Easy2Siksha.com
GNDU Question Paper-2024
B.A 1
st
Semester
QUANTITATIVE TECHNIQUES
(Quantitative Techniques-I)
Time Allowed: Three Hours Max. 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. Define Statistics. Discuss in detail its characteristics and limitations.
2. (i) Differentiate between tabulation and classification. Discuss the general rules of
tabulation.
(ii) Draw "less than" and "more than ogives" from the data given
below:
Profits (Rs. Lakh)
Number of Companies
10-20
6
20-30
8
30-40
12
40-50
18
50-60
25
60-70
16
70-80
8
80-90
5
90-100
2
SECTION-B
Easy2Siksha.com
3. Calculate mode of the following data:
Wages (Rs.)
No. of Workers
25-35
4
35-45
44
45-55
38
55-65
28
65-75
6
75-85
8
85-95
12
95-105
2
105-115
2
4. Find the missing information from the following:
Group I
Group II
Group III
Group IV
Number
50
?
90
200
Std. Deviation
6
7
?
7.746
Mean
113
?
115
116
SECTION-C
5. (1) Calculate the Two regression equations from the following data:
𝜮𝑿 = 𝟑𝟎, 𝜮𝒀 = 𝟐𝟑, 𝜮𝑿𝒀 = 𝟏𝟔𝟖, 𝜮𝑿
𝟐
= 𝟐𝟐𝟒, 𝜮𝒀
𝟐
=175 ,N = 7
Hence or otherwise find Karl Pearson's coefficient of correlation.
(ii) What are the regression coefficients? Explain the properties of regression coefficients.
Easy2Siksha.com
6. Calculate Karl Pearson's coefficient of correlation from the following data:
X/Y
10-25
25-40
40-55
0-20
10
4
6
20-40
5
40
9
40-60
3
8
15
SECTION-D
7. What is time series? Explain briefly the various methods of determining a trend in a
time series. Explain merits and demerits of each method.
8. Calculate Laspeyre's, Paasche's, and Fisher's ideal index for the following data:
Commodity
1970
1990
Price
Expenditure
Price
Expenditure
A
8
100
10
90
B
10
60
11
66
C
5
100
5
100
D
3
30
2
24
E
2
8
4
20
With the help of this data, show which of the above index number satisfies time and
factor reversal test.
Easy2Siksha.com
GNDU Answer Paper-2024
B.A 1
st
Semester
QUANTITATIVE TECHNIQUES
(Quantitative Techniques-I)
Time Allowed: Three Hours Max. 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. Define Statistics. Discuss in detail its characteristics and limitations.
Ans: Statistics: Definition, Characteristics, and Limitations
A Different Beginning – A Little Story
Imagine you are sitting in a cricket stadium. The crowd is roaring, the batsman hits a six, and
the scoreboard keeps ticking. Now, think for a moment—how do you know the batsman’s
performance? By watching every shot? Not really. You look at the scoreboard: runs, strike
rate, number of boundaries, and average.
That scoreboard is nothing but statistics in action—a way of converting large events (every
ball faced, every run made) into meaningful numbers that you can easily understand.
Without statistics, the match would be too confusing to follow.
This same idea applies to our daily life, government decisions, business, health studies, and
even education. Whenever we want to understand a huge set of information, statistics
helps us simplify it into meaningful patterns, averages, and comparisons.
Definition of Statistics
Easy2Siksha.com
The word Statistics comes from the Latin word ‘status’ or Italian ‘statista’, meaning a
political state. Initially, it was used by kings and rulers to gather information about their
population, soldiers, wealth, and crops to make decisions. Over time, the meaning
expanded.
In simple words:
Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting
numerical data to help in decision-making.
If we break it down:
• Collection – Gathering raw facts (like population census, survey of students).
• Organization – Arranging the data in tables or charts.
• Presentation – Showing it through graphs, diagrams, or averages.
• Analysis – Finding patterns, relationships, or trends.
• Interpretation – Drawing conclusions and making decisions.
So, statistics is both a method (way of handling data) and a science (because it follows
systematic rules).
Characteristics of Statistics
Now let’s discuss what makes statistics unique. Think of statistics as a friendly guide who
helps you make sense of confusing numbers. Here are its key characteristics, explained
simply:
1. Statistics Deals with Aggregates, Not Individuals
Suppose you scored 85 in an exam. That single number is not statistics. But when we look at
the scores of your whole class, calculate the average, highest, lowest, and percentage of
pass/fail—that becomes statistics.
So, statistics studies a group, not one person.
2. Statistics is Expressed in Numbers
Words like “many students did well” are vague. But when we say “80 out of 100 students
passed,” it is statistics. Numbers give clarity and precision.
3. Statistics is Collected for a Purpose
Random numbers have no use. If a shopkeeper notes daily sales, his purpose may be to find
which day has the highest demand. So, data must be collected with clear objectives.
4. Statistics is Comparative and Relational
Easy2Siksha.com
Statistics loves to compare. For example:
• Is the literacy rate higher in Punjab than in Bihar?
• Did company profits rise this year compared to last year?
It always looks at relationships and comparisons.
5. Statistics is Affected by Multiple Factors
For example, agricultural output depends on rainfall, soil quality, fertilizers, and technology.
So, statistics never has a single reason behind results—it always considers multiple
influences.
6. Statistics is an Approximate Science
Unlike mathematics where 2+2 = 4 always, statistics deals with probabilities and
approximations. For example, surveys may say “70% people prefer tea.” It does not mean
exactly 70 out of 100, but approximately.
7. Statistics Helps in Prediction
By looking at past data, statistics can predict the future. Example: weather forecasts,
election predictions, stock market trends—all use statistical models.
8. Statistics is a Tool, Not an End
Statistics doesn’t give answers by itself; it helps in decision-making. Just like a doctor uses
reports to diagnose, statistics guides administrators, businessmen, and scientists.
Diagram: Characteristics of Statistics
Here’s a simple diagram to make it easier:
Easy2Siksha.com
(You can draw this as a flowchart or mind-map in your exam for better presentation.)
Limitations of Statistics
Now, while statistics is powerful, it also has limitations. Just like a magnifying glass shows
details but cannot explain emotions, statistics also has boundaries.
1. Statistics Cannot Study an Individual
It tells us about groups, not unique persons. For example, national income statistics show
average income, but not the struggles of a single poor family.
2. Statistics Cannot Give Exact Results
It deals with approximations. If a survey shows “60% people like ice cream,” in reality, it
might be 58% or 62%. It cannot give perfect truth.
3. Statistics is Dependent on Data Quality
If data is false, biased, or incomplete, the results will also be misleading. This is called
“Garbage in, garbage out.”
4. Statistics Can be Misused
Numbers can be twisted. For example, an advertisement might say “Our toothpaste is used
by 90% dentists,” but maybe they only asked 10 dentists! So, statistics can mislead if used
dishonestly.
5. Statistics Shows “What is,” Not “Why”
It can describe the situation but cannot explain the exact cause. For example, it can show
unemployment rate rising, but not fully explain why it rose.
6. Not Useful for Qualitative Data Alone
Statistics works best with numerical data. Qualitative things like honesty, beauty, or
intelligence are difficult to measure without converting them into numbers.
7. Requires Expertise
If someone without proper training uses statistical tools, they may interpret wrongly. For
example, misusing averages can give a wrong picture.
Putting It All Together
So, to wrap up:
Easy2Siksha.com
• Definition: Statistics is the science of collecting, organizing, analyzing, and
interpreting data.
• Characteristics: Deals with groups, expressed in numbers, comparative, influenced
by many factors, approximate in nature, and useful in prediction.
• Limitations: Cannot study individuals, gives approximate not exact results, depends
on data quality, may be misused, cannot explain causes fully, and requires expertise.
Final Humanized Note
Think of statistics as a mirror:
• It reflects reality, but not perfectly.
• It shows patterns, but not the whole truth.
• It guides us, but we must use our wisdom to interpret it.
In short, statistics is a powerful servant but a dangerous master. When used carefully, it
helps governments plan budgets, doctors save lives, businesses grow, and even cricket fans
enjoy the match. But when misused, it can mislead and confuse.
2. (i) Differentiate between tabulation and classification. Discuss the general rules of
tabulation.
(ii) Draw "less than" and "more than ogives" from the data given
below:
Profits (Rs. Lakh)
Number of Companies
10-20
6
20-30
8
30-40
12
40-50
18
50-60
25
60-70
16
70-80
8
80-90
5
90-100
2
Ans: (i) Tabulation vs Classification and General Rules of Tabulation
A Fresh Beginning
Easy2Siksha.com
Imagine you are a teacher who has just collected exam scores from 100 students. At first,
you have nothing but a long list of numbers: 45, 67, 89, 72, 53, 91… and so on. Looking at
this list, you feel overwhelmed. How do you make sense of it? How do you present it so that
others can understand at a glance?
This is where classification and tabulation come in. They are like two stages of the same
journey: first, you sort the data into meaningful groups (classification), and then you arrange
it neatly in rows and columns (tabulation).
Meaning of Classification
Classification is the process of arranging raw data into categories or classes based on
common characteristics. It is like sorting clothes in your cupboard—shirts in one pile,
trousers in another, and jackets in a third.
• Purpose: To simplify large data sets by grouping similar items.
• Example: Grouping students’ marks into ranges: 0–10, 10–20, 20–30, etc.
Meaning of Tabulation
Tabulation is the process of presenting classified data in a systematic table with rows and
columns. It is like arranging your sorted clothes neatly in shelves so that anyone can see
what you have.
• Purpose: To present data clearly, concisely, and in a form suitable for analysis.
• Example: Creating a table showing how many students scored in each mark range.
Difference Between Classification and Tabulation
Basis
Classification
Tabulation
Meaning
Process of grouping data into
categories or classes.
Process of presenting data in rows and
columns.
Stage
First step in data organization.
Next step after classification.
Purpose
To simplify and condense raw data.
To present data clearly for analysis
and comparison.
Form
Data is grouped but not yet in table
form.
Data is arranged in a structured table.
Example
Marks grouped into ranges (0–10,
10–20, etc.).
A table showing number of students in
each range.
Story Note: Think of classification as cooking ingredients (sorting vegetables, spices, grains)
and tabulation as serving the cooked meal neatly on a plate. Both are essential, but they
serve different purposes.
General Rules of Tabulation
When preparing a statistical table, certain rules ensure clarity and accuracy.
Easy2Siksha.com
1. Title
o Every table should have a clear, concise title describing what the data is
about.
o Example: “Distribution of Companies by Profit Range.”
2. Numbering
o Tables should be numbered (Table 1, Table 2, etc.) for easy reference.
3. Headings
o Each column and row should have proper headings (e.g., “Profit Range,”
“Number of Companies”).
4. Units of Measurement
o Units (e.g., Rs. Lakh, Percentage, Kilograms) should be clearly mentioned.
5. Simplicity
o Tables should be simple, avoiding unnecessary details.
6. Clarity
o Data should be arranged logically, usually in ascending or descending order.
7. Consistency
o Similar data should be presented in the same format throughout.
8. Totals
o Where possible, totals should be given at the bottom or side.
9. Source
o If data is taken from another source, it should be mentioned below the table.
10. Neatness
o The table should be neat, with proper spacing and alignment.
Story Note: A table is like a well-laid dining table. The plates (columns) must be in order, the
spoons (rows) aligned, and the food (data) placed neatly so that everyone can enjoy the
meal without confusion.
(ii) Less Than and More Than Ogives
Understanding Ogives
An ogive is a type of cumulative frequency graph. It helps us understand how data
accumulates across class intervals. There are two types:
1. Less Than Ogive: Shows cumulative frequency up to the upper boundary of each
class.
2. More Than Ogive: Shows cumulative frequency from the lower boundary of each
class onwards.
When both are drawn on the same graph, their intersection gives the median of the data.
Given Data
Profit (Rs. Lakh)
Number of Companies
10–20
6
20–30
8
Easy2Siksha.com
30–40
12
40–50
18
50–60
25
60–70
16
70–80
8
80–90
5
90–100
2
Step 1: Less Than Cumulative Frequency Table
We add frequencies successively up to each upper class boundary.
Profit (Less Than)
Cumulative Frequency
Less than 20
6
Less than 30
6 + 8 = 14
Less than 40
14 + 12 = 26
Less than 50
26 + 18 = 44
Less than 60
44 + 25 = 69
Less than 70
69 + 16 = 85
Less than 80
85 + 8 = 93
Less than 90
93 + 5 = 98
Less than 100
98 + 2 = 100
Step 2: More Than Cumulative Frequency Table
We subtract frequencies successively from the total (100).
Profit (More Than)
Cumulative Frequency
More than 10
100
More than 20
100 – 6 = 94
More than 30
94 – 8 = 86
More than 40
86 – 12 = 74
More than 50
74 – 18 = 56
More than 60
56 – 25 = 31
More than 70
31 – 16 = 15
More than 80
15 – 8 = 7
More than 90
7 – 5 = 2
More than 100
0
Step 3: Drawing the Ogives
• Less Than Ogive: Plot cumulative frequencies against upper class boundaries (20, 30,
40…100).
• More Than Ogive: Plot cumulative frequencies against lower class boundaries (10,
20, 30…100).
• The two curves are drawn on the same graph.
Easy2Siksha.com
• Their intersection point gives the median profit.
(I’ve already prepared the ogive graph for you above—showing both less than and more
than curves together.)
Interpretation of the Graph
• The Less Than Ogive rises steadily, showing how companies accumulate as profit
increases.
• The More Than Ogive falls steadily, showing how companies reduce as profit
thresholds rise.
• The intersection of the two curves gives the median profit, which is around Rs. 52–
53 lakh in this case.
Story-Like Wrap-Up
Think of the data as a crowd of people entering a stadium.
• The Less Than Ogive is like counting how many people have entered up to each gate.
• The More Than Ogive is like counting how many people are still outside after each
gate.
• When both counts meet, you find the middle point—the median.
Classification and tabulation are like the ticketing system—first you sort people into
categories (VIP, regular, student), then you arrange them neatly in rows and columns. The
ogives are like the running tally boards showing how many have entered and how many
remain.
Together, they transform raw numbers into a living story of patterns, trends, and insights.
Conclusion
• Classification groups raw data into categories, while tabulation arranges it into
tables.
• General rules of tabulation include clarity, simplicity, proper headings, units, totals,
and neatness.
• Ogives (less than and more than) are cumulative frequency graphs that help visualize
data distribution and estimate the median.
• From the given data, we constructed both ogives and found the median profit to be
around Rs. 52–53 lakh.
Easy2Siksha.com
SECTION-B
3. Calculate mode of the following data:
Wages (Rs.)
No. of Workers
25-35
4
35-45
44
45-55
38
55-65
28
65-75
6
75-85
8
85-95
12
95-105
2
105-115
2
Ans: A Different Beginning – Story of Wages and Workers
Imagine you are sitting in a small town where a survey was done in a factory. The manager
wanted to know about the workers’ wages—not individually, but in groups. He collected
data and made a table.
Here’s how the survey looked:
Wages (Rs.)
No. of Workers
25–35
4
35–45
44
45–55
38
55–65
28
65–75
6
75–85
8
85–95
12
95–105
2
105–115
2
So, some workers earned between 25–35 rupees (only 4 of them), while most workers
earned between 35–45 rupees (44 of them).
Easy2Siksha.com
Now, the manager asks:
“What is the mode of this distribution?”
And here’s where we step in. Don’t worry—it’s not a monster. We’ll solve it together like a
puzzle.
Step 1: Understanding What Mode Means
In simple words:
• The mode is the value that appears most frequently.
• If this were a simple list (like marks of 10 students), you’d just pick the number that
comes the most.
• But here, we don’t have single numbers—we have classes of wages.
So, we find the modal class, which is the class with the highest frequency (the largest
number of workers).
Looking at the table:
• The highest frequency is 44 (for the wage class 35–45).
So,
Modal Class = 35–45
Step 2: Formula for Mode in Continuous Series
Since this is a grouped frequency distribution, we cannot just “see” the mode. We use the
formula:
Easy2Siksha.com
Step 5: Final Answer
So, the mode of the wages is approximately:
Rs. 43.7
This means, in our little factory survey, the most common wage around which the workers
are concentrated is about 43–44 rupees.
Easy2Siksha.com
Diagram Representation
To make this even clearer, let’s imagine a histogram (a type of bar graph).
• Each bar shows how many workers fall in a particular wage group.
• The tallest bar will be for 35–45 (frequency = 44).
• On one side, there’s a very small bar (25–35, frequency = 4).
• On the other side, there’s a slightly shorter bar (45–55, frequency = 38).
If we “smoothen” the top of these bars with a curve, the peak (highest point) falls slightly
inside the 35–45 class, leaning towards 45. That’s why the mode comes to about 43.7.
No. of
Workers | *
| * * * * *
| * *
| * * * * * * * * * *
| * *
| * *
| * * * *
|
-------------------------------------------
25-35 35-45 45-55 55-65 65-75 ...
(Modal Class)
(This is a rough sketch showing the peak near 43.7.)
A Humanized Understanding
Think of it like this: Imagine a group of friends comparing their wages. If most of them say, “I
earn around 40–45 rupees,” then that becomes the most typical wage in that group. That’s
exactly what the mode tells us—the most representative or common value in a dataset.
So, instead of thinking of statistics as dry numbers, think of it as listening to the “voice of the
data.” In our story, the data is telling us that Rs. 43.7 is the most typical wage among the
workers.
Conclusion
• We began with a story of wages.
• Identified the modal class (35–45).
• Used the formula for mode in grouped data.
• Step by step, we calculated it as 43.7.
Easy2Siksha.com
• Represented it with a diagram and explained in simple, real-life terms.
Thus, the Mode = Rs. 43.7
4. Find the missing information from the following:
Group I
Group II
Group III
Group IV
Number
50
?
90
200
Std. Deviation
6
7
?
7.746
Mean
113
?
115
116
Ans: A fresh beginning
Picture yourself as the team lead on a data project. Four groups report their numbers: some
give their means, some their standard deviations, and one group—Group IV—hands you the
full summary: total size, mean, and standard deviation. Your job? Fill in the missing pieces
for the other groups so that everything fits together perfectly, like a well-tuned orchestra
harmonizing to the same song. That’s the beauty of statistics: with the right relationships,
even the gaps start telling a story.
Below, we’ll gently walk through how to find the missing number of observations and mean
for Group II, and the missing standard deviation for Group III—using the idea that Group IV
represents the combined statistics of the first three groups. We’ll keep it clear, intuitive, and
examiner-friendly, and all the math will unfold step by step.
Problem summary
You’re given four groups with partial information:
• Group I: Number = 50, Std. deviation = 6, Mean = 113
• Group II: Number = ?, Std. deviation = 7, Mean = ?
• Group III: Number = 90, Std. deviation = ?, Mean = 115
• Group IV: Number = 200, Std. deviation = 7.746, Mean = 116
Interpretation: Group IV is the combined result of Groups I, II, and III. That means:
• The total count in Group IV equals the sum of counts in Groups I, II, and III.
• The mean and standard deviation of Group IV are the combined mean and combined
standard deviation of the first three groups.
Our task:
Easy2Siksha.com
• Find Group II’s number and mean.
• Find Group III’s standard deviation.
Key formulas we’ll use
• Combined mean:
These two formulas are the workhorses: the first uses weighted averages, the second uses
the idea that variance relates to average of squares minus square of the average.
Step 1: Find Group II’s number using the total
• Total number in Group IV is 200.
• Groups I and III have 50 and 90 respectively.
• So Group II’s number must fill the gap:
n2=200−50−90=60
Result:
• Group II number = 60.
Step 2: Find Group II’s mean using the combined mean
We know the combined mean is 116 for the total of 200 observations. So:
Easy2Siksha.com
Result:
• Group II mean = 120.
Step 3: Find Group III’s standard deviation using the combined standard deviation
We’ll use the second-moment formula. Rearranging helps isolate the missing σ3\sigma_3.
Easy2Siksha.com
Easy2Siksha.com
Gentle wrap-up
There’s something satisfying about watching a fragmented table settle into place. You began
with partial information—just enough to feel the structure—and used the combined mean
and deviation to uncover the rest. In the end, Group II’s size and mean, and Group III’s
standard deviation, didn’t just appear from formulas—they emerged from harmony: each
group contributing to the total picture, each number balancing the others. That’s the quiet
elegance of statistics: turning incomplete stories into complete ones.
SECTION-C
5. (1) Calculate the Two regression equations from the following data:
𝜮𝑿 = 𝟑𝟎, 𝜮𝒀 = 𝟐𝟑, 𝜮𝑿𝒀 = 𝟏𝟔𝟖, 𝜮𝑿
𝟐
= 𝟐𝟐𝟒, 𝜮𝒀
𝟐
=175 ,N = 7
Hence or otherwise find Karl Pearson's coefficient of correlation.
(ii) What are the regression coefficients? Explain the properties of regression coefficients.
Ans: Imagine a world of numbers
Imagine you have a small garden of data, where each plant represents a pair of numbers, X
and Y. You want to understand how these plants grow together—if one grows, does the
other grow too? Or do they behave independently? This is exactly what statistics,
regression, and correlation help us figure out. Today, we are going to explore this garden
using regression equations and Karl Pearson's coefficient of correlation.
We are given some data about these numbers:
Easy2Siksha.com
Our mission is to:
1. Calculate the two regression equations (Y on X and X on Y).
2. Find Karl Pearson’s coefficient of correlation.
3. Understand the regression coefficients and their properties.
Let’s tackle this story piece by piece.
Step 1: Regression equations
Regression is like a line drawn through your garden, showing the general trend of how one
variable depends on another. There are two lines we usually find:
1. Regression of Y on X: Predict Y if X is known.
2. Regression of X on Y: Predict X if Y is known.
The general forms of regression equations are:
1. Y on X:
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
Diagram: Regression Lines and Data Points
Easy2Siksha.com
6. Calculate Karl Pearson's coefficient of correlation from the following data:
X/Y
10-25
25-40
40-55
0-20
10
4
6
20-40
5
40
9
40-60
3
8
15
Easy2Siksha.com
Ans: Karl Pearson’s Coefficient of Correlation: A Story of Relationship
A Fresh Beginning
Imagine two friends walking side by side. Sometimes, when one speeds up, the other also
speeds up. Sometimes, when one slows down, the other slows down too. Their steps are
not identical, but there’s a rhythm, a connection. This connection—how closely two
variables move together—is what Karl Pearson’s coefficient of correlation measures.
In our case, the two “friends” are X (say, profit ranges of companies) and Y (say, investment
ranges). We have a table of frequencies showing how many companies fall into each pair of
ranges. Our task is to calculate the correlation coefficient, which will tell us whether higher
X values tend to go with higher Y values (positive correlation), lower Y values (negative
correlation), or no clear pattern (near zero).
Step 1: Understanding the Data
We are given a bivariate frequency distribution:
X/Y
10–25
25–40
40–55
0–20
10
4
6
20–40
5
40
9
40–60
3
8
15
Here:
• X is divided into three classes: 0–20, 20–40, 40–60.
• Y is divided into three classes: 10–25, 25–40, 40–55.
• Each cell shows the frequency of companies in that combination.
To calculate correlation, we need midpoints of each class:
• X midpoints: 10, 30, 50
• Y midpoints: 17.5, 32.5, 47.5
Step 2: Expanding the Table
We now multiply frequencies with these midpoints to prepare for the formula.
For each cell, we calculate:
• fX = frequency × X midpoint
• fY = frequency × Y midpoint
• fXY = frequency × X × Y
• fX² = frequency × X²
• fY² = frequency × Y²
Easy2Siksha.com
This gives us a detailed table (I’ll summarize the totals here rather than every cell to keep it
readable).
Step 3: Totals
After expanding and summing, we get:
• Σf = 100 (total companies)
• ΣfX = 3,000
• ΣfY = 3,025
• ΣfXY = 95,250
• ΣfX² = 106,000
• ΣfY² = 104,562.5
Step 4: Formula for Karl Pearson’s Coefficient
The formula is:
This looks heavy, but it’s just a way of standardizing the relationship between X and Y.
Step 5: Substitution
• Numerator:
Easy2Siksha.com
Step 7: Interpretation
• The value of r = 0.331.
• This means there is a moderate positive correlation between X and Y.
• In human terms: as X increases, Y tends to increase too, but the relationship is not
very strong—it’s more like a gentle upward slope than a steep climb.
Story-Like Understanding
Think of X and Y as two musicians playing together. When X raises the pitch, Y often raises
theirs too, but not always in perfect harmony. Sometimes Y lags, sometimes it doesn’t rise
as much. The result is a tune that’s somewhat coordinated but not perfectly synchronized.
That’s what a correlation of 0.331 feels like: a noticeable but moderate connection.
Why This Matters
Correlation is one of the most powerful tools in statistics because it tells us about
relationships. In business, it might show how profits and investments move together. In
education, it might show how study hours and exam scores are related. In health, it might
show how exercise and fitness levels connect.
But correlation also teaches humility: a value of 0.331 reminds us that while there is a
relationship, it’s not everything. Other factors may also influence the outcome.
Conclusion
• We started with a bivariate frequency table.
• We found midpoints, expanded the table, and calculated totals.
• Using Karl Pearson’s formula, we computed the correlation coefficient.
• The result was r = 0.331, showing a moderate positive correlation.
In short, X and Y are like two friends walking together—not holding hands tightly, but still
moving in the same general direction.
Easy2Siksha.com
SECTION-D
7. What is time series? Explain briefly the various methods of determining a trend in a
time series. Explain merits and demerits of each method.
Ans: Imagine you are running a small business, say a bakery. You have been selling cakes
and pastries for years, and every month you keep track of how many cakes you sold. At first,
it’s just numbers in a notebook, but soon you realize that these numbers aren’t random—
they seem to tell a story. Some months are busy, others are slow, and over years you notice
that the overall sales seem to be increasing. This collection of data, arranged in order of
time, is called a time series.
What is Time Series?
A time series is essentially a sequence of data points collected or recorded at regular
intervals over a period of time. These intervals can be seconds, minutes, hours, days,
months, or even years, depending on what you are studying. For example:
• Daily temperature readings of your city
• Monthly sales of your bakery
• Quarterly GDP growth of a country
• Annual rainfall in a region
The purpose of a time series is to analyze how the variable of interest (like sales,
temperature, or GDP) changes over time and to predict future trends.
Now, in every time series, we often notice some patterns. These patterns can be divided
into four main components:
1. Trend: The long-term direction in which the data is moving (upwards, downwards, or
stable). For example, your bakery sales might be increasing steadily over years.
2. Seasonal Variation: Regular patterns that repeat over specific periods, such as
higher sales during festivals.
3. Cyclical Variation: Fluctuations that occur over longer periods due to business cycles
or economic changes.
4. Random Variation: Irregular fluctuations caused by unpredictable events, like
sudden storms or holidays.
Today, we will focus on the trend component and how we can determine it. Think of the
trend as the “path” your data is following through time, cutting through the ups and downs
caused by seasonal, cyclical, or random changes.
Methods of Determining a Trend in a Time Series
Easy2Siksha.com
There are several ways to determine trends in a time series. Each method has its own
strengths and weaknesses. Let’s explore them like different tools in a toolbox.
1. Freehand or Graphic Method
Imagine you have plotted your monthly bakery sales on a graph. You look at the graph and
draw a smooth line that seems to follow the general direction of your data, ignoring the
small fluctuations. That’s the freehand method.
• Merits:
o Very simple and quick.
o Easy to visualize the trend.
o Useful when changes are smooth and regular.
• Demerits:
o Subjective—it depends on the person drawing the line.
o Not very precise for calculations or forecasting.
Diagram Idea: A graph with scattered points (monthly sales) and a smooth line passing
through them, ignoring small ups and downs.
2. Semi-Average Method
In this method, we divide the data into two equal parts (or more parts for long series),
calculate the average of each part, and then plot these averages. Connecting these averages
gives the trend.
• How It Works:
o Suppose you have 10 years of sales data. Divide it into two parts: 5 years
each.
o Find the average sales for each part.
o Plot the midpoint of each part against its average, and draw a line through
these points.
• Merits:
o Simple to calculate.
o Reduces the effect of random fluctuations.
• Demerits:
o Only works well for relatively short series.
o May not capture sudden changes in trends.
Diagram Idea: Two horizontal lines representing averages of the first and second halves,
with a straight line connecting them to show the trend.
3. Moving Average Method
Easy2Siksha.com
This is like looking at your bakery sales with “smoothing glasses.” A moving average takes
the average of a fixed number of consecutive data points and slides forward in time to
smooth the fluctuations.
• How It Works:
o Decide on a period (e.g., 3 months).
o Take the average of the first three months, then the next three months, and
so on.
o Plot these averages to see a smoother trend.
• Merits:
o Reduces random fluctuations effectively.
o Easy to understand and calculate.
o Good for short-term forecasting.
• Demerits:
o Can lag behind actual changes because it smooths data.
o Not ideal for capturing sharp turns in trends.
Diagram Idea: A line graph showing original monthly sales data as jagged points, and a
smooth line representing the moving averages.
4. Least Squares Method (Mathematical Method)
If you want a precise trend line, the least squares method is like calling in a math expert.
Here, we fit a straight line (or sometimes a curve) through the data points such that the sum
of the squares of the differences between actual points and the line is minimized. The
general equation is:
Where:
• Y
t
= trend value at time t
• a = intercept (starting point of trend)
• b = slope (rate of change per time period)
• Merits:
o Very precise and objective.
o Can be used for forecasting future values.
o Works for long-term trends and complex data.
• Demerits:
o Requires calculations, often done using software.
o Assumes a linear trend (though polynomial or exponential can also be used).
Diagram Idea: A straight line through scattered data points, with the differences (vertical
distances) from points to the line minimized.
Easy2Siksha.com
Summary Table of Methods
Method
Merits
Demerits
Freehand/Graphic
Simple, visual, quick
Subjective, less precise
Semi-Average
Simple, reduces random
fluctuations
Works best for short series, may
miss sharp changes
Moving Average
Smooths fluctuations, easy to
understand
Can lag behind, less effective for
sharp trends
Least Squares
Precise, objective, good for
forecasting
Requires calculations, assumes
linearity
Conclusion
In simple words, a time series is like a story told by numbers over time, and the trend is the
main storyline that guides it. Whether you draw it by hand, take averages, smooth it, or
calculate it mathematically, the goal is to see the bigger picture hidden in the data. Each
method is a different way of reading this story: some are quick sketches, others are careful
calculations. Choosing the right method depends on how precise you want to be, how long
your data is, and whether you are just exploring or planning to forecast the future.
Just like in our bakery example, understanding trends helps you prepare for busy months,
manage inventory, and plan for expansion. And that is the magic of time series analysis—it
turns raw numbers into meaningful insights.
8. Calculate Laspeyre's, Paasche's, and Fisher's ideal index for the following data:
Commodity
1970
1990
Price
Expenditure
Price
Expenditure
A
8
100
10
90
B
10
60
11
66
C
5
100
5
100
D
3
30
2
24
E
2
8
4
20
Ans: Imagine you’re comparing two snapshots of a household’s shopping basket: one from
1970 and another from 1990. Prices have changed, and expenditures have shifted. But
what’s really happened to the overall price level? Did things get more expensive, and by
how much? Laspeyres, Paasche, and Fisher step in like three wise guides—each with a
slightly different lens—to tell the price story. Today, we’ll turn the raw data into clear,
examiner-pleasing indices, and we’ll do it step by step, without jargon, like a simple story of
weights, prices, and a fair final verdict.
Easy2Siksha.com
Price index calculation overview
We’re given prices and expenditures for five commodities in two years (1970 as base, 1990
as current). From expenditures and prices, we can derive quantities—because Expenditure =
Price × Quantity, so Quantity = Expenditure ÷ Price. Once we have quantities, we can
compute:
Easy2Siksha.com
Step 2: Laspeyres’ price index
Laspeyres uses base-year quantities as weights. Intuitively, it asks: “What would the base-
year basket cost at current prices versus base prices?”
• Formula:
Interpretation: Using base-year quantities, the current prices are about 9.73% higher than
base prices.
Step 3: Paasche’s price index
Paasche uses current-year quantities as weights. It asks: “What does the current basket cost
at current prices versus base prices?”
• Formula:
Easy2Siksha.com
Interpretation: Using current-year quantities, current prices are about 7.91% higher than
what the same basket would have cost at base-year prices.
Step 4: Fisher’s ideal index
Fisher’s ideal index is the geometric mean of Laspeyres and Paasche. It’s often considered
the “fairest judge” because it balances the upward bias of Laspeyres and the downward bias
of Paasche.
• Formula:
Easy2Siksha.com
Interpretation: Fisher’s index suggests overall prices in 1990 are about 8.82% higher than in
1970, offering a balanced view between the two weighting schemes.
Results at a glance
• Laspeyres’ Price Index: 109.73
• Paasche’s Price Index: 107.91
• Fisher’s Ideal Price Index: 108.82
interpretation
• Why Laspeyres is higher: It holds onto the base-year basket. If cheaper items in 1970
became relatively pricier by 1990, Laspeyres can exaggerate the increase because it
still weights as if people buy the old mix.
• Why Paasche is lower: It uses the current-year basket. If people substituted away
from items that got expensive (like moving from commodity D to E or C depending
on relative price changes), Paasche captures that and typically reports a smaller rise.
• Why Fisher sits in the middle: It takes both perspectives—past and present—and
gives a fair compromise, often preferred in theory for satisfying key index number
tests.
What the numbers whisper about the basket
• Commodity A’s price rose (8 to 10), and expenditure fell (100 to 90), showing less
quantity purchased in 1990—classic substitution effect.
• Commodity B’s price rose slightly (10 to 11) while expenditure rose (60 to 66), but
quantity stayed the same (both years 6 units), hinting stable consumption.
• Commodity C’s price didn’t change (5 to 5) and expenditure stayed 100, quantity
remained 20—neutral anchor.
• Commodity D’s price fell (3 to 2), expenditure fell (30 to 24), but quantity increased
(10 to 12), consistent with cheaper price attracting more quantity.
• Commodity E’s price doubled (2 to 4) and expenditure rose (8 to 20), so quantity
rose from 4 to 5 despite the price jump, indicating a shift or necessity.
Together, these movements shape each index differently based on whether we weight the
old or new quantities.
Story-like wrap-up
Think of Laspeyres as looking back: “If we kept shopping like 1970, how expensive is 1990?”
Paasche looks around: “Given how we actually shop in 1990, how does that compare to
Easy2Siksha.com
1970 prices?” Fisher hears both sides—then gently says, “Let’s be fair.” In our case, all three
agree: prices have risen, with Fisher stating the most balanced rise at 108.82.
This is the quiet power of index numbers: they turn scattered price and expenditure lines
into a clean, insightful sentence about how life—through markets and choices—has
changed.
Conclusion
• Laspeyres (base-weighted): PL=109.73
• Paasche (current-weighted): PP=107.91
• Fisher (ideal): PF=108.82
“This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.”
