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GNDU QUESTION PAPERS 2021
BA/BSc 6
th
SEMESTER
ECONOMICS
(Quantave Methods for Economists)
Time Allowed: 3 Hours Maximum Marks: 100
Note: Aempt Five quesons in all, selecng at least One queson from each secon. The
Fih queson may be aempted from any secon. All quesons carry equal marks.
SECTION – A
1.Explain the following with the help of hypothecal examples:
(a) Adjoint of a matrix
(b) Transpose of a matrix
(c) Product of two matrices
(d) Symmetric matrix
(6¼ × 4 = 25)
2.(a) If
nd
(b) Dierenate:
w.r.t.
(c) What are the steps to nd maxima and minima?
Find maximum and minimum values of the funcon:
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(6 + 6 + 13 = 25)
SECTION – B
3.(a) Dene arithmec mean. Discuss its merits and demerits. Also state its important
properes.
(b) Calculate geometric mean of the following distribuon:
x
2
3
4
5
6
7
8
f
2
4
6
2
3
2
1
(18 + 7 = 25)
4.(a) Calculate mean deviaon from mean for the following data:
Marks
0–10
10–20
20–30
30–40
40–50
50–60
60–70
No. of Students
5
6
9
15
7
6
4
(b) Compare mean deviaon and standard deviaon as measures of variaon.
Which of the two is a beer measure and why?
SECTION – C
5.(a) Explain the concept of skewness. Also show graphically the posions of mean,
median, and mode in posively and negavely skewed distribuons.
(b) Calculate Karl Pearson’s coecient of skewness for the following data:
Measurement
10
11
12
13
14
15
Frequency
3
5
10
8
5
1
6.(a) Explain the meaning and signicance of correlaon. Explain the types of correlaon.
Also discuss various methods of calculang correlaon.
(b) Obtain the two lines of regression from the following data:
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X
-4
5
6
7
9
12
Y
2
3
4
6
5
8
SECTION – D
7.What are index numbers? Disnguish between weighted and unweighted index
numbers.
Also explain various tests of consistency of index numbers.
8.From the following data, esmate the expectaon of life at age 33
(using Newton’s method of interpolaon):
Age (years)
10
15
20
25
30
35
Expectaon of life (years)
35.3
32.4
29.2
26.1
23.2
20.5
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GNDU ANSWER PAPERS 2021
BA/BSc 6
th
SEMESTER
ECONOMICS
(Quantave Methods for Economists)
Time Allowed: 3 Hours Maximum Marks: 100
Note: Aempt Five quesons in all, selecng at least One queson from each secon. The
Fih queson may be aempted from any secon. All quesons carry equal marks.
SECTION – A
1.Explain the following with the help of hypothecal examples:
(a) Adjoint of a matrix
(b) Transpose of a matrix
(c) Product of two matrices
(d) Symmetric matrix
Ans: Matrices are one of the most important concepts in mathematics, especially in algebra.
At first glance, they may look like a collection of numbers arranged in rows and columns, but
in reality, they are powerful tools used in engineering, physics, computer graphics,
economics, and even mobile applications. To truly understand matrices, we must become
familiar with some basic operations and types, such as the adjoint, transpose, product of
matrices, and symmetric matrices.
Let us explore each of these concepts in a simple, story-like manner with hypothetical
examples so that the ideas feel natural and easy to understand.
(a) Adjoint of a Matrix
Imagine a school organizing a quiz competition. The teacher prepares a table showing the
marks given by three judges to three participants. The matrix looks like this:
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Now suppose the teacher wants to perform deeper calculations using this matrix — perhaps
to find its inverse. To do that, we first need something called the adjoint of the matrix.
What is the Adjoint?
The adjoint (also called adjugate) of a matrix is obtained by following two main steps:
1. Find the cofactor of each element of the matrix.
2. Take the transpose of the cofactor matrix.
In simple terms, you can think of the adjoint as a “helper matrix” that makes advanced
calculations possible.
Hypothetical Example
Suppose we take a smaller matrix to make the process easier:
Step 1: Find cofactors
For a 2×2 matrix:
• Cofactor of 1 → 4
• Cofactor of 2 → −3
• Cofactor of 3 → −2
• Cofactor of 4 → 1
So the cofactor matrix becomes:
Step 2: Transpose it
After swapping rows and columns:
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Simple Understanding
Think of the adjoint like a mirror-adjusted assistant that reorganizes the matrix so we can
unlock more complex operations such as finding inverses. Without the adjoint, many matrix
problems would be difficult to solve.
(b) Transpose of a Matrix
Let us imagine a cinema hall where seats are arranged in rows and columns. Suppose the
seating chart is:
Here, rows represent seat rows, and columns represent seat numbers.
Now suppose the manager wants the chart flipped so that rows become columns — maybe
for a different viewing perspective.
What is a Transpose?
The transpose of a matrix is formed by converting rows into columns and columns into
rows.
It is usually written as
.
Example
Original matrix:
Transpose:
Everyday Analogy
Imagine taking a photograph and rotating it sideways. Nothing changes except the
orientation. Similarly, the numbers remain the same — only their positions change.
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Why is Transpose Important?
• Used in computer graphics
• Helps in solving linear equations
• Important in data science and statistics
So, transpose is basically a neat rearrangement that often simplifies mathematical work.
(c) Product of Two Matrices
Now let us step into a small business scenario.
Suppose a stationery shop sells notebooks and pens across two branches. The number of
items sold is represented by matrix , and the prices are represented by matrix .
(2 notebooks and 3 pens sold)
(Notebook costs ₹50, pen costs ₹10)
To find the total money earned, we multiply the matrices.
Rule for Matrix Multiplication
Matrix multiplication is possible only when:
Number of columns in the first matrix = Number of rows in the second matrix.
Calculation
So the shop earned ₹130.
Important Note
Matrix multiplication is not commutative, meaning:
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This is unlike normal multiplication where .
Simple Analogy
Think of matrix multiplication like matching locks and keys. If the shapes don’t align, the
multiplication cannot happen.
( (d) Symmetric Matrix**
Finally, let us imagine a friendship chart in a classroom.
If Rahul is friends with Aman, Aman is also friends with Rahul. The relationship works both
ways.
This type of balanced relationship is exactly what a symmetric matrix represents.
Definition
A matrix is symmetric if:
In other words, the matrix remains unchanged even after taking the transpose.
Example
If we transpose it, we get the same matrix.
Why?
Because:
• Element (1,2) = Element (2,1)
• Element (1,3) = Element (3,1)
• Element (2,3) = Element (3,2)
Everything mirrors perfectly across the diagonal.
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Real-Life Interpretation
Symmetric matrices are widely used in:
• Physics
• Machine learning
• Network relationships
• Graph theory
They often represent systems where interactions are mutual.
Conclusion
Matrices may initially seem intimidating, but once we break them into smaller ideas, they
become much easier to understand.
• The adjoint acts like a helper matrix that allows us to perform advanced calculations.
• The transpose simply rearranges rows and columns for better mathematical
handling.
• The product of matrices helps us combine information, just like calculating total
sales.
• A symmetric matrix represents balance and mutual relationships, staying unchanged
even after transposition.
By connecting matrices with real-life examples — classrooms, shops, friendships, and
seating charts — we realize that mathematics is not just about numbers. It is a language
used to organize and interpret the world around us.
2.(a) If
nd
(b) Dierenate:
w.r.t.
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(c) What are the steps to nd maxima and minima?
Find maximum and minimum values of the funcon:
Ans: This problem has three parts: two straightforward differentiation tasks and one
application of calculus to find maxima and minima. Let’s go through each carefully, in a way
that feels natural and easy to follow.
Part (a) Differentiate
Step 1: Apply the power rule The power rule says:
Step 2: Differentiate each term
•
•
•
•
•
Final Answer:
Part (b) Differentiate
Step 1: Expand the expression First multiply
:
Now multiply by :
Step 2: Differentiate term by term
•
•
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•
Final Answer:
Part (c) Maxima and Minima
We are asked to find the maximum and minimum values of:
Step 1: Differentiate to find critical points
Step 2: Solve
Divide through by 6:
Factorize:
So, critical points are:
Step 3: Second derivative test
• At :
negative → maximum
• At :
positive → minimum
Step 4: Find values of the function
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• At :
So, maximum value = -3 at x = 1.
• At :
So, minimum value = -128 at x = 6.
Steps to Find Maxima and Minima (General Method)
1. Differentiate the function:
.
2. Solve
to find critical points.
3. Use the second derivative test:
o If
, the point is a minimum.
o If
, the point is a maximum.
4. Substitute values back into the original function to find the actual maximum or
minimum values.
Summary
• (a)
• (b)
• (c) For
:
o Maximum value = at
o Minimum value = at
In simple words: Differentiation helps us find slopes, and maxima-minima analysis tells
us where a curve reaches its highest and lowest points. It’s like reading the “story” of a
function—where it rises, where it falls, and where it peaks.
SECTION – B
3.(a) Dene arithmec mean. Discuss its merits and demerits. Also state its important
properes.
(b) Calculate geometric mean of the following distribuon:
x
2
3
4
5
6
7
8
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f
2
4
6
2
3
2
1
Ans: (a) Arithmetic Mean
Definition of Arithmetic Mean
The Arithmetic Mean (A.M.), commonly called the average, is the sum of all observations
divided by the total number of observations.
In simple words:
Arithmetic Mean = Total of all values ÷ Number of values
Formula (for individual data):
Where:
•
= Arithmetic Mean
• = Sum of all observations
• = Total number of observations
For a frequency distribution:
Where:
• = Frequency
• = Product of frequency and value
Example:
If five students scored 40, 50, 60, 70, and 80 marks:
Sum = 300
Number of students = 5
Arithmetic Mean = 300 ÷ 5 = 60
So, we can say the average marks of the class are 60.
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Merits (Advantages) of Arithmetic Mean
Arithmetic Mean is one of the most widely used measures in statistics because of its
simplicity and usefulness.
1. Simple to Understand and Easy to Calculate
Even a beginner can calculate it without difficulty. You just need addition and division.
2. Uses All Observations
Unlike some other averages, arithmetic mean considers every value in the dataset.
Therefore, it gives a balanced representation.
3. Rigidly Defined
Its value is fixed and does not change based on personal judgment. Anyone calculating it
correctly will get the same answer.
4. Useful for Further Mathematical Calculations
Arithmetic Mean is very important in advanced statistical methods such as variance,
standard deviation, and regression.
5. Helps in Comparison
It makes comparison between different groups easier. For example, comparing average
income of two cities.
Demerits (Disadvantages) of Arithmetic Mean
Although very useful, arithmetic mean is not perfect.
1. Affected by Extreme Values
If one value is extremely high or low, it can distort the average.
Example:
Marks = 40, 45, 50, 55, 100
The average becomes higher because of one unusually high score.
2. Not Suitable for Highly Skewed Data
When data is unevenly distributed, the arithmetic mean may not represent the true picture.
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3. Cannot Be Used for Qualitative Data
You cannot find the mean of qualities like beauty, honesty, or intelligence.
4. May Give Unrealistic Values
Sometimes the mean may not actually exist in the dataset.
Example: Average number of children = 2.4
But no family has 2.4 children!
Important Properties of Arithmetic Mean
These properties make arithmetic mean very powerful in statistics.
1. Sum of Deviations is Zero
The total of deviations from the mean is always zero.
This means positive and negative differences cancel each other.
2. Least Squares Property
The sum of the squares of deviations from the mean is minimum compared to any other
value.
This is why arithmetic mean is considered the most reliable average.
3. Linear Transformation
If each value is increased, decreased, multiplied, or divided by a constant, the mean changes
accordingly.
If every student gets 5 extra marks, the mean also increases by 5.
4. Combined Mean Can Be Calculated
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If two groups are merged, we can easily find their combined average.
Now let us move to the second part of the question.
(b) Calculation of Geometric Mean
What is Geometric Mean?
The Geometric Mean (G.M.) is another type of average that is especially useful when
dealing with:
• Growth rates
• Population increase
• Financial returns
• Ratios and percentages
Instead of adding values, we multiply them and then take the root.
Formula for frequency distribution:
Antilog
Step 1: Create the Table
x
f
log x
f log x
2
2
0.3010
0.6020
3
4
0.4771
1.9084
4
6
0.6021
3.6126
5
2
0.6990
1.3980
6
3
0.7782
2.3346
7
2
0.8451
1.6902
8
1
0.9031
0.9031
Step 2: Find the Totals
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Step 3: Apply the Formula
Antilog
Antilog
Looking at the antilog table:
Final Answer
Geometric Mean = 4.19 (approximately)
Conclusion
Arithmetic Mean is the simplest and most commonly used measure of central tendency. It is
easy to compute, uses all observations, and forms the foundation for many statistical
calculations. However, it is sensitive to extreme values and may sometimes give misleading
results.
On the other hand, Geometric Mean is ideal when dealing with growth, percentages, and
multiplicative data. By carefully applying logarithms and formulas, we calculated the
geometric mean of the given distribution as 4.19.
4.(a) Calculate mean deviaon from mean for the following data:
Marks
0–10
10–20
20–30
30–40
40–50
50–60
60–70
No. of Students
5
6
9
15
7
6
4
(b) Compare mean deviaon and standard deviaon as measures of variaon.
Which of the two is a beer measure and why?
Ans: Part (a) Calculate Mean Deviation from Mean
We are given the data:
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Marks
0–10
10–20
20–30
30–40
40–50
50–60
60–70
No. of Students
5
6
9
15
7
6
4
Step 1: Find class midpoints
Midpoints are the average of class limits:
• 0–10 → 5
• 10–20 → 15
• 20–30 → 25
• 30–40 → 35
• 40–50 → 45
• 50–60 → 55
• 60–70 → 65
Step 2: Multiply frequency (f) by midpoint (x)
Class
Midpoint (x)
f
fx
0–10
5
5
25
10–20
15
6
90
20–30
25
9
225
30–40
35
15
525
40–50
45
7
315
50–60
55
6
330
60–70
65
4
260
Step 3: Calculate mean
So, mean ≈ 34.04
Step 4: Find deviations from mean
x
5
15
25
35
45
55
65
f
5
6
9
15
7
6
4
\
x – mean\
29.04
19.04
9.04
0.96
10.96
20.96
30.96
f·\
x – mean\
145.20
114.24
81.36
14.40
76.72
125.76
123.84
Step 5: Mean deviation from mean
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Final Answer: Mean deviation from mean ≈ 13.11
Part (b) Compare Mean Deviation and Standard Deviation
Mean Deviation
• Definition: Average of absolute deviations from mean (or median).
• Simpler to calculate.
• Less sensitive to extreme values.
• Sometimes less precise because it ignores the direction of deviations.
Standard Deviation
• Definition: Square root of average of squared deviations from mean.
• More mathematically rigorous.
• Sensitive to extreme values (outliers).
• Widely used in statistics, economics, and sciences.
• Forms the basis of variance analysis, probability distributions, and inferential
statistics.
Which is Better?
• Standard deviation is considered a better measure because:
o It uses squared deviations, giving more weight to larger differences.
o It is mathematically compatible with advanced statistical methods.
o It connects directly to concepts like normal distribution and variance.
• Mean deviation is useful for simpler, descriptive analysis, but standard deviation is
the preferred measure in research and applied statistics.
Summary
• Mean deviation from mean for given data = 13.11
• Comparison:
o Mean deviation → simple, less sensitive, descriptive.
o Standard deviation → precise, widely used, mathematically powerful.
• Better measure: Standard deviation, because it captures variation more effectively
and is essential for advanced statistical analysis.
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SECTION – C
5.(a) Explain the concept of skewness. Also show graphically the posions of mean,
median, and mode in posively and negavely skewed distribuons.
(b) Calculate Karl Pearson’s coecient of skewness for the following data:
Measurement
10
11
12
13
14
15
Frequency
3
5
10
8
5
1
Ans: (a) Concept of Skewness
Imagine you are observing the marks of students in a class.
• If most students score average marks and only a few score very high or very low, the
data looks balanced.
• But if many students score low marks and only a few score high marks, the data
stretches toward the higher side.
• Similarly, if many students score high marks and only a few score low marks, the
data stretches toward the lower side.
This stretching or asymmetry is called skewness.
Definition:
Skewness is the measure of the degree and direction of asymmetry in a frequency
distribution.
In simple words, skewness tells us whether data is tilted to the left or right.
There are three main types of skewness:
1. Symmetrical Distribution
2. Positively Skewed Distribution
3. Negatively Skewed Distribution
Let us focus mainly on positive and negative skewness since the question asks about them.
Positively Skewed Distribution (Right-Skewed)
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A distribution is called positively skewed when the tail on the right side is longer.
Why does this happen?
This usually occurs when:
• Most values are small.
• A few extremely large values pull the average upward.
For example, consider income data in a city:
• Many people earn moderate salaries.
• A few people earn extremely high incomes.
These high incomes stretch the distribution toward the right.
Position of Mean, Median, and Mode
In a positively skewed distribution:
Mode < Median < Mean
Why?
• Mode represents the most frequent value — located near the peak.
• Median lies in the middle.
• Mean gets pulled toward the extreme high values.
So, the mean is always the largest in positive skewness.
Negatively Skewed Distribution (Left-Skewed)
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A distribution is called negatively skewed when the tail extends toward the left side.
Why does this happen?
This occurs when:
• Most values are high.
• A few extremely low values drag the average downward.
For example:
If an exam is very easy, most students score high marks, but a few score very low. Those low
scores stretch the curve toward the left.
Position of Mean, Median, and Mode
In negatively skewed data:
Mean < Median < Mode
Reason:
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• The low extreme values pull the mean downward.
• The mode remains near the highest peak.
So, here the mean is the smallest.
Quick Comparison Table
Type of Distribution
Tail Direction
Relationship
Positive Skewness
Right
Mode < Median < Mean
Negative Skewness
Left
Mean < Median < Mode
(b) Karl Pearson’s Coefficient of Skewness
Now let us move toward the numerical part.
Formula:
Karl Pearson’s coefficient of skewness is:
However, if the mode is not clearly defined, we use:
But in this question, the mode is clearly visible, so we will use the first formula.
Given Data
Measurement
Frequency
10
3
11
5
12
10
13
8
14
5
15
1
Step 1: Calculate Mean
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We use:
Let us calculate :
X
f
fX
10
3
30
11
5
55
12
10
120
13
8
104
14
5
70
15
1
15
Now sum them:
Mean:
Step 2: Find the Mode
The value with the highest frequency is the mode.
Here:
Frequency 10 is highest
Corresponding measurement = 12
So,
Step 3: Calculate Standard Deviation
We use the shortcut formula:
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After calculating deviations and squaring them (step-by-step calculation), we get
approximately:
Step 4: Karl Pearson’s Skewness
Final Answer
Karl Pearson’s Coefficient of Skewness = +0.23
Interpretation
Since the skewness is positive, we conclude:
The distribution is slightly positively skewed.
This means:
• Most measurements cluster around the lower-middle values.
• A few higher values stretch the distribution slightly toward the right.
• However, because the value is close to zero, the distribution is almost symmetrical.
Conclusion
Skewness is an essential statistical concept because it helps us understand the shape and
behavior of data rather than just its numerical values. A positively skewed distribution
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shows that extreme high values are influencing the average, while a negatively skewed
distribution indicates the presence of extreme low values.
Understanding the positions of mean, median, and mode makes it easier to visually
interpret the data and quickly identify the direction of skewness.
In the numerical example, we carefully calculated the mean, mode, and standard deviation
before applying Karl Pearson’s formula. The result, +0.23, tells us that the dataset is only
slightly skewed, meaning it is fairly balanced.
6.(a) Explain the meaning and signicance of correlaon. Explain the types of correlaon.
Also discuss various methods of calculang correlaon.
(b) Obtain the two lines of regression from the following data:
X
-4
5
6
7
9
12
Y
2
3
4
6
5
8
Ans: Meaning of Correlation
Correlation is a statistical tool used to measure the degree of relationship between two
variables. If one variable changes, correlation helps us understand how the other variable is
likely to change. For example, the relationship between hours studied and exam marks, or
between rainfall and crop yield.
In simple words: Correlation tells us how two things move together—whether they rise
and fall in sync, or move in opposite directions.
Significance of Correlation
• Prediction: Helps predict one variable based on another (e.g., predicting sales from
advertising expenditure).
• Decision-making: Useful in economics, business, and social sciences for planning and
policy.
• Understanding Relationships: Shows whether variables are connected strongly,
weakly, or not at all.
• Research: Provides insights into cause-effect relationships (though correlation itself
does not prove causation).
Types of Correlation
1. Positive Correlation: Both variables move in the same direction.
o Example: More hours of study → higher marks.
2. Negative Correlation: Variables move in opposite directions.
o Example: More price → less demand.
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3. Zero Correlation: No relationship between variables.
o Example: Shoe size and intelligence.
4. Perfect Correlation: Change in one variable is exactly matched by change in another
(correlation coefficient = +1 or -1).
5. Partial Correlation: Relationship between two variables while controlling the effect
of others.
Methods of Calculating Correlation
1. Scatter Diagram Method: Plot values on a graph; the closeness of points to a straight
line shows correlation.
2. Karl Pearson’s Coefficient of Correlation (r):
o Measures strength and direction of linear relationship.
o Formula:
3. Spearman’s Rank Correlation:
o Used when data is in ranks.
o Formula:
where = difference in ranks.
4. Concurrent Deviation Method:
o Based on direction of deviations from mean.
o Simpler but less precise.
(b) Regression Lines for Given Data
We are given:
X
-4
5
6
7
9
12
Y
2
3
4
6
5
8
Regression lines show the relationship between X and Y. There are two lines:
• Regression of Y on X (predicting Y from X).
• Regression of X on Y (predicting X from Y).
Step 1: Calculate Means
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Step 2: Calculate Deviations and Products
X
Y
X - X
Y - Ȳ
(X - X)(Y - Ȳ)
(X - X)^2
(Y - Ȳ)^2
-4
2
-9.83
-2.67
26.26
96.6
7.13
5
3
-0.83
-1.67
1.39
0.69
2.79
6
4
0.17
-0.67
-0.11
0.03
0.45
7
6
1.17
1.33
1.56
1.37
1.77
9
5
3.17
0.33
1.05
10.05
0.11
12
8
6.17
3.33
20.56
38.07
11.09
Step 3: Regression Coefficients
• Regression of Y on X:
• Regression of X on Y:
Step 4: Regression Equations
• Regression of Y on X:
• Regression of X on Y:
Summary
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• Correlation: Measures the relationship between two variables.
• Types: Positive, negative, zero, perfect, partial.
• Methods: Scatter diagram, Pearson’s coefficient, Spearman’s rank, concurrent
deviation.
• Regression Lines for Given Data:
o Regression of Y on X:
o Regression of X on Y:
SECTION – D
7.What are index numbers? Disnguish between weighted and unweighted index
numbers.
Also explain various tests of consistency of index numbers.
Ans: Meaning of Index Numbers
An index number is a statistical measure that shows how much a variable (such as price,
quantity, or value) has changed over time compared to a fixed reference point called the
base year.
Think of the base year as a starting line in a race. We usually assign it a value of 100. Any
movement above or below 100 tells us whether there has been an increase or decrease.
For example:
• If the price index today is 120, it means prices have increased by 20% compared to
the base year.
• If it is 90, prices have decreased by 10%.
In everyday life, index numbers help governments track inflation, businesses plan
production, and individuals understand changes in the cost of living.
Key Features of Index Numbers
1. Specialized Average:
Index numbers are not ordinary averages. They combine data from multiple items to
give a single, clear picture of change.
2. Comparison Tool:
They help compare economic conditions across different time periods or locations.
3. Easy Interpretation:
Since the base year equals 100, it becomes simple to understand the percentage
change.
4. Wide Application:
Index numbers are used in measuring inflation, wage adjustments, stock market
trends, and industrial growth.
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So, in simple terms, an index number is like a thermometer for the economy—it tells us
whether things are getting “hotter” (more expensive) or “cooler” (cheaper).
Weighted vs. Unweighted Index Numbers
Now that we understand index numbers, let us explore two major types: unweighted and
weighted index numbers.
Unweighted Index Numbers
An unweighted index number treats all items equally, regardless of their importance.
Imagine you are calculating the average marks of a student who scored:
• English – 95
• Mathematics – 95
• Music – 95
Here, each subject carries equal weight, even though Mathematics might be considered
more important academically.
Similarly, in an unweighted price index, the price of salt is given the same importance as the
price of a car—which is not very realistic.
Characteristics:
• Simple to calculate
• Requires less data
• Suitable when items are of similar importance
• Less accurate in real-life economic analysis
Example:
If the prices of five commodities increase, we simply average the percentage change
without considering how much people actually spend on each item.
This method is easy but sometimes misleading.
Weighted Index Numbers
A weighted index number assigns importance (or weight) to each item based on its
relevance.
Let us take a household example. Suppose a family spends:
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• ₹5000 on rent
• ₹3000 on food
• ₹500 on stationery
If stationery prices rise, the overall cost of living will not increase much because the
spending on it is low. But if rent increases, the impact is huge.
Weighted index numbers capture this reality.
Characteristics:
• More realistic and accurate
• Reflect actual consumption patterns
• Widely used in economics
• Slightly complex to calculate
Common Examples:
• Consumer Price Index (CPI)
• Wholesale Price Index (WPI)
These indices help governments design policies and adjust salaries.
Difference Between Weighted and Unweighted Index Numbers
Basis
Unweighted Index
Weighted Index
Importance
All items treated equally
Items assigned importance
Accuracy
Less accurate
More accurate
Complexity
Easy to calculate
More complex
Practical Use
Limited
Widely used in economics
Realism
Ignores spending patterns
Reflects real-life consumption
In short:
Unweighted index numbers are simple but less reliable, while weighted index numbers
are detailed and dependable.
Tests of Consistency of Index Numbers
Creating an index number is not enough—it must also be reliable and scientifically sound.
Economists use certain tests to check whether an index number is consistent.
Let us understand the three major tests.
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1. Unit Test
This test states that the index number should not be affected by the units of measurement.
For example:
• Sugar measured in kilograms
• Milk measured in liters
If we convert kilograms into grams, the index number should remain unchanged.
A good index number is independent of measurement units.
2. Time Reversal Test
This test ensures logical consistency when time periods are reversed.
The rule is simple:
If an index measures the change from year A to year B, then reversing the comparison (from
B to A) should give the reciprocal result.
Formula Idea:
If
, the test is satisfied.
Example:
If prices doubled from 2020 to 2024, then moving backward should show prices becoming
half.
This test checks mathematical correctness.
3. Factor Reversal Test
The factor reversal test states that when we multiply the price index by the quantity index,
the result should equal the value index.
In simpler words:
Price Change × Quantity Change = Total Value Change
This ensures that the index reflects the real relationship between price and quantity.
Why is it important?
Because both price and quantity together determine total expenditure.
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If an index passes this test, it is considered highly reliable.
Why Are These Tests Important?
Without consistency tests, index numbers could produce misleading results. Imagine a
government increasing salaries based on a faulty inflation index—this could harm the
economy.
These tests ensure that index numbers are:
• Logical
• Accurate
• Scientific
• Trustworthy
Among various formulas, Fisher’s Ideal Index is famous because it satisfies both the time
reversal and factor reversal tests, making it one of the best methods.
Conclusion
Index numbers play a crucial role in understanding economic changes. They simplify
complex data and present it in a form that anyone can interpret easily.
To summarize:
• Index numbers measure changes in variables like prices and quantities over time.
• Unweighted index numbers are simple but ignore the importance of items.
• Weighted index numbers are more realistic because they consider consumption
patterns.
• Tests of consistency—Unit Test, Time Reversal Test, and Factor Reversal Test—
ensure that index numbers are reliable and scientifically valid.
In today’s rapidly changing economy, index numbers act like a compass, guiding
governments, businesses, and consumers toward better decisions.
8.From the following data, esmate the expectaon of life at age 33
(using Newton’s method of interpolaon):
Age (years)
10
15
20
25
30
35
Expectaon of life (years)
35.3
32.4
29.2
26.1
23.2
20.5
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Ans: Interpolation is a mathematical technique used to estimate unknown values that lie
between known data points. In this problem, we are asked to estimate the expectation of
life at age 33 using Newton’s forward interpolation method. Let’s go step by step in a clear
and engaging way.
1. The Given Data
Age (years)
10
15
20
25
30
35
Expectation of life (years)
35.3
32.4
29.2
26.1
23.2
20.5
We want to estimate the expectation of life when age = 33.
2. Step 1: Identify the Interval and Step Size
• The data is given at equal intervals of 5 years.
• Step size, .
• We choose the starting point
.
3. Step 2: Construct the Forward Difference Table
We calculate successive differences of the expectation of life values.
Age (x)
y (Expectation)
Δy
Δ²y
Δ³y
Δ⁴y
Δ⁵y
10
35.3
-2.9
0.3
-0.2
0.1
-0.1
15
32.4
-3.2
0.1
-0.1
0.0
20
29.2
-3.1
0.0
-0.1
25
26.1
-2.9
-0.1
30
23.2
-2.7
35
20.5
4. Step 3: Newton’s Forward Interpolation Formula
The formula is:
Where:
•
= first value (at age 10 → 35.3)
•
Here, ,
, .
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5. Step 4: Substitute Values
•
•
•
•
•
•
Now substitute step by step:
6. Step 5: Simplify Each Term
• First term:
• Second term:
• Third term:
• Fourth term:
• Fifth term:
• Sixth term:
7. Step 6: Add Them Up
Final Answer
The expectation of life at age 33 is approximately:
years
Explanation in Simple Words
Newton’s interpolation method is like filling in the missing pieces of a puzzle. We know the
expectation of life at ages 30 and 35, but we want to estimate it at age 33. By using
differences between known values and applying Newton’s formula, we can predict the value
in between.
Here, the calculation shows that at age 33, the expectation of life is about 22.37 years. This
makes sense because it lies between the given values at ages 30 (23.2 years) and 35 (20.5
years).
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“This paper has been carefully prepared for educaonal purposes. If you noce any
mistakes or have suggesons, feel free to share your feedback.”
