices or quantities are used. * Satisfied by all index methods except simple (unweighted) aggregative method.

(2) Time Reversal Test:- Interchanging of time subscripts of price/quantity gives the reciprocal of the original formula. $\rightarrow P_{01} \times P_{10} = 1$ (Base year '1', Base year '0') $\rightarrow q_{01} \times q_{10} = 1$ * Not satisfied by Laspeyre & Pasche.

(3) Factor Reversal Test:- If $p$ and $q$ factors in price/quantity index formula are interchanged so that a quantity/price index formula is obtained, the product of two indices should give true value ratio. $P_{01} \times q_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0} = V_{10}$ * Satisfied by Fisher Index only.

(4) Circular Test:- (extension of TRT) If the index for year 2018 is based on 2017 and Another index for 2017 based on 2016 then index for 2018 with base year 2016 should be directly obtained. $\rightarrow$ $P_{01} \times P_{12} \times P_{20} = 1$ * Satisfied by:

• simple geometric mean of price relatives
• Kelly's fixed Base method (simple)

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🔷 TOPIC (ix) — DEMOGRAPHY — CENSUS, ITS FEATURES AND FUNCTIONS

Syllabus topic (ix) of 10 · runs until Topic (x)

Covers: Demography & demographic transition · the population cycle · types of demography · four stages of Indian demographic history · Census — features & functions · background of the Census of India · Census 2011 data · ➕ added: national headline figures and Nagaland's negative growth 🎯 Asked in the papers: 2024 · Q79 — a feature of the census → Simultaneity · 2022 · Q40 — highest negative decadal growth → Nagaland · 2022 · Q36 — demographic sex ratio

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Demography - census, its features and functions

Demography: Derived from two Greek words

• "Demos" — "the people"
• "Graphy" — Recording / writing / measuring something.

Study of population of a country or place. Term "Demography" $\rightarrow$ Guillard (1855) Father of "Demography" $\rightarrow$ John Graunt (Natural & Political observation (1662))

Deals with $\rightarrow$ Fertility, Marriage, Mortality, Migration, Social mobility. Mainly deals with: Migration, Birth rate, Death rate.

Demographic Transition :-

Shift from high birth rate and high infant death rate to low birth rate and low infant death rate with minimum education & technology.

Demographic Transition Model (Population cycle) :-

It shows the growth rates in populations and effect on population. Divided in four stages:

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Stage 1 - High Fluctuating (High stationary) $\rightarrow$ Both Birth rate and Death rate are both high. * Population growth slow and fluctuating.

Stage 2 :- Early Expanding : $\rightarrow$ Birth rate remains high, Death rate falls. * Population begins to rise steadily.

Stage 3 :- Late Expanding :- $\rightarrow$ Birth rate starts to fall. Death rate starts to fall. * Population rising.

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Stage 4 :- Low Fluctuating :- (low stationary) $\rightarrow$ Birth rate and Death rate both low. * Population steady. e.g.: USA, Sweden, Japan, Britain.

Stage 5 :- Declining stage. BR < DR (Germany, Hungary).

• Model Assumes all countries pass through all four stages.
• Assumes fall in death rate in stage (2) is because of Industrialisation.
• Countries that grew as a consequence of emigration did not pass through early stages of model (USA, Canada, Australia).

Demographic Dividend: When the working population of a country is more than non-working population. A.K.A (Demographic Bonus) Demographic Burden: Working population is less than non-working population. Doubling time: It is number of years required to double the population of a country / Area at a given growth rate.

Father of "Demographic studies" — Karl Marx

Types of Demography :-

1. Formal Demography :- Mathematical study / statistical Analysis - of numbers (population). e.g.:- No. of males / No. of Females, No. of employed / unemployed people.

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2. Social Demography :- Deals with the changes and consequences due to population. e.g.: Birth rate, Death rate, emigration etc.

Four distinct stages of Indian Demography History :-

1. Period of stagnant Population (1901-1921) (Population more or less stagnant) $\downarrow$
2. Period of steady Growth (1921-1951) (More than 1% Growth rate / year) $\times$ Fertility $\downarrow$ Mortality. $\downarrow$
3. Period of Rapid High Growth (1951-1981) (Growth rate > 2% / year) $\rightarrow$ Period of population Explosion $\downarrow$
4. Period of High Growth with definite signs of slowing down. (1981-2011) 2.22% (1971) $\rightarrow$ 1.64% (2011) $\rightarrow$ world (1.23% GR)

* 1921 $\rightarrow$ Year of Great Demographic Divide. (After this, there was continuous increase in population).

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Census

Complete enumeration method which systematically records information about the members of given population. e.g; Housing, culture, Business, Agriculture etc.

Features of Census :-

(1) Sponsorship :- National Govt / State Govt / Local Bodies. (2) Defined Territory :- Clear Demarcation of Area. (3) Well Defined Periodicity :- Done after regular intervals to compare and Analyse Information. (4) Universality :- Include each person without the territory without redundancy. (5) Compilation and Publication :- Filtering and presentation of useful data and publishing it for potential users. (6) Dissemination :- Getting right data for right people. (7) International simultanity :- compared with other countries.

[!fix] ⚠️ DEFINITION CHECK — "Simultaneity" 2024·Q79 asked for a feature of the census and the answer was "Simultaneity." The gloss above ("compared with other countries") is not the standard meaning. ⭐ Simultaneity = every person is enumerated with reference to the SAME point in time (a single reference moment), so the count is a true snapshot. That is what makes it a census feature. ❗ Also absent from this list: "Individual Enumeration" — each person is recorded separately — which is one of the six standard UN census features and a likely option.

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Functions :-

(1) For Administrative purpose and Policy Making: Used to analyse population of area for employment programs, housing, Education, social welfare schemes, economic aspects etc. (2) For Research purpose: Used the solution and interpretation of scientific problems. (3) For social and economic planning of the country: * Gender classification, Urban Vs Rular classification etc. (4) Utility in Business and Industrial sector: Demand and Supply Analysis.

Background of Census of India:

(U.S.A (1790) First modern census)

• In 1830 (Dacca) First census was done $\downarrow$ (Henry Walter) Father of undivided India
• 1872 $\rightarrow$ First Indegenious census (Lord Mayo) $\rightarrow$ V.G of India
• 1881 $\rightarrow$ First synchronous population census. (Lord Rippon) $\rightarrow$ V.G of India 1st. Census commissioner $\downarrow$ W.C Plowden (Father of Indian census 1931 $\rightarrow$ First caste based census of India. (Hutton - commissioner) $\rightarrow$ Caste in India (1946).

1961 $\rightarrow$ MHA $\rightarrow$ RGCCI (Registrar General & census commissioner of India) $\downarrow$ Present - Vivek Joshi

15th census (1872) (7th after independence) Census 2011 (Our census, our future) RGCCI - Dr. Chandramauli Mascot: Women Enumerator (stick figure)

• No. of states and UTs $\rightarrow$ (35)
• Districts $\rightarrow$ 640 (Increased by 47 from 2001)
• Towns $\rightarrow$ 7,933 ($\uparrow$ by 2772 from 2001)
• No. of villages $\rightarrow$ 6,40,930 ($\uparrow$ 2342 from 2001)
• Total Population $\rightarrow$ 1,21,05,69,573 $\hookrightarrow$ Urban (68.8%) $\hookrightarrow$ Rural (31.2%)
• Child Sex Ratio (0-6 year) $\rightarrow$ 919 $\hookrightarrow$ Rural (923) $\hookrightarrow$ Urban (905)
• Sex Ratio $\rightarrow$ 940 / thousand males.
• Density (Pop) $\rightarrow$ 382 person / $Km^2$.
• Decadal pop. Growth $\rightarrow$ 17.64%.
• India's total pop. of world $\rightarrow$ 17.5%.
• Literacy rate $\rightarrow$ 74.04%.
• India pop. $>$ USA + Indonesia + Brazil + Pakistan + Bangladesh.

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Population wise Rank:-

China (19.4% WP) $\xrightarrow{2030}$ India (17.5% WP) $>$ USA (4.5%) $>$ Indonesia $>$ Brazil

Area wise Rank:-

Russia $>$ Canada $>$ China $>$ US $>$ Brazil $>$ Australia $>$ India (7th rank)

➕ Census 2011 — the national headline figures

These were NOT in the original notes, which jump straight to the state-wise ranks. The paper asks the headline numbers too.

Other decadal-growth extremes (2001–11):

• Highest growth — state: Meghalaya (27.9%) · UT: Dadra & Nagar Haveli (55.9%)
• Lowest / negative — state: ⭐ Nagaland (−0.58%)

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As per 2011 census!

Population wise

Biggest-state :- UP (16.49% of Total Population) $>$ Maharashtra $>$ Bihar Smallest-state :- Sikkim (0.05%) Biggest-UT :- Delhi Smallest-UT :- Lakshadeep

Mnemonic: U S Se Dil Lagi bhool jani Padgi

Area wise:

Largest-state :- Rajasthan (3,42,240 $Km^2$) Smallest-state :- Goa (3702 $Km^2$) Largest-UT :- Andaman & Nicobar (8249 $Km^2$) Smallest-UT :- Lakshadeep (32 $Km^2$)

Mnemonic: Roz Ghar Ana Ladke

Sex Ratio (940) $\rightarrow$ overall:

Highest-state :- Kerala (1084) $>$ Tamil Nadu (995) Lowest-state :- Haryana (877) Highest-UT :- Puducherry (1038) Lowest-UT :- Daman & Diu (618)

Mnemonic: Kash Har Ghar mein Daughter hoti

[!fix] ⚠️ FIGURE CHECK — sex ratio The heading above gives the overall 2011 sex ratio as 940. That was the PROVISIONAL figure; the FINAL Census 2011 figure is 943 females per 1,000 males. Use 943 in the exam. ⭐ Also note the convention trap: 2022·Q36 gave the ratio the other way round — (men ÷ women) × 100 = 96.70. Read the options to see which convention is intended. ❗ Missing here: Census 2011's standout fact — Nagaland was the ONLY state with NEGATIVE decadal growth (−0.58%). 2022·Q40 asked exactly this.

Literacy Rate $\rightarrow$ (74.04%):

$\hookrightarrow$ M 82.14% $\hookrightarrow$ F 65.46%

Highest-state :- Kerala (93.91%) $>$ Mizoram (91.58%) Smallest-state :- Bihar (63.82%) Highest-UT :- Lakshadeep (92.28%) Lowest-UT :- Dadra & Nagar Haveli (77.65)

Mnemonic: Koun Banega Lakhpati - Haveli

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Urbanization wise:

Most-state :- Goa (62.17%) Least-state :- Himachal Pradesh (10.04%) Most-UT :- Delhi (97.50%) Least-UT :- Andaman & Nicobar (35.67%)

Mnemonic: Ghar Ho tou Delhi k Andhar

District wise:-

Population wise:

• Biggest - Thane (Mumbai)
• Lowest - Dibang valley

Sex Ratio:-

• Highest :- Mahe (1184) Puducherry
• Lowest :- Daman (534) (D&D)

Literacy Rate:

• Highest :- Serchipp (Mizoram) 98.76%
• Lowest :- Ali Rajpur (M.P) (36.10%)

Census day - 1st April (FY) National Census Day - 9th Feb Population Day - 11th July

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🔷 TOPIC (x) — VITAL STATISTICS

Syllabus topic (x) of 10 · runs until the end of the syllabus

Covers: Objectives & sources · key terms (density, CBR, CDR, natural increase, life expectancy, IMR, neonatal, MMR) · Fertility — CBR, GFR, SFR, TFR · Mortality — CDR, SDR, STDR · ⭐ GRR · ⭐ NRR · ➕ added: NRR ≤ GRR, ASFR modal age, MMR denominator conventions 🎯 Asked in the papers: 2024 · Q80 — match the measures of mortality/fertility

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Vital Statistics

Study and measurement of vital factors related to health and growth of community. It includes:-

• Marriage
• Births
• Deaths
• Sickness
• Migration and so on.

Objectives:-

• To implement and evaluate health related schemes (National Health Programs).
• To determine community health related infections, epidemics and find solution.
• To use the data as primary tool in research activities.
• Administrative purposes.

"Branch of Biometry that deals with data and law of human mortality, morbidity & demography".

Sources of Vital Statistics

• Civil Registration system.
• National Surveys.
• Health Surveys.
• Sample Registration system.

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Key Terms

1. Population Density: Population per square Area / Region (Pop. / $Km^2$)
2. Crude Birth Rate: Number of Births / 1000 people.
3. Crude Death Rate: Number of Deaths / 1000 people.
4. Natural Increase: (CBR - CDR) per year in percentage (%).
5. Life Expectancy: How long people in certain countries are expected to live.
6. Replacement Rate: The rate that is needed to replace both parents.
7. Age dependency: Percentage of population that depend on economic support (e.g; pension for elderly, school for young).
8. Age Dependency Ratio: $\frac{\text{Number of dependents } (0-14 \text{ \& } 65+)}{\text{Pop. } (15-65) \text{ age}}$
9. Abortion Ratio: Number of Abortions per hundred pregnancies.
10. Abortion Rate: Number of Abortions per thousand women (15-44) age.

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11. Emigration Rate: Number of Emigrants per thousand population per year.

12. Immigration Rate: No. of Immigrants per thousand population per year.

13. Brain Drain: Emigration of highly trained, skilled people from country.

14. Effective literacy rate: Number of Literates with age more than or equal to 7 to the population with age more than or equal to 7 in year. $\text{Eff. L.R} = \frac{\text{No. of Literates } (Age \ge 7 \text{ years})}{\text{Total Pop } (Age \ge 7 \text{ yrs}) \text{ in a year}} \times 100$

15. Infant mortality Rate: $\frac{\text{Number of Infant deaths in a year}}{\text{No. of live births in a year}} \times 1000$

16. Neonatal Mortality Rate: $\frac{\text{Deaths under one month}}{\text{No. of live birth}} \times 1000$

17. Post-neonates Mortality Rate: $= \frac{\text{Death between 1}^{\text{st}} \text{ and 11 complete months}}{\text{No. of live births}} \times 1000$

18. Maternal Mortality Rate: $\frac{\text{No. of death of mother due to maternity}}{\text{No. of live births}} \times 1,00,000$

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Fertility

"Natural capacity to give birth to an offspring"

Fecundity:- capacity to bear (how many) child births.

Fertility Rate / Natality Rate / Birth Rate:-

Measure of rate of growth in population due to births in a specific period.

• Expressed in per thousand women per year.
• Child Bearing Age (India) $\rightarrow$ (15 - 49) years.

Methods to calculate Birth rate:-

1. Crude Birth Rate (CBR)
2. General Fertility Rate (GFR)
3. Specific Fertility Rate (SFR)
4. Total Fertility Rate (TFR)

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(i) Crude Birth Rate (CBR)

$CBR = \frac{\text{No. of Live Births}}{\text{Population of that region in that time period}} \times 1000$

$CBR = \frac{B_t}{P_t} \times 1000$

Q,, No. of live births in particular area is 3000, If the total population is 3 lac. Calculate crude Birth rate. Sol:- $CBR = \frac{B_t}{P_t} \times 1000 = \frac{3000}{3,00,000} \times 1000$ $= \frac{3000}{300} = \text{==10==}$

Q,, Find the crude birth rate for given data.

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Sol:- $CBR = \frac{B_t}{P_t} \times 1000$ $= \frac{250}{\text{Males + Females}} \times 1000$ $= \frac{250}{9500 + 10000} \times 1000 = \frac{250}{19500} \times 1000$ $= \frac{2500}{195} = \text{==12.82==}$

(ii) General Fertility Rate (GFR)

$GFR = \frac{\text{No. of live Births}}{\text{No. of women (15-49)}} \times 1000$

$GFR = \frac{B_t}{\sum_{15}^{49} P_t} \times 1000$

For the Above Data: $GFR = \frac{250}{7500} \times 1000 = \frac{2500}{75} = \text{==33.33==}$

(iii) Specific Fertility Rate (SFR):-

Fertility Rate taking the factors which affect it like; Marriage, Age, migration etc. $SFR = \frac{\text{No. of births in specific class}}{\text{Total women in that class}} \times 1000$

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ASFR = $\frac{\text{No. of Births in that particular age group}}{\text{Total No. of women in that group}} \times 1000$

(iv) Total Fertility Rate (TFR)

(a) For equal intervals = $\frac{(\sum \text{ASFR}) \times \text{interval}}{1000}$ (b) For unequal intervals = $\frac{(\sum (\text{ASFR} \times \text{interval}))}{1000}$

Case (b):- unequal intervals $TFR = \frac{\sum (\text{ASFR}) \times i}{1000} = \frac{1574.99}{1000} = \text{==1.574==}$

Quick Recap

1. $CBR = \frac{B_t}{P_t} \times 1000$
2. $GFR = \frac{B_t}{\sum_{15}^{49} P_t} \times 1000$
3. $SFR = \frac{\text{No. of Births (specific class)}}{\text{Total women (section)}} \times 1000$
4. $TFR = \begin{cases} (i) = \text{interval} \Rightarrow \frac{(\sum \text{ASFR}) \times i}{1000} \\ (ii) \neq \text{interval} \Rightarrow \frac{(\sum \text{ASFR} \times i)}{1000} \end{cases}$

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Mortality (Death Rate):-

Number of deaths per thousand population in a particular area in specified time period.

Methods of calculating Death Rate:-

(1) Crude Death Rate (CDR) (2) Specific Death Rate (SDR) (3) Standardised Death Rate (STDR)

(1) Crude Death Rate (CDR):-

$CDR = m = \frac{\text{No. of Annual Deaths}}{\text{Annual mean population}} \times 1000$

Q,, Find the Male, Female and crude Death Rate from the given data.

Sol: $\text{Crude male Death Rate} = \frac{\text{No. of Male Deaths}}{\text{Total Male pop.}} \times 1000$ $= \frac{810}{1,00,000} \times 1000 = \frac{810}{100} = \text{==8.1==}$

$\text{Crude Female Death Rate} = \frac{\text{No. of Female Deaths}}{\text{Total Female pop}} \times 1000$ $= \frac{510}{85,000} \times 1000 = \frac{510}{85} = \text{==6==}$

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$\text{Crude Death Rate} = \frac{\text{No. of Deaths}}{\text{Total Pop.}} \times 1000$ $= \frac{1320}{\text{Male pop + Female Pop}} \times 1000 = \frac{1320 \times 1000}{1,00,000 + 85,000}$ $= \frac{1320}{1,85,000} \times 1000 = \frac{1320}{185} = \text{==7.135==}$

2. Specific Death Rate (SDR):-

$SDR = m_i = \frac{\text{No. of Deaths (specific class)}}{\text{Total Pop. of that class}} \times 1000$

$SDR_{(0-20)} = \frac{120}{(15+15)1000} \times 1000 = \frac{120 \times 1000}{30,000}$ $= \frac{120}{30} = \frac{12}{3} = \text{==4==}$

$SDR_{(20-40)} = \frac{200}{40000} \times 1000 = \frac{200}{40} = \frac{20}{4} = \text{==5==}$

$SDR_{(40-60)} = \frac{250}{25000} \times 1000 = \frac{250}{25} = \text{==10==}$

$SDR_{(60-80)} = ?$ $SDR_{(80-\text{Above})} = ?$

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3. Standardized Death Rate :- (STDR)

$STDR = \frac{\sum m_i \times S_{pop}(i)}{\sum S_{pop}(i)}$

$m_i \rightarrow$ specific Death Rate $S_{pop}(i) \rightarrow$ Standard Population of $(i^{\text{th}})$ age group.

Q Find the standardized Death Rate for the Given Data.

$STDR = \frac{\sum m_i \times S_{pop}(i)}{\sum S_{pop}(i)} = \frac{2,71,700}{37,000} = \frac{2717}{370}$ $= \text{==7.34==}$

Gross Reproductive Rate (GRR):

To get better view about the rate of population, gender of new born baby is taken into consideration.

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"Female children are potential future mothers and result in population increase"

Methods to calculate GRR:-

(1) If female births to "1000" are given. $GRR = \frac{\sum_{15}^{49} B_f}{1000}$

Q,, Calculate the GRR for the given data:-

$\therefore GRR = \frac{\sum_{15}^{49} B_f}{1000} = \frac{1200}{1000}$ $GRR = \text{==1.2== per woman}$

(2) Female Population and female births are given:- $GRR = \sum_{15}^{49} \left[ \frac{B_f}{P_{xf}} \right] \times i$ $\rightarrow$ For unequal intervals

Q Calculate the GRR for the data given.

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(3) If total Fertility Rate and Male : Female ratio is given:- $GRR = \frac{TFR \times B_f}{\sum B}$

Q,, If TFR = 1500 per thousand, Male : Female ratio is = 65:35 for new Born babies. Calculate GRR. Sol:- $GRR = \frac{TFR \times B_f}{\sum B} = \frac{1500 \times 35}{1000}$ (per 1000) $= \frac{15 \times 35}{1000} = \frac{525}{1000} = \text{==0.525== per woman.}$

Points to remember:

(i) If $GRR > 1 \rightarrow$ Population $\uparrow$ inspite of low birth rate (ii) If GRR &lt; 1 \rightarrow Population $\downarrow$ inspite of low mortality (iii) If $GRR = 1 \rightarrow$ Population stagnant (population of females replacing themselves)

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⭐ 1. NRR is ALWAYS less than or equal to GRR

$\mathbf{NRR \le GRR}$

Why: GRR counts the daughters a woman would bear assuming she survives her whole reproductive span — it IGNORES mortality. NRR applies actual survival rates, so some of those daughters are removed. You can only ever lose women, never gain them — so NRR can never exceed GRR. They are equal only in the impossible case of zero mortality.

⭐ 2. ASFR has its modal value between 20 and 25

The Age-Specific Fertility Rate rises from age 15, ⭐ peaks in the 20–25 age group (the modal childbearing age), then falls away to nearly zero by 49.

⚠️ 3. The MMR denominator — know both conventions

The notes above give Maternal Mortality Rate per 1,00,000 live births — that is the official WHO / SRS convention and is correct. ⚠️ But some textbooks and question papers state MMR "per 1,000 live births" — the 2024 paper did exactly this. Read the options and take whichever convention the question offers. The idea being tested is only that MMR is measured against live births, not against total population.

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Net Reproductive Rate (NRR) :

Net Reproductive Rate is the extension or modified form of GRR where mortality / survival rates is also taken in Account.

(i) If Female births, female population and survival rate is given :- $NRR = i \times \sum \left( \frac{B_f}{P_{xf}} \times 1000 \right) \times S$

• $i \rightarrow$ interval
• $B_f \rightarrow$ Female births
• $S \rightarrow$ Survival Rate
• $P_{xf} \rightarrow$ Female Population

Q,, Calculate the net Reproductive rate for the given Data.

$NRR = i \times \frac{\sum \left( \frac{B_f}{P_{xf}} \times 1000 \right) \times S}{1000}$ $= 5 \times \frac{473.56}{1000} = 5 \times 0.473 = \text{==2.365==}$ Per Woman.

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Trick: for calculating Survival Rate from Mortality Rate or Mortality Rate from Survival Rate :-

$S.R = 10^{\text{no. of digits in mortality rate}} - \text{Mortality Rate}$ or $M.R = 10^{\text{no. of digits in survival rate}} - \text{Survival Rate}$

e.g:- If Survival Rate = 6, M.R = ? $M.R = 10^1 - 6 = 10 - 6 = \text{==4==}$

e.g:- If M.R = 63 what is survival Rate $S.R = 10^2 - 63 = 100 - 63 = \text{==37==}$

Important points About NRR:-

(i) If NRR > 1, Population will increase inspite of high death rate. (ii) If NRR < 1, Population will decrease inspite of high birth rate. (iii) If NRR = 1, Population will be stagnant.

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Case 3 of trick!

Q,, $\frac{32146123}{2043212} = \frac{3214}{204} \xrightarrow{(-4)} \frac{3200}{200}$

$= \text{==16==}$ $16 \times 4 = \text{==64==} \rightarrow \text{factor to subtract.}$

$3214 - 64 = 3150$

$\frac{3150}{200} = \frac{315}{20} = \frac{31.5}{2} = \text{==15.75==}$ (Calculator = 15.73)

Q,, Number of men is 151,781,326 and number of women is 156,964,212. Which options represent demographic sex Ratio. (PAA JKSSB 2020)

(a) 97.6 (b) 96.7 (c) 98.2 (d) 95.3

Sol:- $\text{Sex Ratio} = \frac{\text{No. of males}}{\text{No. of Females}} \times 100$

$= \frac{151,781,326}{156,964,212} \times 100$

$= \frac{151}{156} = \frac{150}{160} = \frac{15}{16}$ , $\frac{15}{16} \times 4 = \frac{15}{4} = \text{==3.75==}$

$= \frac{151 + 3.75}{160} = \frac{154.75}{160} \times 100$ $= \frac{154.75}{16} \times 10 = \frac{154.75 \times 5}{8} = 19.34 \times 5 = \text{==96.70==}$ (Calculator = 96.69)

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