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GNDU Question Paper-2023
Bachelor of Commerce
(B.Com) 1
st
Semester
BUSINESS STATISTICS
Time Allowed: Three Hours Maximum Marks: 100
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. (a) Statistics is not a Science, it is a scientific method. Discuss the statement.
(b) Find the missing frequencies in the following distribution, for which it is known that
Median = 48.25 and Mode = 44.
Class Interval
Frequency
10-20
3
20-30
7
30-40
?
40-50
?
50-60
?
60-70
20
70-80
16
80-90
4
Total
150
2. Find out the Arithmetic Mean, Mode and Median for the following distribution:
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Mid
Points
(Weight in
lb)
95
105
115
125
135
145
155
165
175
No. of
Students
4
2
18
22
21
19
10
3
2
SECTION-B
3. Calculate Karl Pearson's Correlation Coefficient from the following Table:
X
12
9
8
10
11
13
7
Y
14
8
6
9
11
12
3
4. What is meant by Dispersion? What are the methods of computing measures of
dispersion? Illustrate the practical utility of these methods.
SECTION-C
5. The following table gives the data with regard to the prices and consumption of a few
selected items for the years 2012 and 2021:
Year
Articles
I
II
III
IV
P
Q
P
Q
P
Q
P
Q
2012
12.50
9
9.63
4
7.75
6
5.00
5
2021
12.75
9
7.75
6
8.80
10
6.50
7
Calculate the
(i) Laspeyre's Index,
(ii) Paasche's Index and
(iii) Fisher's Index.
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Also, prove that the time reversal test is satisfied by Fisher's formula but not necessarily
by the Laspeyre's and Paasche's Index Number.
6. What is Index Numbers ? Explain the methods of construction of Index Numbers.
SECTION-D
7. The data below give the index of Industrial production from 2011 to 2020:
Year
Index of Production
(Lakh Tonnes)
2011
109.2
2012
119.8
2013
129.7
2014
140.8
2015
153.8
2016
153.2
2017
152.6
2018
163.0
2019
175.3
2020
184.3
Find the trend line and predict the index of production for the year 2022 by semi-average
method.
8.(a) Discuss the Theorems of Probability.
(b) A stock list has 20 items in a lot. Out of which, 12 are non-defective and 8 defective. A
customer selects 3 items from the lot.
(i) What is the probability that all the Three items are not-defective ?
(ii) What is the probability that out of these three items, two are non-defective and
one is defective?
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GNDU Answer Paper-2023
Bachelor of Commerce
(B.Com) 1
st
Semester
BUSINESS STATISTICS
Time Allowed: Three Hours Maximum Marks: 100
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. (a) Statistics is not a Science, it is a scientific method. Discuss the statement.
Ans: A Fresh Beginning: The Tale of Two Friends – Science and Statistics
Once upon a time, there were two close friends: Science and Statistics.
Science was like an explorer—curious, adventurous, always looking for truth about the
world. He wanted to know why the stars shine, how plants grow, or why some people fall
sick while others don’t. But Science had a little problem: he was often surrounded by
enormous amounts of messy, confusing information.
That’s when Statistics came in. Statistics was not an explorer, but rather an organizer. He
could take the mountain of information Science found and turn it into neat patterns, clear
conclusions, and useful results. Without Statistics, Science would feel lost in a jungle of
numbers. But without Science, Statistics wouldn’t even have a purpose—he would just be
playing with numbers without meaning.
This little friendship story already gives us a clue about the statement:
“Statistics is not a science, it is a scientific method.”
Now let’s break this down and understand it in depth.
Is Statistics a Science?
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When we call something a science, it usually means a systematic body of knowledge that
discovers and explains truths about nature or society. Physics, chemistry, and biology are
sciences because they have:
1. A subject matter (what they study).
2. Theories and laws (Newton’s laws, Mendel’s laws, etc.).
3. Experiments to test those laws.
Now think about Statistics. Does it have natural laws? Does it explain why a plant grows or
why the Earth moves around the Sun?
The answer is no.
Statistics does not give us universal laws about the physical or social world. Instead, it
provides us tools and techniques to study those worlds. That’s why many scholars argue:
Statistics is not a science in itself.
Then What is Statistics?
Statistics is best understood as a scientific method—a way of doing things systematically. It
is like the grammar of science. Just as grammar does not tell stories but helps us tell stories
clearly, Statistics does not explain natural phenomena but helps scientists explain them
better.
For example:
• A doctor may conduct a trial for a new medicine. Statistics helps decide whether the
medicine really works or not.
• An economist may want to study poverty trends. Statistics helps organize surveys,
interpret results, and check accuracy.
• A business may want to know customer preferences. Statistics helps in designing
questionnaires and analyzing responses.
In each case, Statistics doesn’t itself give the answers—it helps others arrive at reliable
answers.
Why Call it a Method?
Let’s imagine you are baking a cake. The recipe is the method—it tells you step by step how
to collect ingredients, measure them, mix them, and finally bake. Similarly, Statistics gives a
step-by-step recipe for handling data:
1. Collection of Data – Just like gathering ingredients.
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2. Classification & Organization – Arranging them neatly, like sorting flour, sugar, and
eggs.
3. Presentation – Using tables, graphs, charts (just like setting the cake beautifully on a
plate).
4. Analysis – Finding patterns, averages, or relationships (like tasting the cake to see if
it’s sweet enough).
5. Interpretation – Finally giving meaning: Does the data support the theory? Does the
cake taste good?
This stepwise procedure is why Statistics is called a scientific method rather than an
independent science.
But Wait—Some People Do Call It a Science
Here’s where the story gets interesting. Some thinkers argue that Statistics is a science
because it has its own principles, formulas, and theories (like probability, regression,
sampling theory). These are studied systematically, and there is research devoted purely to
statistical methods.
However, notice the difference:
• Physics or Chemistry aim to discover truths about the world.
• Statistics aims to improve methods of studying those truths.
So while it has the flavor of a science, its true role is more of a method supporting other
sciences.
A Simple Analogy
Think of Science as a detective. The detective’s job is to solve mysteries. Now, what does
Statistics do? It provides the detective with magnifying glasses, fingerprint kits, and DNA
tests. These tools do not solve the case on their own, but they help the detective arrive at
the truth.
This is exactly why Statistics is better understood as a scientific method rather than a full-
fledged science.
Final Thoughts
So, coming back to the statement:
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“Statistics is not a science, it is a scientific method.”
This makes sense because:
• Statistics does not create natural or social laws.
• It provides methods—like collection, classification, analysis, and interpretation of
data—that help sciences and other fields in their investigations.
• It is more of a toolbox for all sciences rather than an independent branch of science.
But remember, without this “toolbox,” most sciences would struggle to confirm their
findings. That is why statistics, even if not a science in itself, is one of the most
indispensable companions of science.
(b) Find the missing frequencies in the following distribution, for which it is known that
Median = 48.25 and Mode = 44.
Class Interval
Frequency
10-20
3
20-30
7
30-40
?
40-50
?
50-60
?
60-70
20
70-80
16
80-90
4
Total
150
Ans: The Beginning of the Story
Imagine you are an explorer walking into a mysterious cave where treasures (answers) are
hidden. The cave has different chambers (class intervals), and in each chamber, there are
certain numbers of jewels (frequencies). Some chambers have exact numbers written
outside, but in a few chambers, the labels have faded, and we don’t know how many jewels
are inside.
Your mission is simple: discover the missing jewels using two guiding clues given by the
guardian of the cave:
1. The Median of all the jewels = 48.25
2. The Mode of all the jewels = 44
3. The total number of jewels = 150
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With this information, we set out on our mathematical adventure.
Step 1: The Data We Have
Here’s the mysterious cave (our table) with the chambers and jewels:
Class Interval
Frequency
10–20
3
20–30
7
30–40
? (let’s call it x)
40–50
? (let’s call it y)
50–60
? (let’s call it z)
60–70
20
70–80
16
80–90
4
Total
150
So the missing frequencies are x, y, z.
We already know:
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Class Interval
Frequency
10–20
3
20–30
7
30–40
32
40–50
40
50–60
28
60–70
20
70–80
16
80–90
4
Total
150
Wrapping Up the Story
So, our cave puzzle is solved! We uncovered the hidden jewels (frequencies) by carefully
following the clues left by the guardian (Median and Mode conditions). Notice how both
Median and Mode pointed to the same class (40–50), which gave us the perfect equations
to unlock the missing pieces.
What seemed like a tricky riddle at the start turned out to be a systematic adventure where
each clue led to the next until the mystery was resolved.
And the beauty of this problem is: it teaches us how statistics is not just about numbers,
but about solving puzzles logically and patiently.
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2. Find out the Arithmetic Mean, Mode and Median for the following distribution:
Mid
Points
(Weight in
lb)
95
105
115
125
135
145
155
165
175
No. of
Students
4
2
18
22
21
19
10
3
2
Ans: A little story about a classroom of weights — and how mean, median and mode find
their places
Imagine a lively class of 101 students lined up during a fitness check. The teacher asks them
to step onto the scale and notes only the mid-point of each weight-class (so each recorded
value stands in for everyone in that class) and how many students landed in that class. Your
job is to play the polite statistician: find the Arithmetic Mean (the centre of gravity of all the
weights), the Median (the weight that sits at the middle of the sorted line), and the Mode
(the weight class that’s the most popular — the crowd favourite).
Here’s the data (mid-points = representative weight of each class, and frequency = number
of students):
• Mid-points (lb): 95, 105, 115, 125, 135, 145, 155, 165, 175
• Frequencies: 4, 2, 18, 22, 21, 19, 10, 3, 2
1) Arithmetic Mean — the “balance point”
Think of the mean as placing all the students on a seesaw: where would the seesaw
balance? For grouped data using mid-points, compute:
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SECTION-B
3. Calculate Karl Pearson's Correlation Coefficient from the following Table:
X
12
9
8
10
11
13
7
Y
14
8
6
9
11
12
3
Ans: The Story of X and Y: A Friendship Tested by Karl Pearson
Imagine for a moment that numbers are like people. They live together in pairs, where one
is called X and the other is called Y. They walk side by side, and our job is to find out:
“How strong is the friendship between X and Y? Do they always walk in the same
direction, or sometimes one goes left while the other goes right?”
That’s exactly what Karl Pearson’s Correlation Coefficient (r) tells us. It measures the
degree of relationship between two variables. Its value always lies between -1 and +1.
• If r = +1, it means X and Y are best friends forever: when X increases, Y always
increases perfectly.
• If r = -1, they are like perfect enemies: when X goes up, Y goes down exactly.
• If r = 0, they have no relationship at all: X does its own thing, Y doesn’t care.
So today, we are detectives trying to find out what kind of relationship exists between these
two sets of numbers:
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Step 3: Build the Table
Now we’ll create a table with deviations. This step is like laying down the clues for solving
the mystery of correlation.
X
Y
x=X−𝒙
y=Y−𝒚
xy
x2x
2
y2y
2
12
14
2
5
10
4
25
9
8
-1
-1
1
1
1
8
6
-2
-3
6
4
9
10
9
0
0
0
0
0
11
11
1
2
2
1
4
13
12
3
3
9
9
9
7
3
-3
-6
18
9
36
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Step 5: Interpret the Result
So the correlation coefficient between X and Y is approximately 0.95.
What does this mean?
• Since the value is very close to +1, it shows a strong positive relationship.
• This means whenever X increases, Y also tends to increase.
• They are almost like best friends: if one smiles, the other smiles too.
Making It Fun for the Examiner
Imagine you’re narrating this in your answer sheet:
“These two variables, X and Y, seem to walk hand in hand. With a correlation of 0.95, they
are like best friends in a classroom—if one studies harder, the other also studies harder.
Their friendship is almost perfect, though not 100%.”
This storytelling style will make even the examiner smile because you’ve not only given the
correct answer but also explained it in a way that feels alive.
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4. What is meant by Dispersion? What are the methods of computing measures of
dispersion? Illustrate the practical utility of these methods.
Ans: Dispersion – A Story Beyond Averages
Imagine you and your friends are going to a cricket coaching camp. At the end of the month,
the coach decides to check the performance scores of each player out of 100. When he
calculates the average score, he finds it is 60 marks.
Now, at first glance, this sounds fair. But here comes the twist:
• One player scored 95, another scored 90, and another scored only 20.
• Another group of players scored close to the average, like 58, 61, 62.
If you only look at the average (60), you will never know the spread of the marks. Some
students are very high, some are very low. This “spread” or “scattering” of data is what we
call Dispersion in statistics.
So, in the simplest sense:
Dispersion tells us how much the data is scattered around an average.
It gives depth to the story of numbers, beyond just the average.
Why is Dispersion Important?
Let’s continue with our cricket camp example. Suppose the coach has to select players for
the state-level team.
• If the marks of all students are tightly packed around 60 (say between 58 to 62), then
he knows the performance level is consistent.
• But if the marks range from 20 to 95, then the average 60 is misleading, because the
team has both very weak and very strong players.
So, dispersion helps in decision-making. It shows:
1. Consistency – Are all players equally good?
2. Fairness – Is the average really representing everyone?
3. Risk/Variability – How uncertain is the outcome?
This is why economists, businessmen, teachers, and even policymakers rely on dispersion to
understand the true picture of data.
Methods of Computing Dispersion
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Now, let’s enter the toolbox of a statistician. There are several methods to measure
dispersion. Each method tells the “scattering story” of data from a different angle.
1. Range
The Range is the simplest method. It is just the difference between the highest value and
the lowest value.
Formula:
Range=L−S
(where L = largest value, S = smallest value)
Example:
Marks of students: 20, 45, 50, 60, 95
Range = 95 – 20 = 75
Utility:
• Very quick to calculate.
• Helps in weather forecasting (difference between max. and min. temperatures).
• But, it only considers the two extreme values, ignoring all middle data.
2. Quartile Deviation (Interquartile Range)
Think of this like cutting your data into four equal parts. Quartiles divide the data into Q1
(25%), Q2 (median – 50%), and Q3 (75%).
The middle 50% of data lies between Q1 and Q3.
Formula:
Example:
Suppose the marks are arranged in ascending order and Q1 = 40, Q3 = 80.
Quartile Deviation = (80 – 40)/2 = 20
Utility:
• Used in income studies to avoid influence of extreme rich or poor.
• Focuses on the central bulk of data.
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3. Mean Deviation
This method measures how far each observation is, on average, from the mean (or median).
Formula:
Example:
If average score = 50 and students’ marks are 40, 50, 60 → deviations are 10, 0, 10 → mean
deviation = (10+0+10)/3 = 6.67
Utility:
• Better than range because it considers all values.
• Used in business to study fluctuations in demand, cost, etc.
• However, dealing with absolute values makes it lengthy.
4. Standard Deviation (SD)
This is the most popular and scientific measure. Instead of absolute differences, it uses
squared differences from the mean.
Formula:
Example:
Marks = 40, 50, 60 → Mean = 50 → Deviations = -10, 0, +10
Squares = 100, 0, 100 → Average = 200/3 ≈ 66.7 → SD = √66.7 ≈ 8.16
Utility:
• Widely used in finance to measure risk of investments.
• Used in research to study variability in experimental results.
• Very accurate but requires more calculation.
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5. Coefficient of Variation (CV)
This is a relative measure of dispersion, useful for comparing variability between two sets
of data with different units or averages.
Formula:
Example:
• Company A: Mean salary = ₹20,000, SD = ₹2,000 → CV = 10%
• Company B: Mean salary = ₹50,000, SD = ₹10,000 → CV = 20%
Even though Company B pays more, salaries are less consistent.
Utility:
• Helps in comparing stability across industries, investments, or regions.
• Economists use it to compare inequality in incomes of different countries.
Practical Utility of Dispersion
Now, let’s step into the real world and see how dispersion helps:
1. In Business:
Companies use standard deviation to study fluctuations in sales, profits, or stock
prices. A business with low variability in profits is more reliable than one with high
ups and downs.
2. In Education:
Teachers don’t just look at average marks. If the class average is 60 but half the
students failed, the dispersion tells the true story.
3. In Economics:
Government studies income inequality using range, quartile deviation, and
coefficient of variation. High dispersion means high inequality.
4. In Weather Reports:
Meteorologists use dispersion to compare temperature variations in different cities.
5. In Sports:
Selectors check not only the batting average of a player but also his consistency
(dispersion). A player who always scores between 40–60 is more reliable than one
who scores 0 in one match and 100 in the next.
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Conclusion
So, dispersion is like the second chapter of a story that averages begin. If the average tells
us “what is typical,” dispersion tells us “how much the reality differs from the typical.”
• Range is quick but rough.
• Quartile deviation focuses on the middle crowd.
• Mean deviation is fairer as it considers all data.
• Standard deviation is the king – precise and powerful.
• Coefficient of variation helps compare across different groups.
Without dispersion, averages are like a movie trailer without the full film—you never know
the real story. That’s why, in every field—from cricket to business, from weather to
economics—dispersion is the mirror that reflects the truth of data.
SECTION-C
5. The following table gives the data with regard to the prices and consumption of a few
selected items for the years 2012 and 2021:
Year
Articles
I
II
III
IV
P
Q
P
Q
P
Q
P
Q
2012
12.50
9
9.63
4
7.75
6
5.00
5
2021
12.75
9
7.75
6
8.80
10
6.50
7
Calculate the
(i) Laspeyre's Index,
(ii) Paasche's Index and
(iii) Fisher's Index.
Also, prove that the time reversal test is satisfied by Fisher's formula but not necessarily
by the Laspeyre's and Paasche's Index Number.
Ans: A little story about baskets, prices and fair averages — with numbers you can follow
Imagine two markets separated by nine years: Market 2012 (our base year) and Market
2021 (our current year). A caring shopkeeper sells four items (Articles I, II, III and IV). To
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know how “prices” have changed between the two markets we put the items into a single
basket and compute three well-known price indices: Laspeyres (base-weighted), Paasche
(current-weighted) and Fisher (a fair geometric blend). I’ll walk you through everything step-
by-step — like telling the story of how each index is built — and then prove the important
time-reversal property for Fisher (and show why Laspeyres and Paasche need not satisfy it).
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Think of Laspeyres as a nostalgic shopper who insists on buying the exact items and
quantities they bought in 2012 — their measurement tends to overweight items that were
abundant in the base year even if people later buy less of them. Paasche is the modern
shopper who measures price change using today’s shopping list — it can underweight goods
that were important in the past. Fisher is the peacemaker: it takes the geometric mean of
the two viewpoints and produces a balanced, “time-neutral” index that behaves nicely when
you reverse time.
In our neat example, Laspeyres gives 103.83, Paasche 104.23, and Fisher 104.03 — all telling
the same story (prices rose), but with slightly different emphases. Algebraically and
numerically we also see why Fisher satisfies the time-reversal test while Laspeyres and
Paasche need not.
That’s the story of the basket: clear calculations, a gentle moral about weighting, and a neat
algebraic proof showing why Fisher is the only one among the three guaranteed to be time-
reversible.
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6. What is Index Numbers ? Explain the methods of construction of Index Numbers.
Ans: What is Index Numbers? A Story-like Explanation
Imagine one day you open your fridge and see some vegetables, fruits, and groceries. You
think about how much money you used to spend on these items a year ago and compare it
with today’s prices. You notice that the same basket of goods which cost you ₹1,000 last
year is now costing you ₹1,200. Immediately, you realize that prices have gone up by 20%.
Now, what you did here is a very simple, everyday example of Index Numbers.
Index Numbers are nothing but a special kind of statistical tool that helps us measure
changes over time. These changes could be in prices, quantities, production, wages,
imports, exports, or anything else that keeps varying.
In short:
Index Numbers = A thermometer of the economy.
Just like a thermometer tells us whether the temperature is rising or falling, Index Numbers
tell us whether prices, production, or other factors are going up or down.
Formal Definition
Index numbers are statistical measures that show changes in the level of a variable or a
group of related variables over two or more periods of time.
For example:
• Price Index Numbers tell us whether prices are rising (inflation) or falling (deflation).
• Quantity Index Numbers tell us whether the production or sales are increasing or
decreasing.
• Value Index Numbers show changes in value, i.e., price × quantity.
Why Index Numbers are Important?
Think of the government, businessmen, and common people:
• The government uses index numbers to make policies about inflation, wages, and
trade.
• Businessmen use them to decide about investments, production, and pricing.
• Common people are indirectly affected, because their salaries, pensions, and DA
(dearness allowance) are linked with index numbers.
For example, if the Consumer Price Index (CPI) shows that prices of essential goods have
increased, the government may increase salaries or pensions to balance the cost of living.
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So, index numbers are like a mirror of the economy—showing us whether life is getting
costlier or cheaper, and whether businesses are growing or shrinking.
Methods of Construction of Index Numbers
Now comes the main part: How do we actually construct index numbers?
Let’s imagine again that you are comparing prices of some items—say rice, sugar, milk, and
oil—in two years. To make sense of the change, we need methods. Statisticians have
developed several methods for constructing index numbers.
Here are the main methods, explained like a story.
1. Simple Aggregate Method
This is the easiest method.
What we do here is just take the sum of prices of selected items in the current year and
divide it by the sum of prices in the base year, then multiply by 100.
Formula:
2. Simple Average of Price Relatives Method
This method is slightly better.
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Here, we calculate the price relative of each item. Price relative means the ratio of
current price to base year price, multiplied by 100. Then we take the average of all these
relatives.
Formula:
So, prices increased on average by 31.67%.
This is better than the simple aggregate method because it gives equal weight to each
percentage change.
3. Weighted Index Numbers
Now let’s get more realistic. In real life, we don’t spend equal amounts on all items. For
example, a family spends more on food than on clothes or gadgets.
So, statisticians introduced weights to give more importance to the goods that matter more.
There are several formulas under this method:
(a) Laspeyres’ Index Number
Here, the weights are the quantities of the base year.
Formula:
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This means we see how much the current year’s basket of goods costs compared with if we
had bought them at base year prices.
Advantage: More realistic because it considers actual current consumption.
Disadvantage: Difficult to get current year quantities every time.
(c) Fisher’s Ideal Index
This is considered the “best” method. Fisher was a famous economist who said: “Why not
take the middle path?”
So, Fisher’s Index is the geometric mean of Laspeyres’ and Paasche’s indices.
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Conclusion: Why Index Numbers Feel Like a Story of Economy
If you think carefully, index numbers are like a diary of our economy. They record whether
our life is becoming costlier or cheaper, whether production is growing or declining, and
whether trade is expanding or shrinking.
From a student’s perspective, you can remember them like this:
• Simple Aggregate → Just total prices comparison.
• Simple Average of Relatives → Average of percentage changes.
• Laspeyres → Base year basket.
• Paasche → Current year basket.
• Fisher → The perfect middle path (geometric mean).
• Value Index → Total value comparison.
So next time when you hear in the news, “Inflation has gone up by 6%,” remember—it is
nothing but an index number speaking to us about the economy’s temperature!
SECTION-D
7. The data below give the index of Industrial production from 2011 to 2020:
Year
Index of Production
(Lakh Tonnes)
2011
109.2
2012
119.8
2013
129.7
2014
140.8
2015
153.8
2016
153.2
2017
152.6
2018
163.0
2019
175.3
2020
184.3
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Find the trend line and predict the index of production for the year 2022 by semi-average
method.
Ans: A Fresh Beginning: The Story of Growing Industries
Think for a moment about a small town. In 2011, the town sets up a few factories. At first,
the production is modest. But as the years go by, more machines are installed, workers gain
experience, and production begins to rise. Like a child growing taller every year, the town’s
industrial output also starts showing steady growth.
Now, as observers, we’re curious: how can we predict what the industrial production might
look like in the future? Can we take the past as a guide? This is where the semi-average
method comes into play.
This method is like dividing a story into two halves: “Act One” (the earlier years) and “Act
Two” (the later years). By studying the average performance in both acts, we can figure out
the overall direction of the story—the trend line. Once we have that trend, it becomes a
ruler that we can stretch forward to make predictions about years that haven’t yet arrived.
So let’s put on our detective hats and carefully investigate.
Step 1: The Data We Have
Here is the information given:
Year
Index of Production (Lakh Tonnes)
2011
109.2
2012
119.8
2013
129.7
2014
140.8
2015
153.8
2016
153.2
2017
152.6
2018
163.0
2019
175.3
2020
184.3
We want to find the trend line equation using the semi-average method, and then use that
equation to predict the production for 2022.
Step 2: Divide the Story into Two Halves
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We have 10 years of data (2011 to 2020). The semi-average method tells us to split this into
two equal halves:
• First Half (2011–2015)
• Second Half (2016–2020)
Step 3: Calculate the Average of Each Half
First Half (2011–2015):
So, the first half’s average index = 130.66.
The middle year of this half = 2013.
Second Half (2016–2020):
So, the second half’s average index = 165.68.
The middle year of this half = 2018.
Step 4: Create Two Key Points
Now we have two “big picture” points that summarize each half:
• Point A = (2013, 130.66)
• Point B = (2018, 165.68)
These points are like the two anchor posts of a straight line.
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Step 5: Find the Slope of the Line
The slope tells us how much the index increases per year.
So, every year, the index rises by about 7 units.
Step 6: Equation of the Trend Line
The general form of the line is:
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The Storytelling Explanation
Think of this as watching a movie about industries. In the first half (2011–2015), the industry
is learning to walk. Production grows steadily from 109.2 to 153.8, giving us an average
“speed” of around 130.66 by the middle year, 2013.
In the second half (2016–2020), the industry begins to run, pushing harder and faster, with
an average of 165.68 around 2018.
When we connect these two averages with a straight line, we’re essentially saying: “This is
the underlying trend of the story. The small ups and downs (like the dip in 2016 and 2017)
are like minor setbacks in a movie, but the overall storyline is upwards.”
By stretching this line forward into the future, we can peek into 2022. And what does the
line whisper to us? It says, “Around 194.7 lakh tonnes.”
Why This Method Feels Magical
The semi-average method is simple yet powerful because:
1. It doesn’t get lost in every small rise or fall—it looks at the bigger picture.
2. It tells us the “average journey” rather than the day-to-day struggles.
3. It gives a clear, easy-to-use formula for prediction.
So, even though industries may face short-term ups and downs (like recessions, booms, or
unexpected events), the trend line acts like a compass, showing the long-term direction.
Conclusion
The predicted Index of Production for 2022 using the semi-average method = 194.7 lakh
tonnes.
In simple words, our detective work shows that if the industries continue their long-term
growth story, production in 2022 will be close to 195 lakh tonnes.
This is the beauty of statistics—it turns dry numbers into a meaningful story of growth,
progress, and the future.
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8.(a) Discuss the Theorems of Probability.
(b) A stock list has 20 items in a lot. Out of which, 12 are non-defective and 8 defective. A
customer selects 3 items from the lot.
(i) What is the probability that all the Three items are not-defective ?
(ii) What is the probability that out of these three items, two are non-defective and
one is defective?
Ans: A different kind of beginning — imagine a market stall
Picture a friendly market stall where probabilities are the shopkeeper’s rules for doing
business. Every rule the shopkeeper follows is a theorem of probability — simple, sensible
laws that make sure customers (that’s us) don’t get fooled when we pick fruits from the
basket or, in your math problem, items from a stock lot. I’ll tell the story of those rules first,
then walk through the problem with the 20-item lot (12 good, 8 defective) like a little scene
so the computations feel natural and obvious.
(a) Theorems (the shopkeeper’s rules) — explained like a story
1. The Axioms (the ground rules of the stall)
Think of the sample space S as the entire box of outcomes in front of the stall: everything
that could possibly happen. Kolmogorov’s axioms are like the stall’s code:
1. Non-negativity: Every probability is a non-negative number. You can’t have negative
chance — that would be like paying someone to take a fruit!
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“This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.”
