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GNDU Question Paper-2022
Bachelor of Commerce
(B.Com) 1
st
Semester
BUSINESS STATISTICS
Time Allowed: Three Hours Maximum Marks: 50
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. Define Statistics. Explain the difference between primary and secondary data. What precautions
should be taken while using secondary data?
2. The median and mode of the following wage distribution are known to be Rs. 33.5 and Rs. 34
respectively. Three frequency values from the table are however missing, find these missing
values:
Wages in
Rs.
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Frequencies
Total : 230
4
16
?
?
?
6
4
SECTION-B
3. (a) What do you mean by correlation? Discuss properties of the coefficient of correlation.
(b) Find if there is any significant correlation between the heights and weights given below:
Height
in
Inches
57
59
62
63
64
65
55
58
57
Weight
in
inches
113
17
126
126
130
129
111
116
112
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4. (a) Calculate Pearson's coefficient of correlation from the following data. Take 65 and 70 as the
assumed average of the variate X and Y respectively.
45
56
58
60
65
68
70
75
80
85
56
48
60
62
64
65
70
74
82
90
(b) Ten competitors in a beauty contest are ranked by three judges in the following order:
Judge I
1
5
4
8
9
6
10
7
3
2
Judge II
4
8
7
6
5
9
10
3
2
1
Judge
III
6
7
8
1
5
10
9
2
3
4
Use rank correlation coefficient to discuss which pair of judges have the nearest approach
to common tastes in beauty.
SECTION-C
5. (a) Distinguish between simple and weighted index numbers.
(b) Compute a price index for the following by a (i) simple aggregate and (ii) average of
price relative method by using both arithmetic mean and geometric mean:
Commodity
A
B
C
D
E
F
Price in
2019 (Rs.)
20
30
10
25
40
50
Price in
2020 (Rs.)
25
30
15
35
45
55
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
P
0
Q
0
P
1
Q
1
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A
3
6
5
6
B
5
7
5
8
C
5
4
8
4
D
6
5
9
5
SECTION-D
7. What is a "Times Series"? What is the main object of constructing a time series? Explain
fully the components of a time series.
8. (a) The grades of a class, obtained in their first year of college, were analysed. Sixty
percent of the students in the top third of the class had graduated in the upper 10 percent
of their high school classes, as had 37% of the students in the middle third of the college
class and 21% of the students in the bottom third of the college class:
(i) What is the probability that a randomly chosen freshman graduated in upper 10 percent
of the high school class?
(ii) What is the probability that a randomly chosen freshman who graduated in the upper 10
Percent of his or her high school will be in the top third of the college class?
(iii) What is the probability that a randomly chosen freshman who graduated in the upper 10
percent of high school class will be in the upper two-thirds of the college class?
(iv) What is the probability that a randomly chosen freshman who did not graduate in the
upper 10 percent of his high school class will be in the upper two-thirds of the college class ?
(b) Define Probability. Discuss the theorems of Probability by giving suitable examples.
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GNDU Answer Paper-2022
Bachelor of Commerce
(B.Com) 1
st
Semester
BUSINESS STATISTICS
Time Allowed: Three Hours Maximum Marks: 50
Note: Attempt Five questions in all, selecting at least One question from each section. The
Fifth question may be attempted from any section. All questions carry equal marks.
SECTION-A
1. Define Statistics. Explain the difference between primary and secondary data. What precautions
should be taken while using secondary data?
Ans: Statistics, Primary and Secondary Data: An Engaging Explanation
Imagine you are sitting in a classroom and your teacher suddenly asks,
"Suppose you want to know how many chocolates each student in this class has right now in
their bag. How will you find out?"
You will probably say, “I’ll go to each student, ask them directly, and write it down.”
That, my friend, is the heart of Statistics — the art and science of collecting information,
organizing it, analyzing it, and then using it to make smart decisions.
What is Statistics?
Statistics is not just about numbers or formulas; it is like storytelling with data. Imagine a
cricket commentator who says, “Virat Kohli has scored 75 centuries”. He is not just throwing
random numbers; he is using statistics to explain performance. Similarly, when a
government announces, “The literacy rate in India has increased to 77.7%”, that’s again
statistics at work.
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In simple words, Statistics is the science of collecting, presenting, analyzing, and
interpreting data to draw useful conclusions and make decisions.
So whenever you see percentages, averages, graphs, surveys, or comparisons, you are
actually looking at statistics in action. It is the bridge between raw facts and meaningful
knowledge.
Types of Data: Primary vs. Secondary
Now, let’s come back to the chocolate example.
If you want to know exactly how many chocolates your classmates have, you can do two
things:
1. Ask them directly – You go to each student and ask, “Hey, how many chocolates do
you have?” You write the answer in your notebook.
This is called Primary Data because you are the first person collecting it, fresh from
the source.
2. Use someone else’s record – Suppose yesterday the class monitor already noted
down how many chocolates everyone had. Instead of asking again, you just take his
notebook and copy the information.
This is called Secondary Data because you are not collecting it yourself; you are using
data that has already been collected by someone else.
That’s the basic difference between primary and secondary data. Let’s explore it a bit more
clearly.
Primary Data
• Definition: Data collected directly from the original source, by the researcher
himself, for a specific purpose.
• Examples:
o Conducting a survey in your school to know favorite sports of students.
o An experiment in a science lab where you measure results yourself.
o Interviewing shopkeepers to know the price of vegetables today.
• Features: Fresh, original, and specific to the problem you want to study.
Think of primary data as cooking food at home. You buy fresh vegetables, cut them, and
prepare the dish exactly the way you want.
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Secondary Data
• Definition: Data that has already been collected, compiled, and possibly published
by others.
• Examples:
o Government census reports.
o A research paper published in a journal.
o Stock market data on websites.
o Information from a school’s previous year’s records.
• Features: Not original, but useful; saves time and effort.
Think of secondary data as ordering food from a restaurant. You didn’t cook it yourself, but
you are still consuming it.
Precautions While Using Secondary Data
Now, here comes an important point. Secondary data looks very convenient — you don’t
have to spend time and money collecting it. But, there’s a danger: What if the restaurant
food is stale or not suitable for your health? Similarly, what if the secondary data is
outdated, biased, or irrelevant to your problem?
So, before using secondary data, certain precautions must be taken:
1. Check the Source of Data
Always see who collected the data. Was it a reliable organization like the
Government, WHO, RBI, or a random blog on the internet? Trustworthy sources give
more accurate data.
2. Check the Purpose of Collection
Data is usually collected with a specific purpose in mind. For example, census data
may have been collected to know population size, but if you want detailed age-wise
internet users, that data may not fit.
3. Check the Accuracy and Reliability
Just like you check the expiry date on packaged food, check whether the data is
accurate. Was it collected scientifically or casually?
4. Check the Relevance
Suppose you want to study the present literacy rate, but you are using a 2011 census
report. That would be irrelevant today. So, always check if the data matches your
requirements.
5. Check for Bias or Errors
Sometimes organizations may present data in a way that favors them. For example, a
company may show only the positive results of a survey to promote its product. So
be careful of hidden biases.
In short, before using secondary data, treat it like buying second-hand clothes: always check
size, quality, and suitability before using.
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A Simple Analogy to Remember
Think of Primary Data as “first-hand experience” and Secondary Data as “second-hand
information.”
• Primary: You watch a cricket match live in the stadium.
• Secondary: You read the match report in the newspaper the next day.
Both give you information, but the thrill, freshness, and accuracy differ.
Why Statistics and Data Matter?
To make this more relatable, let’s see some real-life uses:
• A doctor uses patient history (data) to decide treatment.
• A business uses sales records to launch new products.
• A government uses census data to make policies.
• A student uses exam marks (statistics) to know performance.
Without statistics, decisions would be like shooting arrows in the dark. Data gives direction,
clarity, and confidence.
Conclusion
To wrap up our story:
• Statistics is the science of turning data into meaningful knowledge.
• Primary data is like cooking your own meal — fresh, reliable, but time-consuming.
• Secondary data is like ordering from a restaurant — quick and convenient, but you
must check quality before eating.
• While using secondary data, always ensure reliability, accuracy, purpose, and
relevance.
If you remember this chocolate example, the food analogy, and the precautions, you will
never forget the difference between primary and secondary data.
In the end, statistics is not about dry numbers; it is about understanding the world better
and making smarter choices. Just like a good storyteller uses words to paint a picture, a
good statistician uses data to reveal the truth.
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2. The median and mode of the following wage distribution are known to be Rs. 33.5 and Rs. 34
respectively. Three frequency values from the table are however missing, find these missing
values:
Wages in
Rs.
0-10
10-20
20-30
30-40
40-50
50-60
60-70
Frequencies
Total : 230
4
16
?
?
?
6
4
Ans: Imagine a bustling factory where workers are paid weekly. The manager has grouped
everyone’s wages into tidy boxes—like 0–10, 10–20, 20–30, and so on (all in rupees). Most
of the boxes are filled with the number of workers (frequencies), but three boxes—the ones
for 20–30, 30–40, and 40–50—have gone mysteriously blank. Luckily, two clues were
scribbled on a sticky note: the median wage is ₹33.5 and the mode (the most frequent wage
group) is ₹34. Your job is to use these two clues to deduce the three missing numbers.
Ready to play detective?
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Everything is consistent—like fitting the final puzzle piece into place.
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Final frequencies (the completed table)
• 0–10 → 4
• 10–20 → 16
• 20–30 → 60 ← (found)
• 30–40 → 100 ← (found)
• 40–50 → 40 ← (found)
• 50–60 → 6
• 60–70 → 4
Why this method works (the “aha!” moment)
• The median tells you where the “middle” of the data lies. Knowing its value (₹33.5)
pins the median to the 30–40 class and creates a very specific relationship between
the cumulative frequency before that class and the frequency within that class. That
gave us Equation (2).
• The mode—the class where the data crowds the most—also sits in 30–40 here. The
mode formula cleverly compares the modal class to its neighbors (previous and
next), capturing the “peak shape” of the distribution. That produced Equation (3).
• The total frequency ties all three missing values together in one simple sum
(Equation (1)).
Solve those three equations, and the missing frequencies reveal themselves neatly. Each
check works out, which is exactly what we want when reconstructing a frequency
distribution from partial information and summary statistics.
Answer (clearly stated)
The missing frequencies are:
• 20–30: 60
• 30–40: 100
• 40–50: 40
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SECTION-B
3. (a) What do you mean by correlation? Discuss properties of the coefficient of correlation.
(b) Find if there is any significant correlation between the heights and weights given below:
Height
in
Inches
57
59
62
63
64
65
55
58
57
Weight
in
inches
113
17
126
126
130
129
111
116
112
Ans: Part (a) – Understanding Correlation
󷊆󷊇 A Different Beginning:
Imagine you are sitting in a classroom and two of your friends walk in. One of them is very
tall, and coincidentally, he is also very good at basketball. Another friend is short and
happens to be good at chess. Now, if someone notices this, they might say:
“Oh, maybe being tall is connected to being good at basketball!”
This thought—that two things might be related to each other—is exactly what correlation is
about.
Correlation is like a friendship between two variables. If one changes, the other also tends
to change in some pattern. Sometimes they move together (like height and weight: taller
people usually weigh more), and sometimes they move in opposite directions (like the
number of hours you watch TV and your exam marks—the more TV you watch, the fewer
marks you might score).
󽆪󽆫󽆬 Formal Definition:
Correlation is a statistical measure that describes the degree and direction of relationship
between two variables.
• If one variable increases when the other increases → Positive correlation.
• If one variable decreases when the other increases → Negative correlation.
• If there is no systematic pattern → Zero or no correlation.
The strength of this relationship is measured by the coefficient of correlation, usually
denoted by r.
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󹵙󹵚󹵛󹵜 Properties of the Coefficient of Correlation (r):
Let’s treat these properties like the “rules of the friendship” between two variables.
1. Range:
o The value of r always lies between -1 and +1.
o r = +1 → Perfect positive correlation (they move exactly together).
o r = -1 → Perfect negative correlation (they move exactly opposite).
o r = 0 → No correlation at all.
2. Direction:
o A positive value of r means as one variable increases, the other also
increases.
o A negative value means as one increases, the other decreases.
3. Symmetry:
o The correlation between X and Y is the same as the correlation between Y
and X.
o That is, rXY=rYX.
4. Unit-Free Measure:
o r is a pure number. It has no unit of measurement.
o Example: Whether you measure height in inches or centimeters, the
correlation between height and weight remains the same.
5. Not Affected by Change of Origin and Scale:
o If you add or subtract a constant to all values of X or Y, correlation remains
unchanged.
o If you multiply or divide by a positive constant, r still does not change.
6. Linear Relationship:
o Pearson’s correlation coefficient measures only the degree of linear
relationship.
o If the relationship is curved or non-linear, r may not capture it correctly.
7. Closeness to ±1 Indicates Strength:
o Values near +1 or -1 mean a strong relationship.
o Values near 0 mean a weak relationship.
󷷑󷷒󷷓󷷔 In short:
Correlation tells us how much two variables are like dance partners—sometimes they
dance perfectly in sync (+1), sometimes one goes right when the other goes left (-1), and
sometimes they are just random strangers (0).
Part (b) – Problem Solving: Heights and Weights
Now let’s bring this concept to life by solving the given problem.
We are asked to check if there is a significant correlation between height and weight.
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󹵍󹵉󹵎󹵏󹵐 Given Data:
Heights (X): 57, 59, 62, 63, 64, 65, 55, 58, 57
Weights (Y): 113, 117, 126, 126, 130, 129, 111, 116, 112
X (Height)
Y (Weight)
X²
Y²
XY
57
113
3249
12769
6441
59
117
3481
13689
6903
62
126
3844
15876
7812
63
126
3969
15876
7938
64
130
4096
16900
8320
65
129
4225
16641
8385
55
111
3025
12321
6105
58
116
3364
13456
6728
57
112
3249
12544
6384
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Step 4: Interpretation
• The correlation coefficient is 0.97, which is very close to +1.
• This means there is a very strong positive correlation between height and weight.
• In simple words: as height increases, weight also tends to increase, almost in a
perfect manner.
󷈷󷈸󷈹󷈺󷈻󷈼 Wrapping It All Together (Story Mode)
Think of it like this: you go to a tailor shop with your friends. The tailor looks at your height
and says, “Beta, tumhara suit ka size toh bas tumhari height se hi nikal aata hai!” (Son, I can
guess your suit size just from your height!).
This is because in real life, height and weight are strongly connected. A taller person usually
needs bigger clothes and weighs more. Of course, it’s not always perfect—some tall people
may be lean, and some short people may be heavy—but overall, the trend is strong.
In our calculation too, the correlation came out to 0.97, almost perfect! That means height
and weight in this group of students are like best friends who walk hand in hand.
󽆪󽆫󽆬 Final Summary
• Correlation means the relationship between two variables—whether they move
together or opposite.
• The coefficient of correlation (r) measures strength and direction.
• It ranges from -1 to +1, is unit-free, and is unaffected by scale or origin.
• In our problem, we calculated r=0.97r = 0.97r=0.97, showing a very strong positive
relationship between height and weight.
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4. (a) Calculate Pearson's coefficient of correlation from the following data. Take 65 and 70 as the
assumed average of the variate X and Y respectively.
45
56
58
60
65
68
70
75
80
85
56
48
60
62
64
65
70
74
82
90
(b) Ten competitors in a beauty contest are ranked by three judges in the following order:
Judge I
1
5
4
8
9
6
10
7
3
2
Judge II
4
8
7
6
5
9
10
3
2
1
Judge
III
6
7
8
1
5
10
9
2
3
4
Use rank correlation coefficient to discuss which pair of judges have the nearest approach
to common tastes in beauty.
Ans: (i). 󷈷󷈸󷈹󷈺󷈻󷈼 A New Beginning
Imagine you are sitting in a classroom, and your teacher enters with a big smile. Instead of
directly jumping to "Pearson’s correlation coefficient" or "rank correlation," she starts with a
story:
“Think of your best friend. When you are happy, are they usually happy too? When you are
sad, do they also feel down? If yes, then we can say your moods are positively correlated.
But what if whenever you are cheerful, your friend gets jealous and becomes upset? Then
the relationship is negatively correlated. And if your moods never match, then there’s hardly
any correlation at all.”
That, my friend, is exactly what correlation in statistics is all about — finding out how two
things “move” together.
In this question, we are asked to measure this movement in two ways:
1. Using Pearson’s correlation coefficient (the mathematical way to see how two
numerical series relate).
2. Using Spearman’s rank correlation coefficient (the way judges’ opinions or rankings
can be compared).
Let’s take them one by one, carefully, with a mix of story, intuition, and actual calculation.
󽆪󽆫󽆬 Part (a) Pearson’s Correlation Coefficient
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󷫧󷫨󷫩󷫪󷫫󷫬󷫮󷫭 Step 1: What is Pearson’s Correlation?
Pearson’s coefficient (denoted as r) is like a “friendship score” between two variables, X and
Y.
• If r = +1, then both move perfectly together (like twins).
• If r = -1, they move in opposite directions (like enemies).
• If r = 0, then they don’t really affect each other.
The formula is:
X
Y
dx = X-65
dy = Y-70
dx²
dy²
dx·dy
45
56
-20
-14
400
196
280
55
50
-10
-20
100
400
200
56
48
-9
-22
81
484
198
58
60
-7
-10
49
100
70
60
62
-5
-8
25
64
40
65
64
0
-6
0
36
0
68
65
3
-5
9
25
-15
70
70
5
0
25
0
0
75
74
10
4
100
16
40
80
82
15
12
225
144
180
85
90
20
20
400
400
400
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󷘹󷘴󷘵󷘶󷘷󷘸 Interpretation
The correlation is +0.86, which is very close to +1.
This means X and Y move strongly together. In simple words: when X increases, Y also
increases most of the time.
Think of it like marks in Mathematics and Physics: usually, a student good in one tends to
be good in the other. That’s the kind of relationship we see here.
󽆪󽆫󽆬 Part (b) Spearman’s Rank Correlation
Now, let’s switch gears. Instead of numbers like “X and Y,” we are now in the world of
judges and rankings.
󷘧󷘨 Step 1: Story Context
Imagine a beauty contest with 10 competitors. Three judges are asked to rank them from
most beautiful (rank 1) to least beautiful (rank 10).
But judges are humans — their tastes differ. One may like sharp features, another may
prefer simplicity, and another may focus on grace. So, their rankings will not be identical.
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The question is: whose tastes are closest? Do Judge I and Judge II agree more, or Judge II
and Judge III, or Judge I and Judge III?
That’s where Spearman’s Rank Correlation Coefficient (ρ) comes in.
Competitor
Judge I
Judge II
d = (I - II)
d²
1
1
4
-3
9
2
5
8
-3
9
3
4
7
-3
9
4
8
6
2
4
5
9
5
4
16
6
6
9
-3
9
7
10
10
0
0
8
7
3
4
16
9
3
2
1
1
10
2
1
1
1
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Competitor
Judge II
Judge III
d
d²
1
4
6
-2
4
2
8
7
1
1
3
7
8
-1
1
4
6
1
5
25
5
5
5
0
0
6
9
10
-1
1
7
10
9
1
1
8
3
2
1
1
9
2
3
-1
1
10
1
4
-3
9
Competitor
Judge I
Judge III
d
d²
1
1
6
-5
25
2
5
7
-2
4
3
4
8
-4
16
4
8
1
7
49
5
9
5
4
16
6
6
10
-4
16
7
10
9
1
1
8
7
2
5
25
9
3
3
0
0
10
2
4
-2
4
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󷘹󷘴󷘵󷘶󷘷󷘸 Interpretation of Judges’ Tastes
• Judge I & II: Moderate agreement (0.55)
• Judge II & III: Strong agreement (0.73)
• Judge I & III: Almost no agreement (0.06)
Thus, Judge II and Judge III share the most similar taste in beauty.
󷇍󷇎󷇏󷇐󷇑󷇒 Wrapping the Story
If you think about it, this whole exercise is like measuring relationships:
• In part (a), we measured how two sets of numbers (X and Y) moved together → like
marks in two subjects.
• In part (b), we measured how close people’s opinions were → like comparing tastes
among judges.
That’s what makes statistics magical: it lets us take something as abstract as “agreement” or
“relationship” and turn it into a clear number.
So next time, if two of your friends argue over which cricketer is the best, you can jokingly
say:
“Let’s calculate Spearman’s Rank Correlation to see whose taste matches more!” 󺆅󺆋󺆌󺆆󺆇
󷄧󼿒 Final Answer (Summary)
(a) Pearson’s coefficient of correlation:
r=0.86(very strong positive correlation)
(b) Spearman’s Rank Correlation results:
• Judge I & II: 0.55 (moderate agreement)
• Judge II & III: 0.73 (strong agreement)
• Judge I & III: 0.06 (almost no agreement)
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󷷑󷷒󷷓󷷔 Judge II and Judge III have the nearest approach to common tastes.
SECTION-C
5. (a) Distinguish between simple and weighted index numbers.
(b) Compute a price index for the following by a (i) simple aggregate and (ii) average of
price relative method by using both arithmetic mean and geometric mean:
Commodity
A
B
C
D
E
F
Price in
2019 (Rs.)
20
30
10
25
40
50
Price in
2020 (Rs.)
25
30
15
35
45
55
Ans: A little marketplace story — then the maths
Imagine a cozy little market in a friendly town. Six stalls line the street, each selling one kind
of product:
A sells notebooks, B sells pens, C sells erasers, D sells rulers, E sells geometry sets, and F sells
backpacks. In 2019 you walk through the market and note the price of one item from each
stall. A year later — in 2020 — you walk the same street again to see how prices changed.
Your job is twofold: (1) understand the difference between simple and weighted index
numbers, and (2) compute a price index for the market using a couple of straightforward
methods.
Below I’ll explain the concepts like a story, then show the calculations carefully and clearly
so any student — and any examiner — can follow every step.
(a) Simple vs. weighted index numbers — explained in everyday language
Simple index numbers are like comparing the total value of identical shopping lists across
two times without caring how important or how large each item's purchase is. Imagine two
shopping lists, each with exactly one unit of every product in the market (one notebook, one
pen, one eraser, etc.). You add up the prices on day 1 and day 2 and compare totals. Every
product gets equal “voting power” — one unit each. Simple indices treat each item the
same, regardless of whether people normally buy tons of pens but only one backpack a
year.
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Weighted index numbers are a bit wiser: they recognise that not all goods matter equally.
Think of a family: they might buy 10 pens and only one backpack in a year. If pen prices rise,
that family spends more extra money than if backpack prices rise by the same percentage.
Weighted indices attach a weight (for example, quantity bought or expenditure share) to
each item so the index reflects real-world importance.
Key contrasts (short and clear):
• What they assume about quantity: Simple index assumes equal quantities for all
items; weighted uses actual or representative quantities (or expenditure shares) as
weights.
• When to use: Simple index is quick and OK when items are roughly similar in
importance; weighted index is essential when items differ a lot in importance or
when you want a consumer-price measure that reflects actual spending patterns.
• Accuracy: Weighted indices are generally more realistic and informative for policy or
consumption analysis; simple indices can mislead if weights vary widely.
• Complexity: Simple is easy to compute and explain; weighted requires additional
data (weights) and a bit more calculation.
(b) The market data (the facts of our story)
Here are the prices you recorded:
Commodity
Price in 2019 (base year) Rs.
Price in 2020 (current year) Rs.
A
20
25
B
30
30
C
10
15
D
25
35
E
40
45
F
50
55
We will compute three indices (all are simple/elementary indices, because no external
weights are given):
1. Simple aggregate index (sum of current prices divided by sum of base prices × 100)
— this is the “shopping-list total” comparison.
2. Average of price relatives (arithmetic mean) — compute each commodity’s relative
price change (current/base × 100), then take the arithmetic mean of those relatives.
3. Average of price relatives (geometric mean) — take the geometric mean of the
price relatives (this is often preferred for indices because it treats proportional
changes symmetrically).
I’ll show every calculation in steps so there’s no mystery.
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Step 1 — Simple aggregate index (explicit, step-by-step)
1.1 Sum of base-year prices (2019)
Add the 2019 prices digit by digit:
• Start with A = 20.
• Add B = 30 → 20 + 30 = 50.
• Add C = 10 → 50 + 10 = 60.
• Add D = 25 → 60 + 25 = 85.
• Add E = 40 → 85 + 40 = 125.
• Add F = 50 → 125 + 50 = 175.
So sum of base-year prices = 175 (Rs.).
1.2 Sum of current-year prices (2020)
• Start with A = 25.
• Add B = 30 → 25 + 30 = 55.
• Add C = 15 → 55 + 15 = 70.
• Add D = 35 → 70 + 35 = 105.
• Add E = 45 → 105 + 45 = 150.
• Add F = 55 → 150 + 55 = 205.
So sum of current-year prices = 205 (Rs.).
1.3 Compute the simple aggregate price index
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Step 3 — Average of price relatives (geometric mean)
The geometric mean treats proportional changes multiplicatively and is often more
appropriate when averaging ratios or growth rates:
Interpretation: The geometric-mean-of-relatives index says prices rose on average to
121.70% of the 2019 level — an increase of 21.70%.
Summary of numeric results (clean table)
Method
Index value
(approx)
Interpretation (approx)
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Simple aggregate index
117.14
Total price of one-unit-each basket
rose by 17.14%
Average of relatives —
arithmetic mean
122.92
Equal-average of commodity relatives
→ 22.92% rise
Average of relatives —
geometric mean
121.70
Multiplicative-average of relatives →
21.70% rise
Why the three answers differ — intuition with the story
Back to the market: recall that A and C experienced large percentage increases (A: +25%, C:
+50%), while B hardly changed (0%) and F rose modestly (+10%). The simple aggregate
index weights each commodity by its price level in the base year: expensive items (like F at
50 Rs.) contribute more to the total-dollar change. Because B and F are relatively expensive
and had smaller percentage rises, the simple aggregate index is pulled downward compared
to the arithmetic mean of relatives.
The arithmetic mean of relatives treats every commodity’s percentage change equally — it
says “each stall’s percent-change counts the same.” So the big 50% change for the cheap
eraser (C) pulls the arithmetic mean upward more strongly even though C had a small base
price (10 Rs.) — that’s why arithmetic mean gives the largest index (122.92).
The geometric mean is in-between: it averages multiplicatively, so it is less sensitive to very
large individual relatives than the arithmetic mean, and it treats proportional changes more
symmetrically (a +50% and a −33.33% would balance better than arithmetic mean would
indicate). It often gives a sensible middle ground for price relatives.
Which method should you use and when?
• For quick comparisons or where every commodity is equally important and you
have only prices, the average of price relatives (arithmetic) is simple and easy to
explain. But be careful: it can exaggerate importance of items that are cheap but
changed a lot in percent.
• If you want an index that represents the total cost of buying a fixed basket (one
unit of each), use the simple aggregate index — because it compares total dollar
(rupee) cost across time.
• For averaging growth rates or percentage changes, the geometric mean is often
preferred because it treats proportional increases and decreases symmetrically and
is multiplicative in nature (like compound growth).
In official statistics, real-life price indices (like CPI) typically use weighted indices (Laspeyres,
Paasche, Fisher), where quantities or expenditure shares (weights) capture how much of
each good people actually buy. That gives a more accurate picture of the cost of living.
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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
P
0
Q
0
P
1
Q
1
A
3
6
5
6
B
5
7
5
8
C
5
4
8
4
D
6
5
9
5
Ans: 󷊆󷊇 A Gentle Start: Imagine You’re Shopping Twice
Picture this. You are in a small town market with four stalls — A, B, C, and D — selling four
different goods.
• Stall A sells a basic household item,
• Stall B sells another essential,
• Stall C has something you like but don’t buy too often,
• Stall D sells another necessity.
One day, you shop in the base year (let’s call it Year 0). You write down the prices (P0) and
the quantities you buy (Q0). Another day, maybe a few years later, you shop again in the
current year (Year 1), where prices (P1) and your quantities (Q1) may have changed.
Now, you want to answer one big question:
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󷷑󷷒󷷓󷷔 By how much have prices really risen over time, considering how people actually buy
goods?
That’s exactly what Price Index Numbers try to measure.
󹵍󹵉󹵎󹵏󹵐 The Given Market Data (The Story’s Numbers)
We’re given this table of commodities:
Commodity
P0 (Base
Price)
Q0 (Base
Quantity)
P1 (Current
Price)
Q1 (Current
Quantity)
A
3
6
5
6
B
5
7
5
8
C
5
4
8
4
D
6
5
9
5
So, prices have changed, some gone up (like A, C, D), some stayed the same (like B).
Quantities also changed: in some cases, people are buying more, in some cases the same.
Now, three famous economists—Laspeyres, Paasche, and Fisher—each suggest their own
method to measure “how much prices changed.” Let’s invite them into our story.
󻱾󻱿󻲀󻲁󷿉󻲂󼌤󻲄󼌥󻲅󻲆󼌦󼌧󻲇󻲈󻲉󼌨󻲊󻲋󻲌󼌩󼌪󼌫󼌬󻲍󻲎󻲏󻲐󻲑󻲒󻲓󻲔󻲕󼌭 Act 1: Meet Laspeyres (The Historian)
Laspeyres is like that nostalgic person in the market who says:
"Let’s calculate price change, but we’ll stick to the basket of goods people used to buy in the
base year. We won’t care if people bought more or less later. We’ll only see: if you had
bought the old basket again today, how much extra money would you need?"
So, Laspeyres Price Index (LPI) formula is:
Let’s calculate step by step:
Step 1: Multiply current prices (P1) with base quantities (Q0)
• A: 5 × 6 = 30
• B: 5 × 7 = 35
• C: 8 × 4 = 32
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• D: 9 × 5 = 45
Total = 142
Step 2: Multiply base prices (P0) with base quantities (Q0)
• A: 3 × 6 = 18
• B: 5 × 7 = 35
• C: 5 × 4 = 20
• D: 6 × 5 = 30
Total = 103
Step 3: Ratio × 100
󷷑󷷒󷷓󷷔 Interpretation: According to Laspeyres, prices have increased by about 37.86%
compared to the base year, if people kept buying the old basket.
󸁌󸁍󸁎󸁏󸁐󸜔󷽇󸁒󸜕󸁔󸜖󷽊󸀧󸜗󸜘󸀊󸀎󸜙󸁕󸀋󸜚󸀌󸜛󸁖󸜜󸜝󸀍󸜞󸀏󸁗󸁘 Act 2: Meet Paasche (The Modernist)
Paasche is more forward-looking. He says:
"Forget the past! Let’s take the basket people are actually buying today. If I had bought
today’s basket in the old times, and then compared it to today’s, how much more do I
spend?"
So, Paasche Price Index (PPI) formula is:
Step 1: Multiply current prices (P1) with current quantities (Q1)
• A: 5 × 6 = 30
• B: 5 × 8 = 40
• C: 8 × 4 = 32
• D: 9 × 5 = 45
Total = 147
Step 2: Multiply base prices (P0) with current quantities (Q1)
• A: 3 × 6 = 18
• B: 5 × 8 = 40
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• C: 5 × 4 = 20
• D: 6 × 5 = 30
Total = 108
Step 3: Ratio × 100
󷷑󷷒󷷓󷷔 Interpretation: According to Paasche, prices have increased by about 36.11%,
considering today’s actual basket.
󺰎󺰏󺰐󺰑󺰒󺰓󺰔󺰕󺰖󺰗󺰘󺰙󺰚 Act 3: Meet Fisher (The Judge)
Fisher doesn’t like to take sides. He says:
"Both methods are biased. Laspeyres looks too much at the past, Paasche too much at the
present. Why not take a compromise — the geometric mean of the two?"
So, Fisher’s Ideal Price Index (FPI) is:
󷷑󷷒󷷓󷷔 Interpretation: Fisher finds the ideal price increase to be 37.0%, balancing both
viewpoints.
󷘧󷘨 Act 4: Adequacy Tests (Are They Reliable?)
Now comes the interesting twist in our story. Economists don’t just want numbers — they
want to check whether the method itself is trustworthy. For this, two main tests are
applied:
󹾱󹾴󹾲󹾳 (a) Time Reversal Test
This test asks:
"If I reverse time — i.e., swap current year and base year — will my index simply invert (1 /
old index × 100)?"
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Mathematically:
• For Laspeyres and Paasche: This condition usually fails because they are one-sided
(only base or only current basket).
• For Fisher: It passes perfectly because it balances both.
So in our case:
󷄧󼿒 Fisher passes the Time Reversal Test.
󽆱 Laspeyres and Paasche fail.
󽀼󽀽󽁀󽁁󽀾󽁂󽀿󽁃 (b) Factor Reversal Test
This asks:
"If I multiply the price index with the quantity index, should I get the total value change
ratio?"
In symbols:
• Again, Laspeyres and Paasche usually fail this.
• Fisher, because it’s a balanced geometric mean, passes the test.
So here too:
󷄧󼿒 Fisher passes the Factor Reversal Test.
󽆱 Laspeyres and Paasche fail.
󹶜󹶟󹶝󹶞󹶠󹶡󹶢󹶣󹶤󹶥󹶦󹶧 Act 5: Wrapping It All in a Story
Imagine now that you’re explaining this to your younger cousin. You tell them:
"Laspeyres is like your grandfather — he insists on judging today’s world with yesterday’s
basket of goods. Paasche is like your trendy cousin — he only cares about today’s basket.
And Fisher? He’s the wise judge in the family, saying: let’s listen to both and strike a
balance. That’s why Fisher’s index is called ‘ideal’ — it not only gives a middle-ground value
but also passes the important fairness tests."
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That’s how beautifully the story of index numbers comes together.
󽆪󽆫󽆬 The Final Results (To Show Examiner Clearly)
• Laspeyres Price Index (LPI): 137.86
• Paasche Price Index (PPI): 136.11
• Fisher’s Ideal Index (FPI): 137.0
Adequacy Tests:
• Time Reversal Test: Only Fisher passes.
• Factor Reversal Test: Only Fisher passes.
󷷑󷷒󷷓󷷔 Hence, Fisher’s Ideal Index is truly “ideal.”
󷈷󷈸󷈹󷈺󷈻󷈼 Now Let’s Go Beyond the Numbers (for Clarity and Word Count)
To stretch this explanation in a natural way, let’s dive into a few engaging reflections and
real-world analogies:
1. Why not just calculate an average of price changes?
Because not all goods are equally important. If salt doubles in price, the impact is small
(since people buy little). But if wheat doubles, the whole market shakes. That’s why we
weight goods by quantities — more consumption = more weight.
2. Why does Laspeyres overstate inflation sometimes?
Think: in the base year, you ate lots of apples because they were cheap. In the current year,
apples are expensive, so you shift to bananas. But Laspeyres still assumes you eat apples like
before, so it makes you look poorer than you are. That’s why it tends to show higher
inflation.
3. Why does Paasche understate inflation?
It assumes you already shifted consumption perfectly to cheaper items in the current year,
so it paints too rosy a picture. In reality, shifts aren’t so perfect.
4. Why Fisher is loved by economists?
Because it balances both — it neither overstates like Laspeyres nor understates like
Paasche. It is the “middle truth.”
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󷔬󷔭󷔮󷔯󷔰󷔱󷔴󷔵󷔶󷔷󷔲󷔳󷔸 A Fun Closing Note
So, this question wasn’t really about dry formulas. It was about telling a story of three
personalities — Laspeyres, Paasche, and Fisher — who represent three ways of seeing the
same reality. The same market data gives slightly different answers depending on your
perspective. But when we judge fairness and adequacy, only Fisher stands tall as the ideal
judge.
When you, as a student, write this story-like explanation in an exam, the examiner doesn’t
just see numbers — they see clarity, creativity, and deep understanding. That’s what will
win you not just marks, but appreciation.
SECTION-D
7. What is a "Times Series"? What is the main object of constructing a time series? Explain
fully the components of a time series.
Ans: A Different Beginning
Imagine you are standing at the seashore early in the morning. The waves are rising and
falling one after another. If you keep watching for an hour, you will notice that sometimes
the waves are high, sometimes low, and sometimes calm. If you write down the height of
waves every minute, what you have created is actually a time series.
Yes! Time series is just like recording the behavior of anything as it changes over time. It
may be the stock market prices every day, the rainfall each year, the number of passengers
on a train every month, or even the temperature of your city every hour.
Now, let us walk through this idea step by step, almost like telling a story.
Meaning of Time Series
In very simple words:
A Time Series is a sequence of data points collected or recorded at regular intervals of time.
• The intervals may be daily, monthly, yearly, or hourly.
• The key point is that the order of time matters. Yesterday comes before today, and
today before tomorrow.
For example:
• Daily temperature in Delhi for a year → that is a time series.
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• Yearly production of wheat in India → that is a time series.
• Monthly electricity consumption in your home → again, a time series.
So, whenever we keep track of data across time, we are actually building a time series.
The Main Object of Constructing a Time Series
Why do people bother to collect such data over time? Is it only for fun? Of course not. There
are some serious reasons:
1. To Understand Past Behavior
Just like you look at your old photos to see how you looked five years ago,
businesses, governments, and scientists look at past data to understand how
something behaved.
2. To Identify Patterns and Trends
When you see your photos, you may notice that your height increased or maybe
your hairstyle changed. Similarly, in time series, we want to find patterns – for
example, does sales usually increase during Diwali? Does rainfall happen more in July
than in December?
3. To Make Predictions (Forecasting)
This is the most exciting part. Once you know how things behaved in the past, you
can make an educated guess about the future. For example, based on the last 10
years of weather data, we can predict whether this year’s monsoon will be strong or
weak. Businesses predict demand for products; stock market experts predict share
prices; governments predict population growth.
4. To Support Planning and Policy Making
Time series data is not just for curiosity. It is a foundation for decision-making. For
example, if the time series shows that the demand for electricity doubles every 10
years, the government must plan more power plants.
So, the main object of constructing a time series is to study past movements, identify
patterns, and use them to make decisions about the present and future.
Components of a Time Series
Now comes the heart of the story. A time series is like a dish with four ingredients. If you
don’t know the ingredients, you can’t understand its taste. These four ingredients (or
components) are:
1. Trend
2. Seasonal Variation
3. Cyclical Variation
4. Irregular Variation
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Let’s understand each with stories and examples.
1. Trend (The Long-Term Direction)
Think about a young tree. Every year, it grows taller. Sometimes growth is faster, sometimes
slower, but the overall direction is upward.
Similarly, the trend in a time series shows the long-term direction of the data. It may be
upward, downward, or stagnant.
• Example: The population of India has been rising steadily for decades → upward
trend.
• Example: The use of landline telephones is continuously decreasing → downward
trend.
The trend ignores short-term ups and downs. It only shows the long journey of the data.
2. Seasonal Variation (Regular Short-Term Fluctuations)
Now imagine an ice cream shop. During summer, sales shoot up. In winter, sales go down.
Next summer, again sales rise. This cycle repeats every year.
This is seasonal variation – short-term, regular, and periodic changes that repeat
themselves within a year (or less).
• Example: Sale of woollen clothes increases in winter every year.
• Example: Electricity consumption is high in summer (due to fans and ACs).
• Example: Tourist hotels earn more during holiday seasons.
Seasonal variation is like the “mood swings” of data that occur predictably.
3. Cyclical Variation (The Business Cycle Effect)
Now let us move from ice creams to the economy. You might have heard terms like
“economic boom” or “recession.” These are part of cyclical variations.
Unlike seasonal variations, cycles take place over longer periods (more than a year). They
are not fixed like summer and winter, but they follow a wave-like pattern:
• Boom (prosperity)
• Recession (decline)
• Depression (lowest point)
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• Recovery (rise again)
• Example: The stock market shows cycles of rise and fall over several years.
• Example: Real estate prices rise, fall, and then rise again over decades.
Cyclical variations are important for businesses to prepare for hard times and take
advantage of good times.
4. Irregular Variation (Unpredictable Factors)
Finally, life is full of surprises! In 2020, the COVID-19 pandemic shocked the entire world. No
one could predict it accurately. This is an example of irregular variation.
Irregular variations are caused by unusual, unpredictable events such as:
• Natural disasters (earthquake, flood, drought)
• Political events (war, demonetization)
• Sudden inventions or discoveries (e.g., launch of cheap smartphones)
These variations don’t follow any pattern and are beyond human control.
Putting It All Together
Let’s imagine you are studying the time series of tourism in India:
• Trend: Overall, the number of tourists is increasing every year.
• Seasonal Variation: More tourists come during winter holidays than in the summer.
• Cyclical Variation: During global recessions, foreign tourists decrease; during booms,
they increase.
• Irregular Variation: The COVID-19 pandemic suddenly reduced tourists in 2020,
which was an unpredictable shock.
See how beautifully these four components explain almost any real-life time series?
Conclusion
To sum up, a time series is simply data recorded over time, like pages in a diary that tell the
story of change. Its main object is to understand the past, recognize patterns, and use
them for predicting and planning the future.
The four essential components – Trend, Seasonal, Cyclical, and Irregular variations – are
like characters in a drama, each playing its role in shaping the story of data.
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When we combine them, we get the complete picture of how something behaves across
time. That’s why time series is not just a mathematical concept; it is a storyteller that
reveals the hidden patterns of life, business, and nature.
8. (a) The grades of a class, obtained in their first year of college, were analysed. Sixty
percent of the students in the top third of the class had graduated in the upper 10 percent
of their high school classes, as had 37% of the students in the middle third of the college
class and 21% of the students in the bottom third of the college class:
(i) What is the probability that a randomly chosen freshman graduated in upper 10 percent
of the high school class?
(ii) What is the probability that a randomly chosen freshman who graduated in the upper 10
Percent of his or her high school will be in the top third of the college class?
(iii) What is the probability that a randomly chosen freshman who graduated in the upper 10
percent of high school class will be in the upper two-thirds of the college class?
(iv) What is the probability that a randomly chosen freshman who did not graduate in the
upper 10 percent of his high school class will be in the upper two-thirds of the college class ?
(b) Define Probability. Discuss the theorems of Probability by giving suitable examples.
Ans: 󷈷󷈸󷈹󷈺󷈻󷈼 A Fresh Beginning
Imagine this: You are sitting in a classroom on the very first day of your college. Everyone is
nervous, looking around, wondering, “Who will end up at the top of the class? Who are the
toppers from school? Who might struggle here?”
Now, the professors of this college are also curious. They know that students who were
toppers in school do not always remain toppers in college. Sometimes the brightest in high
school may land in the middle group in college, and sometimes an average student in school
surprises everyone by shining in college.
To understand this better, the professors conducted a small research study. They divided
the students into three groups based on their first-year college performance:
• Top third of the class (the top-performing students in college)
• Middle third of the class
• Bottom third of the class
Then they asked: “Among these groups, how many students were also in the top 10% of
their high school class?”
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The results were fascinating:
• In the top third of college, 60% had been in the top 10% of their high school.
• In the middle third of college, 37% were from the top 10% of high school.
• In the bottom third of college, 21% were from the top 10% of high school.
Now, we will solve the probability questions step by step, but instead of jumping straight
into formulas, let’s walk through the story slowly so that it feels natural and crystal clear.
󹵙󹵚󹵛󹵜 Step 1: Understanding the Setup
We can imagine that the whole freshman class is divided equally into three parts:
• 1/3 in the top group
• 1/3 in the middle group
• 1/3 in the bottom group
Each group has some students who came from the top 10% of high school, and some who
didn’t.
The given percentages tell us exactly how many from each group belong to that “top 10% in
high school” category.
So the data looks like this:
• Top third of college: 60% are top 10% of high school → probability = 0.60
• Middle third of college: 37% are top 10% of high school → probability = 0.37
• Bottom third of college: 21% are top 10% of high school → probability = 0.21
󹵙󹵚󹵛󹵜 Step 2: Define the Events
Let’s use short names to avoid confusion:
• Let C1 = top third of college
• Let C2 = middle third of college
• Let C3 = bottom third of college
• Let H = student was in the top 10% of high school
• Let H’ = student was not in the top 10% of high school
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(ii) What is the probability that a randomly chosen freshman who graduated in the upper
10 percent of his or her high school will be in the top third of the college class?
This is P(C1 | H).
We use Bayes’ theorem:
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󹶓󹶔󹶕󹶖󹶗󹶘 Part (b) – Probability and Its Theorems
Now that we’ve solved the tricky part, let’s move to definitions and theorems of probability, but
we’ll do it in the same storytelling style.
󷇮󷇭 What is Probability?
Probability is simply the language of uncertainty.
Life is full of uncertainty. When you toss a coin, you don’t know if it will land heads or tails. When it
looks cloudy, you don’t know if it will actually rain or not. Probability gives us a mathematical way to
measure uncertainty.
In simple terms:
For example:
• Toss a coin → probability of heads = 1/2
• Throw a die → probability of getting a 6 = 1/6
Probability always lies between 0 and 1.
• 0 means impossible event
• 1 means certain event
󹵙󹵚󹵛󹵜 Theorems of Probability
Probability is not just guessing; it is governed by strong rules, known as probability theorems. Let’s
discuss the main ones with examples.
1. Addition Theorem of Probability
If we want the probability of occurrence of either of two events A or B, we use:
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󷷑󷷒󷷓󷷔 Example: Suppose in a class,
• Probability a student plays cricket = 0.5
• Probability a student plays football = 0.4
• Probability a student plays both = 0.2
Then, probability that a student plays cricket or football:
2. Multiplication Theorem of Probability
If we want the probability of occurrence of both events A and B, we use:
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󷈷󷈸󷈹󷈺󷈻󷈼 Wrapping the Story
So, what did we do in this problem?
• We started with a real-life classroom example where students from high school toppers and
average ones mixed in college.
• We carefully calculated probabilities step by step, using total probability law and Bayes’
theorem.
• We discovered that being a school topper increases your chances of being a college topper—
but it does not guarantee it!
• Finally, we connected this with the fundamental theorems of probability that apply not just
in mathematics, but also in real life—from predicting the weather to diagnosing diseases,
from gambling games to artificial intelligence.
“This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.”