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GNDU Question Paper-2021

Bachelor of Commerce

(B.Com) 1

Semester

BUSINESS STATISTICS

Time Allowed: Three Hours Maximum Marks: 50

Note:-Attempt any FIVE questions, selecting at least ONE question from each section and

the fifth question may be attempted from any section. Each question carries 10 marks.

SECTION-A

1. Find the mean, median and modal ages of married women at first child birth.

Age at the birth of

first child

No. of married

women

162

343

390

256

433

161

355

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2. Explain the meaning and scope of statistics.

SECTION-B

3. (a) Distinguish between correlation and regression. Discuss the properties of regression.

(b) From the following data calculate the rank correlation coefficient after making

adjustment for tied ranks.

4. Goals scored by two teams A and B in a football season were as follows:

Number of goals scored

in match

Number of Matches

A team

B team

By calculating the coefficient of variation in each case, find which team may be considered

more consistent.

SECTION-C

5. (a) What do you understand by Index Numbers? Explain the various pr problems faced

in the construction of Index Numbers.

(b) Compute a price index for the following by

(ii) Average of price

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(ii) Simple aggregate and relative method by using both arithmetic mean and geometric

mean:

Commodity

Price in

2018 (Rs.)

Price in

2019 (Rs.)

6. Find Laspeyre's, Paasche's and Fisher's Ideal Price Index Numbers from the following

data and test their adequacy with respect to Time Reversal and Factor Reversal Tests.

Commodity

SECTION-D

7. Assuming a four-yearly cycle, calculate the trend by the method of moving averages

from the following data relating to the production of tea in India.

Year

Production (in million lbs)

2009

464

2010

515

2011

518

2012

467

2013

502

2014

540

2015

557

2016

571

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2017

586

2018

612

8. (a) Define Probability. Explain various theorems of probability.

(b) The CEO of a company plans to take two of his executives to a conference in Chicago.

He plans to select at random one of the eight executives in finance and one of the ten

executives in marketing. Of the executives in finance, two have already travelled with the

CEO while three of the executives in marketing have travelled with CEO before. What is

the probability that:

(i) Both the executives have travelled with him before?

(ii) At least one of the executives selected has travelled with him before?

(iii) None of the executives chosen has travelled with the CEO before?

(iv) The finance executive has, and the marketing executive has not, travelled with

the CEO. before?

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GNDU Answer Paper-2021

Bachelor of Commerce

(B.Com) 1

Semester

BUSINESS STATISTICS

Time Allowed: Three Hours Maximum Marks: 50

Note:-Attempt any FIVE questions, selecting at least ONE question from each section and

the fifth question may be attempted from any section. Each question carries 10 marks.

SECTION-A

1. Find the mean, median and modal ages of married women at first child birth.

Age at the birth of

first child

No. of married

women

162

343

390

256

433

161

355

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Ans: Imagine a group of women sitting together in a village courtyard, chatting about the

age at which they had their first child. Some say they were very young, some a little older,

but together their experiences form a kind of “picture” of society. Now, as curious learners,

we want to capture that picture in numbers.

But here comes the problem: when the data is huge, like in this case, where we have the

ages of over 2000 women, we can’t just look at everyone individually. We need some smart

ways to summarize the entire story into one or two numbers that can describe the whole

group. This is where the mean, median, and mode come into play. They are like three

storytellers, each showing us the same reality but from different angles.

So let’s dive in, but in a way that feels like we are discovering the secret of this story

together.

Step 1: Understanding the Data

Here is the data given:

Age at first child (x)

No. of women (f)

162

343

390

256

433

161

355

First, let’s find the total number of women (N).

N=37+162+343+390+256+433+161+355+65+85+49+46+40=2372

So, the total group consists of 2372 women.

Step 2: Finding the Mean

The mean is like the average — if we take all ages, multiply them with the number of

women at that age, and then divide by the total women, we’ll get it.

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󷵻󷵼󷵽󷵾 The mean age is about 17.8 years.

Step 3: Finding the Median

The median is the middle value. It’s like asking, “If all 2372 women stood in a line in order of

age, what age would the woman in the middle have?”

The formula is:

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Here,

So we want the 1186th woman in the cumulative frequency (CF) distribution.

Let’s make CF:

Age

162

199

343

542

390

932

256

1188

433

1621

161

1782

355

2137

2202

2287

2336

2382

2422

Now, look carefully:

• The 932nd woman is aged 16.

• The 1188th woman is aged 17.

Since 1186th falls between 933 and 1188, the median = 17 years.

󷵻󷵼󷵽󷵾 The median age is 17 years.

Step 4: Finding the Mode

The mode is the most common value — the age that occurs most frequently. Just like in a

classroom where the most common favorite color becomes the “class favorite,” here the

most common age at first childbirth is the one with the highest frequency.

Looking at the table, the highest frequency is 433 (at age 18).

󷵻󷵼󷵽󷵾 The modal age = 18 years.

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Step 5: Wrapping it all into a Story

So what have we found?

• The mean age is around 17.8 years, which is like saying that if we spread all ages

equally, the “average” comes to about 18.

• The median age is 17 years, meaning half of the women had their first child before

17, and the other half after 17.

• The mode is 18 years, which tells us that 18 was the most popular age for becoming

a mother for the first time.

Now imagine this as a social picture: in this group, most women had their first child between

16 and 18 years. Very few had it as early as 13 or as late as 25. This tells us not just about

numbers, but about society’s norms, family traditions, and perhaps even pressures during

that time.

Final Answer (in one line):

• Mean = 17.8 years

• Median = 17 years

• Mode = 18 years

2. Explain the meaning and scope of statistics.

Ans: Meaning and Scope of Statistics

Imagine you are sitting with your friends in a college canteen. Everyone is discussing who

scored the highest in the last exam, how many people actually passed, how many failed, and

what the average marks of the class might be. At that moment, without realizing it, you are

already using statistics. You are trying to collect numbers (marks of students), organize them

(pass/fail list), and then draw conclusions (who performed better, what is the average).

This small incident explains the basic meaning of statistics: it is a science of collecting,

organizing, presenting, analyzing, and interpreting numerical data. In simple words,

statistics helps us make sense of large amounts of information.

But before we dive deeper, let us understand how the meaning of statistics has changed

over time and then see its vast scope in real life.

The Meaning of Statistics

The word Statistics has two different senses:

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1. Statistics in the Singular Sense (a subject or discipline):

o Here, statistics is treated as a branch of study or a method.

o For example, when we say “Statistics is an important subject in social

sciences,” we are referring to statistics as a science or method of dealing with

data.

o It means the techniques and methods used to collect, classify, present, and

interpret numerical facts.

2. Statistics in the Plural Sense (data):

o Here, statistics means “numerical facts” themselves.

o For example, “The statistics of population shows that India has crossed 1.4

billion.”

o In this sense, statistics refers to the raw numerical data collected about a

subject.

So, whenever you hear the word “statistics,” it could either mean the figures (like literacy

rate, population, exam results) or the methods used to study those figures.

Why Do We Need Statistics?

Let’s say the government wants to know whether poverty has reduced in the last ten years.

Can they ask every single citizen? Of course not. Instead, they collect data from a sample of

households, study it, and then make conclusions about the whole population.

That’s the power of statistics. It allows us to:

• Summarize huge information in a simple form (like average marks of a class).

• Compare (which state has more literacy).

• Forecast (predicting tomorrow’s weather using past data).

• Make decisions (deciding government policies, business investments, sports

strategies, etc.).

The Scope of Statistics

Now that we know what statistics means, let us explore its wide scope. Statistics is not just

about numbers—it is about life, society, economics, business, medicine, and almost

everything around us.

1. In State Administration

The word “statistics” actually comes from the word status or state. In ancient times, kings

and rulers collected data about population, land, crops, and revenue to manage their

kingdoms. Even today, governments depend heavily on statistics to:

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• Conduct census every 10 years.

• Collect data on literacy, employment, health, and poverty.

• Frame policies based on statistical data.

Without statistics, good governance would be impossible.

2. In Economics

Economics and statistics are like best friends. Almost every economic theory, plan, or policy

is backed by statistical data. For example:

• To measure national income, economists use statistical methods.

• Inflation, unemployment rate, poverty ratio—all are statistics.

• Demand and supply curves are based on numerical data.

In short, economics without statistics is like a body without a backbone.

3. In Business and Industry

Businesses use statistics every single day:

• To find out what customers like or dislike.

• To set prices after studying demand and supply.

• To measure production efficiency.

• To check quality control in factories.

Even big companies like Amazon or Flipkart analyze customer data statistically before giving

discounts or recommendations.

4. In Medicine and Biology

Doctors and scientists use statistics to test new medicines. For example, before introducing

a vaccine, they test it on thousands of people and use statistics to analyze its success rate.

Similarly, biologists use it for population studies, genetics, and disease control.

5. In Social Sciences

Subjects like Sociology, Psychology, and Political Science use statistics to understand human

behavior. For example:

• Surveys about voting behavior during elections.

• Studies on crime rates.

• Research on literacy and education.

Without statistical data, social sciences would only remain in the world of theory.

6. In Research and Technology

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Every field of research, whether it is space science or artificial intelligence, uses statistics.

Predictions about climate change, analysis of rocket performance, or even algorithms used

in Google search are based on statistical models.

7. In Daily Life

Even in our daily routine, statistics plays an invisible role:

• Cricket fans check batting averages.

• Students calculate their percentage in exams.

• Weather reports predict rainfall chances.

• Banks calculate interest using statistical formulas.

So, statistics is not limited to textbooks; it’s everywhere around us.

The Limitations of Statistics

To be balanced, we must also remember that statistics has limitations:

• It deals only with numbers, not with qualitative aspects like honesty or beauty.

• If the data collected is wrong, the results will also be wrong (garbage in, garbage

out).

• It cannot establish universal truths—it only shows tendencies or probabilities.

Thus, while statistics is powerful, it must be used carefully.

Conclusion

In simple terms, statistics is the art and science of learning from data. It helps us transform

confusing numbers into meaningful conclusions. Its scope is so wide that it touches every

corner of human life—from governments and businesses to classrooms and cricket fields.

If we look around, life without statistics would be like walking in the dark without a torch. It

guides decision-making, supports research, and helps society grow with clarity.

So, the next time you compare your marks with a friend, check cricket scores, or hear about

India’s population in the news, remember—you are actually dealing with statistics

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SECTION-B

3. (a) Distinguish between correlation and regression. Discuss the properties of regression.

(b) From the following data calculate the rank correlation coefficient after making

adjustment for tied ranks.

Ans: Part (a) Distinguish between Correlation and Regression + Properties of Regression

A Gentle Start with a Story

Imagine two friends, Ravi and Sita. Ravi loves cricket, and Sita loves badminton. One day

they start talking about how their exam marks might be connected. Ravi says, “Whenever I

study a lot, my marks go up, but when I spend time playing cricket, my marks sometimes go

down.” Sita replies, “Same here! I wonder if our sports practice hours and exam scores are

somehow related.”

This small conversation is exactly what statistics tries to do with correlation and regression.

Both are about finding relationships between two variables, but they do so in slightly

different ways.

Correlation

• Meaning: Correlation measures the degree of association between two variables. It

tells us how strongly and in which direction (positive or negative) the variables move

together.

• Example: If the number of hours you study increases, your marks may also increase.

That is a positive correlation. If the number of hours you play video games increases

and your marks fall, that is a negative correlation.

• Key Point: Correlation does not tell us cause-and-effect. It only tells us the strength

and direction of association.

Regression

• Meaning: Regression goes one step ahead. It not only studies the relationship but

also gives an equation so that we can predict the value of one variable from

another.

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• Example: Suppose data shows that “Every extra 1 hour of study increases marks by

5.” Then we can write an equation like:

This is regression – it allows prediction.

Key Differences Between Correlation and Regression

Basis

Correlation

Regression

Purpose

Measures the strength and direction of

association

Establishes a predictive equation

Nature

Symmetrical (correlation between X

and Y is same as between Y and X)

Asymmetrical (regression of Y on X is

different from regression of X on Y)

Range

Lies between -1 and +1

No such fixed range

Focus

Only tells whether variables move

together

Explains how one variable changes when

another changes

Cause-

effect

No assumption of cause-effect

Often assumes one variable depends on

another

Properties of Regression

Now that we understand what regression is, let’s list and explain its properties in a simple

manner.

1. Linearity

o The regression equation is usually a straight line when we deal with simple

linear regression.

o Example: Marks = a + b × (Hours studied). It’s a line, not a curve.

2. Two Regression Lines

o There are two regression equations:

▪ Regression of Y on X (to predict Y from X)

▪ Regression of X on Y (to predict X from Y)

o Both lines generally intersect at a common point: the mean of X and mean of

3. Slope and Correlation Connection

o The slope of the regression line depends on the correlation coefficient. If

correlation is zero, the regression line becomes horizontal (no relation).

4. Regression Coefficients

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o The coefficients (bxy and byx) are constants that tell us how much one

variable changes with the other.

o Important property:

(where r = correlation coefficient)

5. Direction of Regression Coefficients

o Both regression coefficients have the same sign. If correlation is positive,

both are positive. If correlation is negative, both are negative.

6. Use in Prediction

o The most practical property: regression can predict unknown values. If we

know someone’s study hours, we can guess their marks fairly well.

So, in short:

• Correlation is like saying “Ravi and Sita are friends and walk in the same or opposite

direction.”

• Regression is like saying “If Ravi takes 2 steps, Sita takes 3 steps. Let’s write an

equation for it.”

Part (b) Rank Correlation Coefficient with Tied Ranks

Now comes the numerical part. Don’t worry; we’ll do it step by step like a puzzle.

We are asked to calculate Spearman’s Rank Correlation Coefficient (ρ).

The formula is:

where

• D= difference between ranks of X and Y,

• n= number of pairs,

• tie correction is added when ranks are repeated.

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Step 1: Write the data

X: 48, 33, 40, 9, 16, 16, 65, 24, 16, 57

Y: 13, 13, 24, 6, 15, 4, 20, 9, 6, 19

Step 2: Assign ranks for X

We rank X values in descending order (highest value = rank 1).

• 65 → Rank 1

• 57 → Rank 2

• 48 → Rank 3

• 40 → Rank 4

• 33 → Rank 5

• 24 → Rank 6

• 16, 16, 16 → These are tied. Normally they would be ranks 7, 8, 9. So we give them

average rank = (7+8+9)/3 = 8.

• 9 → Rank 10

So X ranks = [3, 5, 4, 10, 8, 8, 1, 6, 8, 2]

Step 3: Assign ranks for Y

Y values in descending order:

• 24 → Rank 1

• 20 → Rank 2

• 19 → Rank 3

• 15 → Rank 4

• 13, 13 → These are tied for ranks 5 and 6 → average rank = 5.5 each

• 9 → Rank 7

• 6, 6 → tied for ranks 8 and 9 → average rank = 8.5 each

• 4 → Rank 10

So Y ranks = [5.5, 5.5, 1, 8.5, 4, 10, 2, 7, 8.5, 3]

Step 4: Calculate d and d²

RankX

RankY

d = (RankX – RankY)

d²

5.5

-2.5

6.25

5.5

-0.5

0.25

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8.5

1.5

2.25

-2

-1

8.5

-0.5

0.25

-1

∑d2=41

Step 6: Apply formula

Final Answer

The Rank Correlation Coefficient = 0.733

This shows a fairly strong positive correlation between the two sets of data.

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4. Goals scored by two teams A and B in a football season were as follows:

Number of goals scored

in match

Number of Matches

A team

B team

By calculating the coefficient of variation in each case, find which team may be considered

more consistent.

Ans: Step 1: Understanding the Problem

We are given data about the number of goals scored by two football teams (A and B) in

different matches during a season. Instead of listing all the goals from every single match,

the data is summarized in a table:

Goals scored in match

Number of matches (Team A)

Number of matches (Team B)

So, for example, Team A scored 0 goals in 27 matches, 1 goal in 9 matches, and so on.

Now, our task is to calculate the Coefficient of Variation (C.V.) for both teams and then

decide which team was more consistent.

Step 2: What is Coefficient of Variation (C.V.)?

Before calculating, let’s recall the meaning.

• Mean (average): tells us the central value of the data (average goals per match).

• Standard Deviation (S.D.): tells us how much the values deviate (spread out) from

the mean.

• Coefficient of Variation (C.V.): compares the standard deviation to the mean and

expresses it as a percentage.

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The formula is:

The smaller the C.V., the more consistent the performance, because it means less variation

compared to the average.

Step 3: Calculations for Team A

We will calculate step by step like detectives collecting clues.

(i) Mean goals for Team A

Where:

• f = frequency (number of matches)

• x = goals scored

So for Team A:

So, on average, Team A scored about 1.06 goals per match.

(ii) Variance and Standard Deviation for Team A

Formula for variance:

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But it is easier to use this shortcut:

So, we need ∑fx2.

Now,

So, standard deviation for Team A = 1.31.

(iii) C.V. for Team A

That’s quite high!

Step 4: Calculations for Team B

Following the same method:

(i) Mean goals for Team B

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So, on average, Team B scored 1.2 goals per match.

(ii) Variance and Standard Deviation for Team B

So, standard deviation for Team B = 1.31.

(iii) C.V. for Team B

Step 5: Comparison

• Team A: C.V. = 123.6%

• Team B: C.V. = 109.2%

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Since lower C.V. means more consistent performance, Team B is more consistent than

Team A.

Step 6: Final Answer in Simple Words

Think of it like two students in a class. Both students have similar average marks, but one

has marks that fluctuate wildly – sometimes very high, sometimes very low – while the

other performs more steadily. Even if their averages are close, the steadier student is

considered more consistent.

Here, both Team A and Team B score on average about 1 goal per match, but Team A’s

performance is much more irregular (sometimes zero, sometimes higher), while Team B is

relatively steadier.

So, the conclusion is clear:

󷵻󷵼󷵽󷵾 Team B is more consistent than Team A in their goal-scoring performance during the

season.

SECTION-C

5. (a) What do you understand by Index Numbers? Explain the various pr problems faced

in the construction of Index Numbers.

Ans: Index Numbers – A Story of Comparing Time and Change

Imagine you walk into your local market. Last year, the price of rice was ₹40 per kg, sugar

was ₹30 per kg, and milk was ₹50 per litre. This year, rice is ₹50, sugar is ₹40, and milk is

₹60. You scratch your head and wonder: “Are things really becoming expensive overall, or is

it just rice and sugar playing tricks on me?”

This little confusion is exactly where the idea of Index Numbers comes into play. Index

Numbers are like a “thermometer” for the economy – they help us measure changes in

prices, quantities, or values over a period of time, and allow us to compare them with a base

year. Just like a thermometer tells us whether it’s hotter or colder today compared to

yesterday, an index number tells us whether prices, production, or wages have risen or

fallen compared to a chosen reference year.

In simple terms, Index Numbers are statistical devices used to measure relative changes in

a group of related variables over time or place.

Why Do We Need Index Numbers?

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Life is never still. Prices change, production levels rise and fall, population grows, and wages

fluctuate. If we want to study these changes one by one, it would be very confusing and

time-consuming. That’s where index numbers become handy – they summarize a lot of data

into a single figure, making comparisons easy.

For example:

• Cost of Living Index tells us how much more (or less) expensive it is to live this year

compared to the past.

• Industrial Production Index shows whether industries are producing more or less

goods than before.

• Wholesale Price Index (WPI) or Consumer Price Index (CPI) helps the government,

economists, and businesses understand inflation.

So, Index Numbers act as guiding lamps that illuminate trends in an otherwise confusing sea

of data.

Problems Faced in the Construction of Index Numbers

Now, just like making a delicious recipe requires careful choice of ingredients, constructing a

reliable index number is not simple. There are several problems that economists and

statisticians face. Let’s walk through these challenges one by one, almost like hurdles in a

race.

1. The Problem of Purpose

The very first problem is: “Why are we making the index number?”

Is it for studying inflation? Is it to understand the cost of living of workers? Or is it to

measure industrial growth?

Different purposes require different types of data. If the purpose is not clearly defined, the

whole index number may become misleading.

For example, an index for farmers (tracking fertilizer, seeds, tools, and crop prices) would be

completely different from an index meant for factory workers (tracking rent, food, and

transport).

2. The Problem of Selecting Commodities

Suppose you are making a price index. Out of thousands of commodities available in the

market, which ones should you include? Should you take only basic necessities like food and

clothes, or also luxury items like smartphones and perfumes?

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If you include too many, the calculation becomes complicated. If you include too few, the

index will not represent reality.

This problem of selection of commodities is one of the biggest challenges.

3. The Problem of Weights

All commodities do not have equal importance in people’s lives. For example, rice and

wheat are consumed daily, while perfumes or jewelry are used occasionally. If both are

given equal importance in the index, the result will be unrealistic.

Hence, weights must be assigned. That means more important commodities should carry

more weight, and less important commodities should carry less. But assigning proper

weights is very difficult and often subjective.

4. The Problem of Base Year

An index number is always compared with a base year, which acts like a “reference point.”

But which year should be chosen?

The base year should be:

• Normal (no war, famine, flood, or economic crisis)

• Recent enough to be relevant

• Representative of typical conditions

If the base year chosen is abnormal, the index number will not give a fair comparison. For

example, if 2020 (COVID-19 pandemic year) is chosen as the base year, it may distort many

comparisons.

5. The Problem of Price Quotation

Prices of the same commodity may differ from place to place and even from shop to shop.

Which price should be used? Wholesale or retail? Urban or rural? Market price or controlled

price?

Getting accurate, reliable, and comparable price data is a constant headache in constructing

index numbers.

6. The Problem of Method of Averaging

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Once prices and quantities are selected, the next question is: “How should we calculate the

index?”

There are various methods – like simple aggregative method, average of relatives, Laspeyres

method, Paasche method, Fisher’s Ideal method, etc. Each method gives slightly different

results. Choosing the right method is not easy.

7. The Problem of Comparability

Another difficulty is comparability. Suppose you make an index for 2015 and one for 2025.

Can you directly compare them? Maybe not – because commodities, tastes, technology, and

habits change over time. For example, in 2005, there were no smartphones, but today they

are a major expense.

Thus, changes in lifestyle make comparison across long periods tricky.

8. The Problem of Accuracy

Finally, index numbers are only approximations. They cannot capture the reality of millions

of people perfectly. Errors in data collection, wrong selection of items, or inappropriate

methods can all reduce accuracy. Yet, despite these limitations, index numbers are still

widely used because they are the best available tools.

Conclusion

So, to wrap up, index numbers are not just dry statistics – they are like storytellers of the

economy. They tell us how life has changed compared to the past, whether we are spending

more, producing more, or living better.

But constructing these storytellers is not easy. From choosing the purpose, commodities,

and base year to assigning weights, collecting price quotations, and deciding on methods –

every step has its own challenges.

Even with these problems, index numbers remain one of the most powerful tools in

economics. Without them, governments cannot design policies, businesses cannot plan

strategies, and even common people cannot understand how their cost of living is changing.

In short, index numbers are like a mirror – maybe not perfect, but clear enough to show us

the reflection of economic reality.

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(b) Compute a price index for the following by

(ii) Average of price

(ii) Simple aggregate and relative method by using both arithmetic mean and geometric

mean:

Commodity

Price in

2018 (Rs.)

Price in

2019 (Rs.)

Ans: Once upon a classroom — imagine six friends (A, B, C, D, E, F) who each sell one kind of

commodity. In 2018 each friend had a price tag on their product; in 2019 those tags

changed. Your job is to measure how much overall prices moved between the two years.

We’ll walk through four commonly used ways to make that comparison, compute each one

carefully, and explain what each method means in plain language so any student (and any

examiner!) enjoys reading it.

Data (given)

Commodities: A, B, C, D, E, F

Prices in 2018 (base year): 20, 30, 10, 25, 40, 50

Prices in 2019 (current year): 25, 30, 15, 35, 45, 55

1. First step — simple bookkeeping (sums and averages)

We’ll start by computing totals and simple averages. Doing these basic steps explicitly avoids

mistakes later.

Sum of base-year prices (2018):

20 + 30 + 10 + 25 + 40 + 50 = 175

Sum of current-year prices (2019):

25 + 30 + 15 + 35 + 45 + 55 = 205

Number of commodities = 6

Average (arithmetic mean) price in 2018:

175 ÷ 6 = 29.1666666667 (≈ 29.1667)

Average (arithmetic mean) price in 2019:

205 ÷ 6 = 34.1666666667 (≈ 34.1667)

We will use these totals and averages in the index formulas below.

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2. Price relatives (individual item percent changes)

A price relative for a commodity shows how that particular price changed: (Price_2019 ÷

Price_2018) × 100. Compute this for each friend:

A: (25 ÷ 20) × 100 = 1.25 × 100 = 125.0

B: (30 ÷ 30) × 100 = 1.00 × 100 = 100.0

C: (15 ÷ 10) × 100 = 1.50 × 100 = 150.0

D: (35 ÷ 25) × 100 = 1.40 × 100 = 140.0

E: (45 ÷ 40) × 100 = 1.125 × 100 = 112.5

F: (55 ÷ 50) × 100 = 1.10 × 100 = 110.0

So the list of price relatives (in percent) is:

[125.0, 100.0, 150.0, 140.0, 112.5, 110.0]

Interpretation: Commodity C shot up 50% (relative 150), B didn’t change at all (relative 100),

A rose 25%, D rose 40%, etc.

3. Method 1 — Average of price (index using averages)

This method first computes the mean price in each year and then compares those means:

Index = (Average price in 2019 ÷ Average price in 2018) × 100

We already computed:

Average 2018 = 29.1666666667

Average 2019 = 34.1666666667

Index = (34.1666666667 ÷ 29.1666666667) × 100

First compute the ratio: 34.1666666667 ÷ 29.1666666667 ≈ 1.1714285714

Index ≈ 1.1714285714 × 100 = 117.1428571429

Rounded sensibly we can write: 117.1429

Interpretation: According to the average-of-price method, overall prices are about 17.14%

higher in 2019 than in 2018.

Why it works: This treats each commodity equally by averaging their prices first, then

comparing those average levels.

4. Method 2 — Simple aggregate method

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This method sums up all prices in each year and compares the totals:

Index = (Sum of prices in 2019 ÷ Sum of prices in 2018) × 100

Sum 2019 = 205

Sum 2018 = 175

Index = (205 ÷ 175) × 100 = 1.1714285714 × 100 = 117.1428571429

Rounded: 117.1429

Observation: For this data set the aggregate method gives the exact same result as the

average-of-price method. That is because averaging is just the sum divided by the same

count; comparing averages equals comparing sums when the number of items is constant.

So both tell us prices rose 17.14%.

5. Method 3 — Relative method using Arithmetic Mean of price relatives

This method produces an index by taking the average (arithmetic mean) of the individual

price relatives we computed earlier.

Price relatives (again): [125.0, 100.0, 150.0, 140.0, 112.5, 110.0]

Arithmetic mean of relatives = (125.0 + 100.0 + 150.0 + 140.0 + 112.5 + 110.0) ÷ 6

Sum of relatives = 125 + 100 + 150 + 140 + 112.5 + 110 = 737.5

Arithmetic mean = 737.5 ÷ 6 = 122.9166666667

Rounded: 122.9167

Interpretation: This implies an average relative change of 122.92, meaning about 22.92%

increase (because index 122.92 corresponds to +22.92%).

Why this can differ from the aggregate index: This approach treats each commodity equally

in terms of relative change, whereas aggregate/sum-based methods effectively weight

items by their price levels (higher priced items contribute more to the sum). If items that

rose most were low-priced or high-priced, the arithmetic-relative mean will reflect the

simple average of percent changes, not the weighted effect on total spending.

6. Method 4 — Relative method using Geometric Mean of price relatives

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The geometric mean is often recommended for averaging ratios (like price relatives)

because it gives a multiplicative average and reduces the influence of very large or very

small relatives. The geometric mean index is computed as:

Geometric mean of relatives = (∏ (price_relative_i))^(1/n)

But because price relatives are given as percentages (like 125 = 1.25×100), the standard

practice is to convert them into ratios (r_i = price_relative_i ÷ 100), take the product of the

ratios, take the n-th root, and then convert back to percentage by multiplying by 100.

Compute step-by-step:

Ratios = [1.25, 1.00, 1.50, 1.40, 1.125, 1.10]

Product of ratios = 1.25 × 1.00 × 1.50 × 1.40 × 1.125 × 1.10

Let’s multiply carefully:

1.25 × 1.00 = 1.25

1.25 × 1.50 = 1.875

1.875 × 1.40 = 2.625

2.625 × 1.125 = 2.953125

2.953125 × 1.10 = 3.2484375

So the product of ratios = 3.2484375

Now take the 6th root (since there are 6 commodities):

Geometric mean ratio = (3.2484375)^(1/6)

Compute this root (numerical):

(3.2484375)^(1/6) ≈ 1.2169677961

Convert back to percentage: 1.2169677961 × 100 = 121.6967796100

Rounded: 121.6968

Interpretation: The geometric-mean relative index is 121.6968, meaning an average

increase of ≈21.70%.

Why geometric mean? It reflects a multiplicative central tendency of growth factors and is

less sensitive to large outliers than the simple arithmetic mean of relatives. In inflation and

index number theory, the geometric mean often has desirable properties (e.g., being

consistent when you chain indices), but it also treats increases and decreases symmetrically

in a multiplicative sense.

7. Summary of computed indices (rounded to 4 decimal places)

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• Average-of-price (or simple aggregate) index = 117.1429 → +17.1429%

• Simple aggregate (same as above) = 117.1429 → +17.1429%

• Relative (Arithmetic mean of price relatives) = 122.9167 → +22.9167%

• Relative (Geometric mean of price relatives) = 121.6968 → +21.6968%

8. Interpretation and story wrap-up — which number to use, and why they differ

Think of the six friends selling their goods. Two ways of asking “how much did prices go up?”

are natural:

1. “If I bought one unit of each friend’s product in 2018 and one unit of each in 2019,

how much more would I pay?” — That’s the aggregate/simple-sum (or average-of-

price) approach. It sums the actual rupee prices and compares totals. Because it

weights each item by its price level, an item with a high price (like F at 50 rupees)

naturally has a larger influence on the aggregate index than a low-priced item (like C

at 10 rupees). For our data, this gives +17.14%.

2. “On average, by what percent did each friend’s price change?” — That’s the relative

approach. If you want the mean of percent-changes, you can take the arithmetic

average of those percent changes: that gave +22.92%. But percent-changes combine

multiplicatively (a 50% rise then a 20% fall is not the same as their arithmetic mean),

so some economists prefer the geometric mean of relatives, which gives +21.70%.

The geometric mean is often considered more appropriate for rates of change

because it respects multiplicative compounding.

Why are they different here? Because the commodities with the largest percent increases

(commodity C: +50%, D: +40%) are relatively low-priced in base year (C = 10) — so they pull

up the simple average of percent changes strongly, but don’t contribute as much to the total

rupee increase (since their base value is small). The aggregate index reflects changes in the

total rupee cost (important for cost-of-living and spending measures), while the simple

mean of relatives looks at the typical percent-change per item (which might be useful in

other contexts).

9. Guidance on which method to report (practical advice)

• If your aim is to measure change in the total amount a consumer would pay for one

unit of each item (or for a fixed basket with quantities equal across items), use the

aggregate or average-of-price method. That's often simple and intuitive.

• If you’re interested in the average percent change across items (each item equally

important in percent-terms), use the arithmetic mean of relatives, but be aware

arithmetic mean can be skewed by big percent changes.

• For a preferable “middle ground” in many statistical contexts — especially when

dealing with rates and multiplicative change — use the geometric mean of relatives.

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It is frequently recommended in index-number theory (think of chained indices or

when combining growth rates).

• Always state clearly which method you used and why — examiners like clear

justification.

10. Final closing (one last story beat)

So the six friends watched their price tags change. If you were paying the bill for one unit of

each friend’s product, your bill rose by about 17.14% from 2018 to 2019. But if you asked,

“on average, how much did each friend raise their price in percentage terms?” you’d say

something closer to 22% (arithmetic mean of relatives), or 21.7% if you prefer the geometric

mean that tones down extreme moves.

That’s the tale of prices A–F: the numbers tell complementary stories depending on what

question you ask. Always pick the index that answers the question you truly care about —

and explain your choice clearly.

6. Find Laspeyre's, Paasche's and Fisher's Ideal Price Index Numbers from the following

data and test their adequacy with respect to Time Reversal and Factor Reversal Tests.

Commodity

Ans: Imagine you and I run two little village shops — one in Year 0 (our “base” year) and one

in Year 1 (our “current” year). We sell four items: A, B, C and D. The exam question asks us

to compare the overall price level between Year 0 and Year 1 using three famous price

indices (Laspeyres, Paasche and Fisher), then check two consistency tests those indices are

sometimes judged by: the Time Reversal Test and the Factor Reversal Test. I’ll walk you

through every step as if we’re counting coins on the table — clear, patient, and with the

arithmetic shown so nobody gets lost.

Given data (recap)

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We have prices and quantities in Year 0 (P₀ and Q₀) and Year 1 (P₁ and Q₁):

Commodity

P₀

Q₀

P₁

Q₁

For clarity: P₀ is price in Year 0, P₁ is price in Year 1. Q₀ and Q₁ are quantities in the two

years.

Step 1 — Compute the values (money spent) in each year and some building blocks

First compute the expenditure in Year 0 for each commodity: P0×Q0P_0 \times Q_0P0×Q0.

Now sum them to get total value in Year 0:

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Intuition: Fisher’s geometric-mean construction balances the two directional biases and

therefore makes price and quantity indices consistent with the value change. Laspeyres

anchors on old quantities (biasing upward if expensive items keep heavy weight), Paasche

anchors on new quantities (biasing downward), so neither multiplies cleanly with its own

quantity counterpart to reproduce exactly the change in total spending.

Final interpretation — which index to prefer and why (story wrap-up)

Think back to our two village shops. Laspeyres says: “Let’s carry last year’s shopping bag and

see how much more it costs.” Paasche says: “Let’s look at the new shopping bag and

compare prices.” Each story is plausible, but they lead to slightly different answers because

consumers change what they buy when prices change. Fisher’s index acts like a diplomatic

compromise — it says “let’s take the middle ground between the two stories” by using a

geometric mean. Because of that compromise, Fisher obeys two elegant consistency checks

(time reversal and factor reversal) that the other two generally fail.

From a student / examiner perspective:

• Show your arithmetic clearly (we computed each P×QP \times QP×Q and summed).

• Present the three index values with interpretation: Laspeyres ≈ 147.917, Paasche =

147.000, Fisher ≈ 147.458 — each implies a price rise of roughly 47% from Year 0 to

Year 1, with small differences reflecting the basket choice.

• Then show the two tests: Fisher passes both (time reversal and factor reversal) while

Laspeyres and Paasche fail. Explain why this is meaningful — Fisher’s symmetry gives

it attractive logical properties.

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SECTION-D

7. Assuming a four-yearly cycle, calculate the trend by the method of moving averages

from the following data relating to the production of tea in India.

Year

Production (in million lbs)

2009

464

2010

515

2011

518

2012

467

2013

502

2014

540

2015

557

2016

571

2017

586

2018

612

Ans: A Fresh Beginning: The Story of Tea and Its Numbers

India is a land where tea is not just a drink—it is an emotion. From the early morning “chai”

on railway platforms to evening cups shared with friends, tea has become a symbol of

everyday life. But behind this simple cup lies a huge industry that produces millions of

pounds of tea every year.

Now, like every industry, the production of tea doesn’t move in a straight line. Some years it

goes up, some years it goes down. Maybe the weather was favorable in one year, or maybe

rainfall was less in another. To make sense of such fluctuations, economists and statisticians

use a special technique called the Moving Average Method.

It’s like when you want to know how a cricket player is performing—not by just one match

(which may be good or bad)—but by taking his average score across several matches. That

way, you can see his real form, his trend. Similarly, we calculate moving averages to see the

underlying trend in tea production.

In this problem, we are asked to calculate the trend using a four-yearly moving average.

Let’s carefully go step by step, making sure every part is crystal clear.

Step 1: Data at Hand

We are given the following data about tea production in India:

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Year

Production (in million lbs)

2009

464

2010

515

2011

518

2012

467

2013

502

2014

540

2015

557

2016

571

2017

586

2018

612

At first glance, you can see that production generally increases, but not in a perfectly

smooth way. That’s where the moving average helps us.

Step 2: What is a Four-Yearly Moving Average?

Suppose you want to understand the general direction of tea production over time. Instead

of looking at one year, you take the average of four consecutive years. This gives you one

figure that represents the overall situation of those years.

Then, you “move forward” one year and again take the average of the next four years. By

repeating this, we get a smooth trend line.

So, the first average will be of 2009, 2010, 2011, and 2012. The second will be of 2010, 2011,

2012, and 2013, and so on.

Step 3: Calculating the Four-Yearly Moving Averages

Let’s calculate carefully:

1. 2009–2012 = (464 + 515 + 518 + 467) ÷ 4 = 491.0

2. 2010–2013 = (515 + 518 + 467 + 502) ÷ 4 = 500.5

3. 2011–2014 = (518 + 467 + 502 + 540) ÷ 4 = 506.75

4. 2012–2015 = (467 + 502 + 540 + 557) ÷ 4 = 516.5

5. 2013–2016 = (502 + 540 + 557 + 571) ÷ 4 = 542.5

6. 2014–2017 = (540 + 557 + 571 + 586) ÷ 4 = 563.5

7. 2015–2018 = (557 + 571 + 586 + 612) ÷ 4 = 581.5

Now, we can put this in a table:

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Years Covered

Four-Year Moving Average

2009–2012

491.0

2010–2013

500.5

2011–2014

506.75

2012–2015

516.5

2013–2016

542.5

2014–2017

563.5

2015–2018

581.5

Step 4: Positioning the Trend Values

One small technical detail: Since we are using a 4-year average, the trend value does not

belong exactly to one year, but to the middle of the 4 years.

For example:

• The average of 2009–2012 (491.0) is placed between 2010 and 2011.

• The average of 2010–2013 (500.5) is placed between 2011 and 2012, and so on.

This way, the trend values line up in the middle of the data period.

Step 5: Understanding the Trend

Now, if we look at these averages, what do we see?

• The trend starts at 491.0 (around 2010–11).

• It keeps rising slowly: 500.5 → 506.75 → 516.5.

• Then, it grows more quickly: 542.5 → 563.5 → 581.5.

This tells us that tea production in India was steadily increasing during this period. Even

though some years like 2012 had a dip (467), the overall long-term direction is upward.

In simple words, the moving average method has helped us smooth out the “noise” (ups

and downs) and clearly see the rising “melody” of tea production.

Step 6: Why This Matters

You might wonder: Why go through all this trouble? Why not just look at the raw data?

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Here’s the thing: raw data can be misleading. Imagine if you look at 2011 (518) and then at

2012 (467), you may think production is falling badly. But when you take the average of four

years, you realize it’s just a temporary fall—the long-term trend is still upward.

That is why policymakers, economists, and business planners use such statistical tools. For

instance, the tea industry could use this analysis to decide how much land to allocate, how

many workers to employ, or how to plan exports.

A Story-Like Ending

Think of the moving average like the gentle hand of a gardener. The gardener doesn’t get

worried if one branch of the tea plant looks weak one year. Instead, he looks at how the

whole plant has been growing over several years. Similarly, the moving average gives us a

calm and steady picture, instead of being distracted by small ups and downs.

So, the conclusion of our little story is this:

• The trend of tea production in India (2009–2018), calculated by the four-yearly

moving average method, shows a steady upward growth.

• Despite small fluctuations, the production increased from around 491 million lbs

(2010–11) to about 581 million lbs (2016–17).

And that’s how the story of tea, when told with numbers and averages, becomes not just a

mathematical solution but a meaningful insight into India’s growing tea industry.

8. (a) Define Probability. Explain various theorems of probability.

(b) The CEO of a company plans to take two of his executives to a conference in Chicago.

He plans to select at random one of the eight executives in finance and one of the ten

executives in marketing. Of the executives in finance, two have already travelled with the

CEO while three of the executives in marketing have travelled with CEO before. What is

the probability that:

(i) Both the executives have travelled with him before?

(ii) At least one of the executives selected has travelled with him before?

(iii) None of the executives chosen has travelled with the CEO before?

(iv) The finance executive has, and the marketing executive has not, travelled with

the CEO. before?

Ans: Picture probability as a storyteller’s toolkit — it helps us turn uncertainty into tidy

chances we can work with. Let’s walk through the idea slowly, tell the key rules as if they

were tools in a traveller’s satchel, and then use those tools to solve the CEO-and-executives

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problem in a clear, step-by-step way. I’ll keep the language simple and add little examples so

the logic feels intuitive — like a short, enjoyable story you can explain in an exam.

What is Probability?

Probability measures how likely an event is to happen. If we imagine all possible outcomes

of an experiment as pages in a book, probability tells us how many of those pages contain a

particular event compared to the whole book. Formally, when outcomes are equally likely,

the probability of an event A is

Probabilities range from 0 (impossible) to 1 (certain). We often express them as fractions,

decimals, or percentages.

The Fundamental Theorems / Rules (the traveller’s toolkit)

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Now the CEO story (apply the tools)

The CEO will randomly pick:

• one finance executive from 8 finance executives, of whom 2 have travelled with him

before;

• one marketing executive from 10 marketing executives, of whom 3 have travelled

with him before.

Because the finance choice and the marketing choice are made from two separate pools

(finance and marketing), the events are independent — selecting someone from finance

doesn’t change the composition of marketing. That lets us multiply appropriate

probabilities.

Because choices are independent, the probability of any combined event is the product of

the finance probability and the marketing probability.

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(i) Both executives have travelled with him before.

Wrap-up — quick sanity checks and interpretation

• All probabilities are between 0 and 1.

• The possibilities partition into: both travelled (3/40), none travelled (21/40), and

exactly one travelled (the remainder 16/40 = 0.4). These add up to 1.

• Using the complement rule made part (ii) much easier — a common exam trick.

Think of the CEO as flipping two separate coins weighted differently: one coin has chance

1/4 of “yes,” the other has 3/10 of “yes.” We multiply to find joint chances and subtract

from 1 when we want “at least one.” That small picture captures almost everything you

need for this problem: pick the right basic probabilities, decide if events are independent,

and use multiplication and complement rules smartly.

“This paper has been carefully prepared for educational purposes. If you notice any mistakes or

have suggestions, feel free to share your feedback.”