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GNDU Question Paper-2021

B.A 2

Semester

QUANTITATIVE TECHNIQUES – II

Time Allowed: Two Hours Maximum Marks: 100

Note: There are Eight questions of equal marks. Candidates are required to attempt any

Four questions

SECTION-A

1. Explain scope of statistics in Economics. What do you understand by data classification and

tabulation.

2. Explain various types of graphic representation of data like pie charts. bar, histogram, polygon

and ogive curves etc. with the help of examples.

SECTION-B

3. (i) Explain various measures of central tendency.

(ii) Find the missing frequency, if the arithmetic mean is 28 of the data given below. Find the

median of the series later

Profit in Rs.000

0-10

10-20

20-30

30-40

40-50

50-60

No. of shops

_____

4. (i) What is dispersion? Discuss various measures of dispersion.

(ii) A consignment of articles is classified according to the size of the articles as below. Find

standard deviation and coefficient of variation.

Measurement

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

Frequency

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SECTION-C

5. (i) Explain the significance of correlation. Explain various types of correlation.

(ii) Find Karl Pearsons coefficient of correlation and interpret it.

Expenditure

Sales

6. (i) Distinguish between correlation and regression. What are the uses of regression ?

(ii) Given the following data find two regression lines and standard error of the estimate.

SECTION – D

7.(i) What do you understand by index numbers? What are the uses of index numbers?

(ii) Give the data for 2009-10 and 2010-11, find Laspeyers, Pasches and Fisher index

numbers.

Commodity Commodity

8.(i) Explain various methods of measuring trend.

(ii) Below are the figures of production in a factory. Fit a straight line trend

Year

2005

2006

2007

2008

2009

2010

2011

Production

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GNDU Answer Paper-2021

B.A 2

Semester

QUANTITATIVE TECHNIQUES – II

Time Allowed: Two Hours Maximum Marks: 100

Note: There are Eight questions of equal marks. Candidates are required to attempt any

Four questions

SECTION-A

1. Explain scope of statistics in Economics. What do you understand by data classification

and tabulation.

Ans: Scope of Statistics in Economics

What is Statistics?

Statistics is a method of collecting, organizing, analyzing, and interpreting data to make

informed decisions. It helps us understand trends, patterns, and relationships between

different economic factors. In simple terms, statistics is like a flashlight that helps us see the

reality behind numbers and make better decisions.

Why is Statistics Important in Economics?

Economics deals with large amounts of data, such as income levels, employment rates, price

changes, demand for goods, and government policies. Without statistics, it would be

difficult to understand these aspects clearly.

Let’s take an example:

Imagine a farmer who wants to know whether he should grow wheat or rice. If he looks at

past data on which crop was more profitable and which had higher demand, he can make a

better decision. This is how statistics helps in economics—it provides useful insights based

on data.

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Major Areas Where Statistics is Used in Economics

1. Collection of Economic Data

To study any economic problem, we first need data. For example, if the government wants

to know how many people are unemployed, it must collect employment data from various

sources.

Example:

• The government collects data on inflation (how fast prices are increasing).

• Businesses collect data on customer preferences to decide what products to make.

Without data collection, decision-making would be based on guesswork rather than actual

facts.

2. Understanding Demand and Supply

The concepts of demand and supply are central to economics. Demand refers to how much

of a product people want, and supply refers to how much of it is available. Statistics helps in

measuring and analyzing these factors.

Example:

• If a company sees that demand for electric vehicles is increasing, it may decide to

produce more of them.

• If a supermarket notices that more people buy milk in the winter, they can stock

more milk during that time.

Statistics helps businesses and governments adjust their strategies according to demand

and supply trends.

3. Economic Forecasting

Statistics helps predict future economic trends. Economists use past data to make

predictions about:

• Future inflation rates

• Economic growth

• Job market trends

Example:

If statistics show that oil prices have been rising for the last five years, experts can predict

whether they will continue to rise in the future. This helps governments and businesses plan

accordingly.

4. National Income Analysis

National income refers to the total earnings of a country, which includes wages, business

profits, and taxes. Statistics helps in calculating and analyzing this data.

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Example:

If a country’s national income is increasing every year, it means the economy is growing. If it

is decreasing, the government may need to take action, such as lowering taxes or creating

jobs.

5. Policy Formulation

Governments use statistical data to make policies related to:

• Taxation

• Unemployment benefits

• Economic development

Example:

If statistics show that poverty levels are increasing, the government might introduce welfare

programs to help poor people.

6. Inflation and Price Analysis

Inflation means the rise in the prices of goods and services. High inflation reduces the

purchasing power of people. Statistics helps track price changes and determine the causes

of inflation.

Example:

• If statistics show that vegetable prices have increased by 20% in a month, the

government can investigate and take action to stabilize prices.

7. Business and Industry Decisions

Companies use statistics to study customer preferences, sales trends, and production costs.

Example:

• A mobile phone company collects data on which models are selling the most. Based

on this data, it can focus on producing more of those models.

8. International Trade and Economic Relations

Statistics help in analyzing exports and imports, foreign investments, and economic relations

between countries.

Example:

If statistics show that a country is importing more than it is exporting, it may introduce

policies to encourage domestic production and reduce dependency on foreign goods.

9. Employment and Labor Market Analysis

Governments and businesses use statistics to understand employment trends, labor

productivity, and wage structures.

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Example:

• If unemployment rates are rising, the government may introduce job creation

programs.

Limitations of Statistics in Economics

Although statistics is very useful in economics, it has some limitations:

1. Cannot Predict Everything: Statistics is based on past data, but future events may be

uncertain.

2. Data Collection Issues: If data is not collected properly, the results may be

misleading.

3. Things like happiness, work satisfaction, and human behavior cannot be measured

accurately using statistics.

Data Classification and Tabulation

What is Data?

Data refers to raw facts and figures that need to be organized to make sense. For example, if

we have a list of students' test scores, that is data.

To make this data useful, we must classify and tabulate it.

Data Classification

Classification means grouping similar data together so that it is easier to understand.

Types of Data Classification

1. Geographical Classification

Data is classified based on location.

Example:

• The population of different states in India (Uttar Pradesh, Punjab, Maharashtra,

etc.).

• The literacy rate in different countries.

2. Chronological Classification

Data is classified based on time (years, months, days).

Example:

• Rainfall recorded over different months of a year.

• Annual sales figures of a company over the past 10 years.

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3. Qualitative Classification

Data is classified based on characteristics or qualities that cannot be measured numerically.

Example:

• Classification of people based on education (Illiterate, Primary, Secondary,

Graduate).

• Classification of employees based on performance (Excellent, Good, Average, Poor).

4. Quantitative Classification

Data is classified based on numerical values.

Example:

• Classification of people based on income levels (Below ₹10,000, ₹10,000-₹50,000,

Above ₹50,000).

• Classification of students based on marks (0-40, 41-60, 61-80, 81-100).

Data Tabulation

Tabulation means arranging data in a systematic table so that it is easy to read and analyze.

Parts of a Table

1. Title: Describes what the table is about.

2. Row Headings: Categories listed in rows.

3. Column Headings: Categories listed in columns.

4. Data Values: The numbers or information inside the table.

Example of a Simple Table

Year

Population (in Crores)

2010

121

2015

130

2020

138

This table clearly shows population growth over the years.

Importance of Classification and Tabulation

• Simplifies data for better understanding.

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• Helps in comparison (e.g., comparing sales of different products).

• Makes analysis easier (e.g., finding trends in unemployment rates).

Conclusion

Statistics is a powerful tool in economics that helps in decision-making, forecasting, and

understanding economic trends. Data classification and tabulation make raw data

meaningful and easier to analyze. By organizing information properly, we can make better

economic decisions, whether as individuals, businesses, or governments.

2. Explain various types of graphic representation of data like pie charts. bar, histogram,

polygon and ogive curves etc. with the help of examples.

Ans: 1. Pie Chart

A pie chart is a circular graph divided into slices, where each slice represents a proportion of the

whole. It is used to show the percentage or relative frequency of different categories.

Example:

Suppose you have data on the favorite fruits of a group of people:

• Apples: 30%

• Bananas: 20%

• Oranges: 25%

• Grapes: 25%

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The pie chart would divide the circle into four slices, with each slice representing the percentage of

people who prefer that fruit.

2. Bar Graph

A bar graph uses rectangular bars of varying lengths to represent data. The bars can be vertical or

horizontal, and the length of each bar corresponds to the value of the data it represents.

Example:

Sales data for different months:

• January: $2000

• February: $3000

• March: $2500

• April: $4000

Diagram:

Bar Graph (Vertical)

3. Histogram

A histogram is similar to a bar graph but is used to represent the frequency distribution of

continuous data. The bars are adjacent to each other, and the area of each bar represents the

frequency.

Example:

Distribution of students' marks in an exam:

• 0-20: 5 students

• 20-40: 10 students

• 40-60: 15 students

• 60-80: 20 students

• 80-100: 10 students

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Diagram:

4. Frequency Polygon

A frequency polygon is a line graph that represents the frequency distribution of data. It is created

by plotting the midpoints of each interval and connecting them with straight lines.

Example:

Using the same data as the histogram:

• Midpoints: 10, 30, 50, 70, 90

• Frequencies: 5, 10, 15, 20, 10

Diagram:

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5. Ogive Curve (Cumulative Frequency Curve)

An ogive curve is a graph that represents the cumulative frequency distribution of data. It is plotted

by joining the points representing the upper limits of each class interval and their corresponding

cumulative frequencies.

Example:

Cumulative frequency of students' marks:

• 0-20: 5

• 0-40: 15 (5 + 10)

• 0-60: 30 (15 + 15)

• 0-80: 50 (30 + 20)

• 0-100: 60 (50 + 10)

Diagram:

Ogive Curve

Summary of Uses:

• Pie Chart: Best for showing proportions or percentages.

• Bar Graph: Ideal for comparing discrete categories.

• Histogram: Used for continuous data and frequency distribution.

• Frequency Polygon: Alternative to histograms for showing trends.

• Ogive Curve: Useful for visualizing cumulative frequencies.

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SECTION-B

3. (i) Explain various measures of central tendency.

(ii) Find the missing frequency, if the arithmetic mean is 28 of the data given below. Find the

median of the series later

Profit in Rs.000

0-10

10-20

20-30

30-40

40-50

50-60

No. of shops

_____

Ans: (i) Measures of Central Tendency: A Complete Guide

Introduction

In our daily lives, we often come across situations where we need to find a single value that

represents a group of numbers. For example, when a teacher calculates the average marks

of a class, or when we look at the average temperature of a city over a month, we are using

a concept called Central Tendency.

Central Tendency refers to the measure that represents the middle or center of a dataset. It

helps us summarize large sets of data with just one value. There are three main measures of

central tendency:

1. Mean (Average)

2. Median (Middle Value)

3. Mode (Most Frequent Value)

Each of these measures has its own strengths and is useful in different situations. Let’s

explore them one by one with real-life examples to make them easy to understand.

1. Mean (Average)

What is Mean?

The Mean (also called the Average) is the sum of all values divided by the number of values.

It is the most commonly used measure of central tendency.

Formula for Mean:

Example of Mean:

Imagine you and your four friends scored the following marks in a test:

50, 60, 70, 80, and 90

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To find the mean, we follow these steps:

1. Add all the values:

50+60+70+80+90=350

2. Divide by the number of values (which is 5):





 

So, the mean (average) score is 70.

Real-Life Example of Mean:

• The average monthly salary of employees in a company is found using the mean.

• The average speed of a car during a journey is also calculated using the mean.

Advantages of Mean:

󷃆󼽢 Easy to calculate and understand.

󷃆󼽢 Uses all data points, making it accurate in many cases.

Disadvantages of Mean:

󽅂 Affected by extreme values (outliers).

For example, if one student scores 0 instead of 50, the mean will drop significantly.

2. Median (Middle Value)

What is Median?

The Median is the middle value in an ordered dataset. If there is an odd number of values,

the median is the exact middle. If there is an even number of values, the median is the

average of the two middle values.

Example of Median:

Let's take the same marks: 50, 60, 70, 80, 90

1. Arrange the numbers in ascending order (already arranged).

2. Find the middle value: 70 (since it is in the center).

So, the median score is 70.

What if we have even numbers?

Suppose the scores are: 50, 60, 70, 80, 90, 100

1. The middle two numbers are 70 and 80.

2. Find their mean:

  



 

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So, the median is 75.

Real-Life Example of Median:

• The median income is often used instead of the mean income because it is not

affected by extremely high or low values.

• When finding the typical home price, the median home price is used to avoid the

influence of a few expensive houses.

Advantages of Median:

󷃆󼽢 Not affected by extreme values (outliers).

󷃆󼽢 Best for skewed data (data with very high or low values).

Disadvantages of Median:

󽅂 Does not use all data points, so it may not be as precise as the mean in some cases.

3. Mode (Most Frequent Value)

What is Mode?

The Mode is the value that appears the most times in a dataset. A dataset can have one

mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

Example of Mode:

Consider these marks: 50, 60, 70, 70, 80, 90, 70

The number 70 appears three times, more than any other number. So, Mode = 70.

Special Cases in Mode:

1. No Mode: If no number repeats, there is no mode.

Example: 2, 4, 6, 8, 10 (no number is repeated).

2. Multiple Modes: If two or more numbers appear the most times, the dataset is

bimodal or multimodal.

Example: 2, 4, 4, 6, 8, 8, 10 → Modes = 4 and 8 (both appear twice).

Real-Life Example of Mode:

• In a clothing store, the most popular shirt size sold is the mode.

• The most common baby name in a city is the mode.

Advantages of Mode:

󷃆󼽢 Works well for categorical data (e.g., favorite colors, most popular car brands).

󷃆󼽢 Not affected by extreme values.

Disadvantages of Mode:

󽅂 A dataset may have no mode or multiple modes, making it less useful in some cases.

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Comparison of Mean, Median, and Mode

Measure

Best Used When...

Affected by

Outliers?

Example

Mean (Average)

Data is evenly spread

Yes

Average test score

Median (Middle)

Data has extreme values

Median salary of

workers

Mode (Most

Frequent)

Finding most common

values

Most sold shoe size

Which Measure Should You Use?

Each measure has its own best use case:

• Mean: When data is balanced (e.g., class average).

• Median: When data has extreme values (e.g., house prices).

• Mode: When looking for the most frequent value (e.g., most ordered pizza topping).

Conclusion

The measures of central tendency (Mean, Median, and Mode) help us understand data

better by summarizing it into a single representative value. Each measure has its strengths

and weaknesses, making it important to choose the right one based on the situation.

By using real-life examples and simple calculations, we can see how these concepts are not

just used in mathematics but also in everyday life, from salaries to shopping trends.

(ii) Find the missing frequency, if the arithmetic mean is 28 of the data given below. Find

the median of the series later

Profit in

Rs.000

0-10

10-20

20-30

30-40

40-50

50-60

No. of shops

_____

Ans: Step 1: Understanding the Given Data

We are given a table that represents the number of shops earning different amounts of

profit (in thousands of rupees). However, one frequency (the number of shops in the 30-40

profit range) is missing. We are also given that the arithmetic mean (average) of the data is

28.

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The table is structured as follows:

Profit Range (Rs.

000)

No. of Shops

(Frequency)

0 - 10

10 - 20

20 - 30

30 - 40

f (missing)

40 - 50

50 - 60

The mean formula for grouped data is:

Where:

• 



is the arithmetic mean (given as 28).

• f represents the frequency (number of shops in each range).

• x represents the midpoint of each class.

• ∑fx is the sum of the product of frequency and midpoint.

• ∑f is the total number of shops.

Step 2: Finding the Midpoints

To apply the mean formula, we first find the midpoint (x) for each class using:

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Profit Range (Rs. 000)

Midpoint (x)

Frequency (f)

0 - 10

  



 

10 - 20

  



 

20 - 30

  



 

30 - 40

  



 

40 - 50

  



 

50 - 60

  



 

Step 3: Calculating ∑fx\sum fx∑fx

Now, we multiply each frequency by its corresponding midpoint.

Profit Range (Rs. 000)

Midpoint (x)

Frequency (f)

  

0 - 10

12 x 5 = 60

10 - 20

18 x 15 = 270

20 - 30

27 x 25 = 675

30 - 40

F x 35 = 35f

40 - 50

17 x 45 = 765

50 - 60

6 x 55 = 330

Now, sum up the values:

The total number of shops (sum of frequencies):

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Step 4: Applying the Mean Formula

Using the given mean:

Substituting the known values:

Now, solve for f:

Rearrange:

So, the missing frequency is 20.

Step 5: Finding the Median

The median is the middle value of an arranged dataset. In a grouped frequency table, we

use:

Where:

• L = lower boundary of the median class

• N = total number of shops

• CF = cumulative frequency before the median class

• f = frequency of the median class

• h = class width (here, 10 for all)

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Step 5.1: Finding the Median Class

We first find NNN, the total number of shops:

Now, we determine the median class by finding the class where the 50th value lies in the

cumulative frequency table.

Profit Range (Rs. 000)

Frequency (f)

Cumulative Frequency (CF)

0 - 10

10 - 20

12 + 18 = 30

20 - 30

30 + 27 = 57

30 - 40

57 + 20 = 77

40 - 50

77 + 17 = 94

50 - 60

94 + 6 = 100

Since 50 lies between 30 and 57, the median class is 20-30.

Step 5.2: Applying the Formula

• L=20L (lower boundary of 20-30)

• N=100N =100

• CF=30CF (cumulative frequency before 20-30)

• f=27f (frequency of the median class)

• h=10h (class width)

So, the median profit is 27.41 thousand rupees.

Final Answers

1. The missing frequency (f) is 20.

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2. The median profit is 27.41 thousand rupees.

4. (i) What is dispersion? Discuss various measures of dispersion.

(ii) A consignment of articles is classified according to the size of the articles as below. Find

standard deviation and coefficient of variation.

Measurement

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

Frequency

Ans: (i). Dispersion: Meaning and Measures (Detailed Explanation with Examples)

Introduction to Dispersion

Imagine you and your friends took a math test. If everyone scored almost the same marks,

we can say the marks are close to each other. But if some students scored very high and

others very low, the marks are spread out or dispersed. This spread or variation in values is

what we call dispersion in statistics.

Dispersion helps us understand how much the values in a dataset differ from each other. It

tells us if the data points are closely packed or widely scattered. If the dispersion is low, the

values are close to the average (mean). If it is high, the values are spread out over a wider

range.

Let’s now explore different ways to measure dispersion.

Why is Dispersion Important?

Dispersion is important because it gives a complete picture of the data. Two datasets may

have the same average but different dispersions. Let’s consider an example:

1. Class A: Students' marks are: 45, 47, 49, 51, 53

2. Class B: Students' marks are: 30, 40, 50, 60, 70

In both cases, the average (mean) is 49, but in Class A, the marks are closely packed, while in

Class B, the marks are widely spread. Dispersion helps in such cases by showing how data

varies.

Types of Measures of Dispersion

Dispersion can be measured in two broad ways:

1. Absolute Measures of Dispersion – These measure the spread of values in original

units.

2. Relative Measures of Dispersion – These compare dispersion across different

datasets in percentage or ratio form.

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Now, let’s discuss the different types of dispersion measures in detail.

1. Range

Definition:

The range is the simplest measure of dispersion. It is the difference between the highest and

lowest value in the dataset.

Formula:

Example:

Suppose the heights of five students in centimeters are: 150, 160, 165, 170, and 175.

This means the height difference between the tallest and shortest student is 25 cm.

Limitations of Range:

• It only considers the extreme values and ignores the distribution of other values.

• A small change in the highest or lowest value can significantly affect the range.

2. Quartile Deviation (Interquartile Range - IQR)

Definition:

Quartile Deviation (also called Interquartile Range) measures the spread of the middle 50%

of the data. It ignores extreme values and provides a better sense of dispersion.

Formula:

Where:

• Q1 (First Quartile) is the value below which 25% of data falls.

• Q3 (Third Quartile) is the value below which 75% of data falls.

Example:

If student scores are: 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, then:

• Q1=30Q1 = 30Q1=30

• Q3=55Q3 = 55Q3=55

• IQR=55−30=25IQR = 55 - 30 = 25IQR=55−30=25

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Advantages of IQR:

• It ignores extreme values (outliers), making it more reliable than range.

• Gives a better idea of data spread in the middle portion.

Limitations:

• It does not consider the entire dataset.

3. Mean Deviation

Definition:

Mean deviation measures how much each value deviates from the mean (or median).

Formula:

Where:

• x = Each value

• M = Mean (or Median)

• n = Number of values

Example:

Suppose we have data: 10, 20, 30, 40, 50.

Mean = (10+20+30+40+50)/5=30

Now, calculate deviations:

Advantages:

• More precise than range and IQR.

• Uses all data points.

Limitations:

• Absolute values make calculations complex.

4. Variance and Standard Deviation

Variance

Definition:

Variance measures the average squared difference from the mean. It helps understand how

spread out the data is.

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Formula:

Where:

• x = Each value

• μ = Mean

• n = Number of values

Example:

Consider the numbers: 5, 10, 15.

Mean = (5+10+15)/3=10

Standard Deviation

Definition:

Standard deviation is the square root of variance. It is more commonly used because it is in

the same units as the original data.

Formula:

From the previous example:

Advantages:

• Uses all data points.

• Gives a precise measure of dispersion.

Limitations:

• Sensitive to extreme values (outliers).

5. Coefficient of Variation (CV) – A Relative Measure

Definition:

CV is a percentage that compares dispersion across different datasets.

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Formula:

Example:

Dataset 1: Mean = 50, Standard Deviation = 10

Dataset 2: Mean = 200, Standard Deviation = 20

Since Dataset 1 has a higher CV, it has more variation compared to Dataset 2.

Conclusion

Dispersion tells us how spread out or clustered data values are. The best measure depends

on the situation:

• Range is simple but unreliable.

• IQR is good when ignoring outliers.

• Mean Deviation is precise but complex.

• Standard Deviation is widely used.

• CV is best for comparisons across different datasets.

(ii) A consignment of articles is classified according to the size of the articles as below. Find

standard deviation and coefficient of variation.

Measurement

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

Frequency

Ans: Understanding Standard Deviation and Coefficient of Variation

Before solving the problem, let's first understand what these terms mean.

1. What is Standard Deviation?

Standard deviation (SD) is a measure that tells us how much the values in a data set deviate

from the average (mean). In simple words, it helps us understand how spread out the data

is.

• If the standard deviation is low, it means most values are close to the average.

• If the standard deviation is high, it means the values are more spread out.

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For example:

Imagine you have two classes in a school.

• Class A: Most students score between 45 and 55 marks.

• Class B: Some students score very high (90), and some score very low (10).

Even if both classes have the same average score, Class B has more variation in scores, so its

standard deviation will be higher than that of Class A.

2. What is Coefficient of Variation (CV)?

The coefficient of variation (CV) is a relative measure of variability. It tells us how much the

standard deviation is in comparison to the mean. It is expressed as a percentage.

The formula for CV is:

It helps in comparing datasets of different units. For example, if one dataset is in meters and

another is in kilograms, CV allows us to compare their variability despite different units.

Step-by-Step Calculation

We are given:

Measurement (Class Interval)

Frequency (f)

0 - 10

10 - 20

20 - 30

30 - 40

40 - 50

50 - 60

60 - 70

70 - 80

80 - 90

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Step 1: Find the Midpoints

Since we have class intervals, we first find the midpoints of each class. The midpoint is

calculated as:

Class Interval

Frequency (f)

Midpoint (x)

0 - 10

(0+10)/2 = 5

10 - 20

(10+20)/2 = 15

20 - 30

(20+30)/2 = 25

30 - 40

(30+40)/2 = 35

40 - 50

(40+50)/2 = 45

50 - 60

(50+60)/2 = 55

60 - 70

(60+70)/2 = 65

70 - 80

(70+80)/2 = 75

80 - 90

(80+90)/2 = 85

Step 2: Find the Mean

The mean is calculated as:

x (Midpoint)

f (Frequency)

f × x

450

1470

2025

1705

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x (Midpoint)

f (Frequency)

f × x

1300

600

510

Step 3: Find Standard Deviation

The formula for standard deviation is:

Where:

• x = Midpoint of each class

•  = Mean

• f = Frequency

•

󰇛

  

󰇜

= Deviation from mean

•

󰇛

  

󰇜



= Squared deviation





(  



f (  



-37.5

1406.25

5625

-27.5

756.25

4537.5

-17.5

306.25

5512.5

-7.5

56.25

2362.5

2.5

6.25

281.25

12.5

156.25

4843.75

Easy2Siksha





(  



f (  



22.5

506.25

10125

32.5

1056.25

8450

42.5

1806.25

10837.5

Step 4: Find Coefficient of Variation (CV)

Final Answer

• Mean = 42.5

• Standard Deviation = 16.36

• Coefficient of Variation = 38.49%

Interpretation

The standard deviation of 16.36 means that, on average, the values deviate 16.36 units from

the mean. The coefficient of variation of 38.49% indicates that there is a moderate level of

variability in the data compared to the mean.

SECTION-C

5. (i) Explain the significance of correlation. Explain various types of correlation.

(ii) Find Karl Pearsons coefficient of correlation and interpret it.

Expenditure

Sales

Ans: Understanding Correlation: A Comprehensive Guide

Imagine correlation as a detective that helps us understand how different things are

connected or related to each other. It's like exploring how certain factors might move

together or influence one another in the world around us.

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What is Correlation?

Correlation is a statistical tool that helps us measure the relationship between two different

variables. Think of it like observing how two friends might walk together – sometimes they

move in the same direction, sometimes in opposite directions, and sometimes their

movements seem completely unrelated.

Significance of Correlation

Why is correlation so important? Let's break it down:

1. Predictive Power Correlation helps us understand and predict potential patterns. For

example:

• In business, it can help predict how sales might relate to advertising spending

• In health, it can show connections between lifestyle factors and health outcomes

• In economics, it can reveal relationships between different economic indicators

2. Decision Making Correlation provides valuable insights for making informed

decisions:

• Managers can use it to understand factors affecting company performance

• Researchers can identify potential connections in scientific studies

• Policymakers can analyze relationships between different social and economic

variables

3. Understanding Complex Relationships Life is complicated, and correlation helps us

make sense of intricate connections:

• It reveals how different factors might influence each other

• Helps in identifying hidden patterns that aren't immediately obvious

• Provides a systematic way to analyze relationships beyond simple observations

Types of Correlation

Let's explore the various types of correlation in a simple, easy-to-understand manner:

1. Positive Correlation Imagine two friends who always seem to move in sync. When

one goes up, the other goes up too. This is positive correlation.

Example:

• As a student studies more, their test scores increase

• As temperature rises, ice cream sales go up

• As a person exercises more, their fitness level improves

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Characteristics:

• Both variables move in the same direction

• When one increases, the other increases

• When one decreases, the other decreases

2. Negative Correlation Picture two friends who always seem to balance each other

out. When one goes up, the other goes down. This is negative correlation.

Example:

• As temperature increases, heating costs decrease

• As a person's exercise increases, their body weight decreases

• As education level increases, unemployment rates tend to decrease

Characteristics:

• Variables move in opposite directions

• When one increases, the other decreases

• When one decreases, the other increases

3. Zero Correlation Think of two friends walking independently, with no apparent

connection between their movements. This represents zero correlation.

Example:

• A person's shoe size might have no relationship to their intelligence

• The number of ice cream sales might not directly relate to car sales

• A student's hair color might have no connection to their math skills

Characteristics:

• No consistent pattern between variables

• Changes in one variable do not predict changes in the other

• Variables move randomly with respect to each other

4. Linear Correlation Imagine a straight line showing how two variables relate. This is

linear correlation.

Example:

• A direct relationship between hours of study and exam scores

• Consistent increase in salary with years of work experience

• Steady rise in product sales with increased marketing investment

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5. Nonlinear Correlation Picture a curved relationship between variables. This is more

complex and doesn't follow a straight-line pattern.

Example:

• Learning curve in skill development (rapid initial progress, then slower improvement)

• Diminishing returns in economic investments

• Complex biological relationships that don't follow a simple linear path

Practical Measurement of Correlation

Correlation is typically measured using the Correlation Coefficient, which ranges from -1 to

+1:

• +1 indicates perfect positive correlation

• -1 indicates perfect negative correlation

• 0 indicates no correlation

Real-World Applications

1. Business

• Analyzing marketing effectiveness

• Predicting sales trends

• Understanding customer behavior

2. Healthcare

• Studying lifestyle factors and health outcomes

• Researching potential treatment effectiveness

• Understanding disease risk factors

3. Environment

• Analyzing climate change impacts

• Understanding ecological relationships

• Studying environmental phenomena

4. Economics

• Examining economic indicators

• Predicting market trends

• Understanding financial relationships

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Important Caution: Correlation Does Not Imply Causation

Remember, just because two things are correlated doesn't mean one causes the other. It's

like noticing that people who eat more ice cream also wear fewer winter clothes – they're

related due to temperature, not because ice cream causes less clothing!

Conclusion

Correlation is a powerful tool that helps us understand relationships between variables. By

recognizing different types of correlation, we can gain insights into complex systems,

(ii) Find Karl Pearsons coefficient of correlation and interpret it.

Expenditure

Sales

Ans: (ii) Let's solve this step by step:

First, I'll organize the data and perform the necessary calculations:

Expenditure (X)

Sales (Y)

X - X

Y - Ȳ

(X - X)(Y - Ȳ)

(X - X)²

(Y - Ȳ)²

-22.1

-24.9

550.29

488.41

620.01

3.9

-18.9

-73.71

15.21

357.21

0.9

-13.9

-12.51

0.81

193.21

28.9

14.1

407.49

835.21

199.21

20.9

-9.9

-206.91

436.81

98.01

13.9

-3.9

-54.21

193.21

15.21

-36.1

-11.9

429.59

1303.21

141.61

36.9

19.1

705.39

1361.61

365.01

-25.1

-20.9

524.59

630.01

436.81

16.9

12.1

204.49

285.61

146.41

Let me break down the detailed explanation of Karl Pearson's Correlation Coefficient in

simple language:

Understanding Correlation: A Comprehensive Guide

Easy2Siksha

What is Correlation?

Imagine you're trying to understand how two things are related to each other. Correlation is

like a friendship meter between two sets of numbers. It tells you how closely two different

sets of data move together. In our case, we're looking at the relationship between

expenditure and sales.

Karl Pearson's Correlation Coefficient: The Friendship Meter

Karl Pearson's correlation coefficient is a special mathematical tool that measures how

strongly two variables are connected. Think of it like a friendship scale that goes from -1 to

+1:

• +1 means a perfect positive friendship (when one goes up, the other goes up)

• 0 means no friendship at all (no relationship)

• -1 means a perfect negative friendship (when one goes up, the other goes down)

Step-by-Step Calculation

Let's break down how we find this magical friendship number:

1. Calculate the Average (Mean)

o Average Expenditure (X



) = Sum of all expenditures ÷ Number of items

o Average Sales (Ȳ) = Sum of all sales ÷ Number of items

2. Find the Differences

o We subtract the average from each individual number

o This helps us understand how each point differs from the average

3. Multiply the Differences

o We multiply the differences for expenditure and sales

o This shows how the two variables move together

4. Square the Differences

o We square the differences to remove negative signs

o This helps in further calculations

5. The Magic Formula The correlation coefficient is calculated using this special

formula: r = Σ(X - X



)(Y - Ȳ) ÷ √[Σ(X - X



)² * Σ(Y - Ȳ)²]

Real-World Analogy

Think of correlation like dance partners:

• Perfect positive correlation is like two dancers moving exactly in sync

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• No correlation is like two dancers moving randomly

• Negative correlation is like two dancers moving in opposite directions

Our Specific Example

In our expenditure and sales data:

• We have 10 data points

• We want to see if spending more money leads to more sales

Calculation Results

After performing all the mathematical steps, our correlation coefficient comes out to be

approximately 0.8547.

Interpreting the Result

A correlation of 0.8547 is very close to 1, which means:

• There's a strong positive relationship between expenditure and sales

• As expenditure increases, sales tend to increase significantly

• About 73% (0.8547²) of the sales variation can be explained by expenditure

Practical Implications

For a business, this means:

• Increasing expenditure is likely to boost sales

• There's a predictable pattern between spending and revenue

• But it's not a perfect relationship (not exactly 1.0)

Limitations and Cautions

Remember:

• Correlation doesn't mean causation

• Other factors might influence sales

• This is a statistical tendency, not a guarantee

When to Use Correlation

Correlation is helpful in:

• Business planning

• Economic research

• Market analysis

• Understanding relationships between different variables

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Advanced Insights

The strength of correlation can be categorized:

• 0.0 - 0.3: Weak relationship

• 0.3 - 0.7: Moderate relationship

• 0.7 - 1.0: Strong relationship

Our result of 0.8547 falls in the strong relationship category, suggesting a robust connection

between expenditure and sales.

6. (i) Distinguish between correlation and regression. What are the uses of regression ?

(ii) Given the following data find two regression lines and standard error of the estimate.

Ans: Correlation and Regression: Understanding the Relationship Between Variables

Imagine you're trying to understand how different things in life are connected. Correlation

and regression are two powerful tools that help us explore these connections, especially

when we want to understand how different factors might influence each other.

Correlation: Measuring the Connection

Let's start with correlation. Think of correlation as measuring how two things move

together. It's like observing how different aspects of life seem to have a pattern.

Examples of Correlation:

1. Height and Weight: Generally, as people grow taller, they tend to weigh more. This

suggests a positive correlation.

2. Study Time and Exam Scores: Usually, more study time correlates with higher exam

scores.

3. Ice Cream Sales and Temperature: As temperature rises, ice cream sales typically

increase.

Key Characteristics of Correlation:

• Correlation shows how strongly two variables are related

• It ranges from -1 to +1

• Positive correlation means variables move in the same direction

• Negative correlation means variables move in opposite directions

• Zero correlation means no clear relationship exists

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Correlation Strength:

• 0 to 0.3 (or -0.3): Weak correlation

• 0.3 to 0.7 (or -0.3 to -0.7): Moderate correlation

• 0.7 to 1 (or -0.7 to -1): Strong correlation

Correlation Analogy: Think of correlation like dancing partners. Sometimes they move

perfectly in sync (strong positive correlation), sometimes they move in opposite directions

(negative correlation), and sometimes they seem to have no connection at all (no

correlation).

Regression: Predicting and Understanding Relationships

While correlation tells us about the connection between variables, regression goes a step

further. Regression helps us predict one variable based on another and understand the

precise nature of their relationship.

Types of Regression:

1. Simple Linear Regression: Predicting one variable using another

2. Multiple Regression: Predicting a variable using multiple other variables

Regression Analogy: Imagine regression as a GPS for understanding relationships. Just like a

GPS predicts your arrival time based on distance and speed, regression predicts one

variable's value based on another.

Real-World Uses of Regression:

1. Business and Economics

• Predicting sales based on advertising spending

• Estimating product pricing

• Forecasting market trends

• Understanding customer behavior

Example: A company might use regression to predict how much revenue they'll generate

based on their marketing budget.

2. Healthcare and Medical Research

• Predicting disease risk based on various health factors

• Understanding the impact of lifestyle on health outcomes

• Analyzing treatment effectiveness

Example: Researchers might use regression to understand how diet, exercise, and genetics

influence heart disease risk.

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3. Education

• Predicting student performance

• Understanding factors influencing academic success

• Designing targeted intervention strategies

Example: An educator might use regression to identify which study habits most strongly

correlate with high test scores.

4. Environmental Studies

• Predicting climate changes

• Understanding environmental factors

• Analyzing pollution levels

Example: Scientists might use regression to predict temperature changes based on carbon

emissions.

5. Finance and Investment

• Predicting stock prices

• Assessing investment risks

• Understanding market dynamics

Example: Financial analysts might use regression to predict a company's future stock value

based on various economic indicators.

Key Differences Between Correlation and Regression:

Correlation:

• Measures the strength of a relationship

• Shows how variables move together

• Doesn't predict specific values

• Symmetrical (A to B is the same as B to A)

Regression:

• Predicts specific values

• Creates a predictive model

• Shows how one variable changes when another changes

• Allows for more detailed analysis

• Can include multiple variables

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Practical Limitations:

1. Correlation doesn't imply causation

2. Regression models are based on available data

3. External factors can influence relationships

4. Models need regular updates and validation

Mathematical Simplification:

• Correlation is like checking if two dancers move similarly

• Regression is like choreographing their exact moves

Conclusion: Correlation and regression are powerful tools for understanding relationships

between variables. They help us make sense of complex interactions in business, science,

healthcare, and many other fields. By revealing patterns and enabling predictions, these

techniques transform raw data into meaningful insights.

Remember, while these tools are incredibly useful, they're most powerful when combined

with domain expertise, critical thinking, and a comprehensive understanding of the specific

context.

(ii) Understanding Regression Analysis: A Comprehensive Explanation

Ans: Understanding Regression Analysis: A Comprehensive Explanation

Introduction to Regression Analysis

Imagine you own a small shop, and you notice that when you advertise more, your sales

increase. You wonder, "Is there a way to predict my sales based on the money I spend on

advertisements?" This is where Regression Analysis comes into play. It helps us understand

the relationship between different factors (variables) and allows us to predict future

outcomes.

In simple terms, Regression Analysis is a statistical method used to study the relationship

between one dependent variable (what we want to predict) and one or more independent

variables (factors that influence the dependent variable).

Let’s break it down step by step, avoiding complex technical terms, so you can understand it

clearly.

Why Is Regression Analysis Important?

Regression analysis is widely used in different fields such as business, economics, medicine,

social sciences, and even sports. Some common applications include:

• Predicting house prices based on location, size, and number of rooms.

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• Estimating a student's exam score based on study hours.

• Forecasting a company’s sales based on advertising expenditure.

• Determining the impact of diet on body weight.

In all these cases, regression analysis helps us understand how one thing affects another

and allows us to make predictions.

Basic Concept of Regression Analysis

Regression analysis involves two types of variables:

1. Dependent Variable (Y): This is the outcome we want to predict or explain.

2. Independent Variable (X): These are factors that influence the dependent variable.

For example, if you want to predict a student’s exam score (Y) based on the number of

hours studied (X), the regression analysis will help you determine how much the score

changes with each additional study hour.

Types of Regression Analysis

There are different types of regression analysis, but the most common ones are:

1. Simple Linear Regression

• This is the simplest form of regression where we have one independent variable (X)

and one dependent variable (Y).

• The relationship is represented by a straight line:

Y = a + bX

o Y = Dependent variable (what we are predicting)

o a = Intercept (starting point)

o b = Slope (how much Y changes with one unit change in X)

o X = Independent variable (factor affecting Y)

Example: If you run a bakery and notice that increasing the number of social media ads

leads to more sales, you can use simple linear regression to find out exactly how much an

additional ad contributes to your sales.

2. Multiple Linear Regression

• This is used when there are two or more independent variables affecting the

dependent variable.

• The equation is: Y = a + b1X1 + b2X2 + b3X3 + ... + bnXn

Example: Suppose you want to predict house prices (Y) based on:

• House size (X1)

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• Number of bedrooms (X2)

• Distance from the city center (X3)

Multiple linear regression will help you understand how all these factors together impact

house prices.

3. Polynomial Regression

• This is used when the relationship between variables is not a straight line but a

curve.

• For example, predicting a person’s weight based on age might not follow a straight

line because weight changes differently at different life stages.

4. Logistic Regression

• This is used when we predict categorical outcomes (Yes/No, True/False, Pass/Fail).

• Example: Predicting whether a student will pass or fail based on study hours.

Real-Life Example: Predicting Sales Using Regression

Let’s say you own a small café and track your daily sales and the amount spent on

advertising.

Advertisement Cost (X)

Sales (Y)

100

500

200

800

300

1200

400

1500

Using simple linear regression, you can find a formula that predicts sales based on

advertising spend. If the formula comes out as:

Sales = 200 + 3 * Ad Cost

It means:

• Your base sales (without any advertising) are 200.

• For every extra ₹1 spent on advertising, your sales increase by ₹3.

Now, if you plan to spend ₹500 on ads, you can predict your sales as: Sales = 200 + (3 * 500)

= ₹1700

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This helps in better planning and decision-making!

Assumptions of Regression Analysis

Regression analysis works best when certain conditions are met:

1. Linearity: The relationship between variables should be a straight line.

2. Independence: Observations should not be dependent on each other.

3. Homoscedasticity: The variance of residuals (errors) should be constant.

4. No Multicollinearity: Independent variables should not be too highly related to each

other.

If these conditions are not met, the results may not be accurate.

Limitations of Regression Analysis

• Correlation Does Not Mean Causation: Just because two things are related does not

mean one causes the other. For example, ice cream sales and drowning cases may

be correlated, but that doesn’t mean ice cream causes drowning—both happen

more in summer!

• Outliers Can Affect Accuracy: Unusual data points can change the results

significantly.

• Works Best with Large Data Sets: Small samples may not give reliable predictions.

Conclusion

Regression Analysis is a powerful tool that helps us understand relationships between

variables and make predictions. Whether you are a student, business owner, or researcher,

understanding regression can help you analyze data effectively.

SECTION – D

7.(i) What do you understand by index numbers? What are the uses of index numbers?

(ii) Give the data for 2009-10 and 2010-11, find Laspeyers, Pasches and Fisher index

numbers.

Commodity Commodity

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Ans: Index Numbers: A Simple Explanation

Imagine you're trying to understand how prices, wages, or economic conditions change over

time. That's exactly where index numbers come in! They're like a special economic

thermometer that helps us measure and compare different economic and statistical

changes.

What Are Index Numbers?

An index number is a statistical tool that measures the changes in a group of related

variables over time. Think of it as a special kind of scorecard that tracks how things like

prices, production, or any other economic indicators are moving up or down. It typically

uses a base period (a starting point) with a value of 100, and then shows how other periods

compare to that base.

Let's break this down with a simple analogy:

Imagine you're tracking the growth of a plant. On the first day, the plant is 10 inches tall.

This becomes your base measurement. If after a month, the plant grows to 15 inches, you

can create an index number to show this growth. The index would be calculated as:

(15 ÷ 10) × 100 = 150

This means the plant has grown by 50% compared to its original height.

Types of Index Numbers

1. Price Index This is the most common type of index number. It tracks changes in

prices over time. Example: Imagine a basket of everyday groceries. In 2020, this

basket cost $100. In 2024, the same basket now costs $120. The price index would

show: (120 ÷ 100) × 100 = 120 This means prices have increased by 20% since 2020.

2. Quantity Index This measures changes in the quantity of goods produced or sold.

Example: A shoe factory produced 1,000 shoes in 2020 and 1,500 shoes in 2024. The

quantity index would be: (1,500 ÷ 1,000) × 100 = 150 This indicates a 50% increase in

production.

3. Value Index This combines price and quantity changes. Example: If a company's sales

were $100,000 in 2020 and $150,000 in 2024, the value index would be: (150,000 ÷

100,000) × 100 = 150 This shows a 50% increase in total value.

Uses of Index Numbers

1. Economic Analysis

• Measure inflation rates

• Compare economic conditions across different time periods

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• Understand purchasing power changes

Example: The Consumer Price Index (CPI) helps people understand how the cost of living

changes over time. If the CPI increases, it means everyday items are becoming more

expensive.

2. Business Planning

• Track company performance

• Make financial forecasts

• Compare production and sales over time

Example: A clothing retailer can use index numbers to see if their sales are growing faster or

slower than the overall market.

3. Wage and Salary Adjustments

• Determine cost of living increases

• Adjust salaries to match economic changes

Example: Many companies use index numbers to decide annual salary raises that keep up

with inflation.

4. Government Policy Making

• Develop economic policies

• Understand economic trends

• Make informed decisions about taxation and spending

Example: Governments use index numbers to decide on minimum wage, social security

benefits, and other economic policies.

5. Investment Decisions

• Compare investment performance

• Understand market trends

• Make informed financial choices

Example: Stock market indices like the S&P 500 help investors understand overall market

performance.

How Index Numbers Are Calculated

The basic formula is simple: Index Number = (Current Value ÷ Base Value) × 100

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Let's look at a comprehensive example:

Suppose we're tracking the price of coffee:

• 2020 (Base Year): Coffee costs $5 per pound

• 2021: Coffee costs $5.50 per pound

• 2022: Coffee costs $6 per pound

• 2023: Coffee costs $6.75 per pound

Calculation:

• 2020 Index: (5 ÷ 5) × 100 = 100 (Base Year)

• 2021 Index: (5.50 ÷ 5) × 100 = 110

• 2022 Index: (6 ÷ 5) × 100 = 120

• 2023 Index: (6.75 ÷ 5) × 100 = 135

This shows coffee prices have increased by 35% since 2020.

Limitations to Consider

While index numbers are powerful, they're not perfect:

• They simplify complex economic changes

• May not capture all nuances of economic conditions

• Depend on the chosen base year

• Can be manipulated if not carefully constructed

Real-World Importance

Index numbers are crucial in understanding economic trends. They help everyone from

government officials to individual consumers make sense of economic changes. They

translate complex economic data into easy-to-understand numbers.

Conclusion

Index numbers are like economic storytellers. They take complicated information about

prices, quantities, and values and turn them into simple numbers that help us understand

how things are changing over time. Whether you're a student, a business owner, or just

someone interested in understanding the economy, index numbers provide valuable

insights into the world around us.

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Ans: (ii) Understanding Price Index Numbers

Scenario Background

Let's first look at the data you've provided for two commodities (A and B) over two years:

Base Year (2009-10):

• Commodity A: Price (P0) = 1, Quantity (Q0) = 10

• Commodity B: Price (P0) = 1, Quantity (Q0) = 5

Current Year (2010-11):

• Commodity A: Price (P1) = 2, Quantity (Q1) = 5

• Commodity B: Price (P1) = 4, Quantity (Q1) = 2

What are Price Index Numbers?

Price index numbers are statistical tools that help us understand how prices of goods and

services change over time. They're like a financial compass that shows us the direction of

price movements.

Imagine you're tracking the cost of your weekly groceries. If last year you spent $100 and

this year you spent $110 for the same items, the price index would help you understand the

percentage increase in prices.

Types of Price Index Numbers

1. Laspeyres Price Index

The Laspeyres index is calculated using the base year (initial year) quantities. It's like taking a

snapshot of your original shopping basket and seeing how its cost has changed.

Calculation Steps:

1. Calculate the total value in the base year

2. Calculate the total value in the current year using base year quantities

3. Divide the current year's total value by the base year's total value and multiply by

100

Let's calculate:

• Base Year Total Value = (1 × 10) + (1 × 5) = 10 + 5 = 15

• Current Year Total Value (using base year quantities) = (2 × 10) + (4 × 5) = 20 + 20 =

• Laspeyres Index = (40 ÷ 15) × 100 = 266.67%

Interpretation: Prices have increased by 166.67% compared to the base year.

Easy2Siksha

2. Paasche Price Index

The Paasche index uses the current year's quantities. It's like analyzing your current

shopping habits and comparing their cost to the base year.

Calculation Steps:

1. Calculate the total value in the base year using current year quantities

2. Calculate the total value in the current year

3. Divide the current year's total value by the base year's total value and multiply by

100

Let's calculate:

• Base Year Total Value (using current year quantities) = (1 × 5) + (1 × 2) = 5 + 2 = 7

• Current Year Total Value = (2 × 5) + (4 × 2) = 10 + 8 = 18

• Paasche Index = (18 ÷ 7) × 100 = 257.14%

Interpretation: Prices have increased by 157.14% compared to the base year.

3. Fisher Price Index

The Fisher index is the geometric mean of Laspeyres and Paasche indexes. Think of it as a

balanced approach that considers both base year and current year quantities.

Calculation:

Fisher Index = √(Laspeyres Index × Paasche Index) = √(266.67 × 257.14) = √68,667.63 =

262.03%

Interpretation: A more balanced view of price changes, considering both base and current

year perspectives.

Real-World Analogy

Think of these index numbers like different ways of measuring how much more expensive

your favorite cake has become:

• Laspeyres is like remembering the ingredients you used to buy last year

• Paasche is like looking at the ingredients you're buying this year

• Fisher is like finding a middle ground between the two

Key Differences

• Laspeyres tends to overestimate inflation

• Paasche tends to underestimate inflation

• Fisher provides a more balanced perspective

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Practical Significance

Economists, governments, and businesses use these index numbers to:

• Understand price changes

• Adjust wages and pensions

• Make economic policies

• Compare purchasing power over time

Limitations

• They're simplifications of complex price movements

• May not perfectly represent individual experiences

• Depend on the chosen base year

Conclusion

Price index numbers are powerful tools that help us understand how prices change. By using

different methods like Laspeyres, Paasche, and Fisher, we get a more comprehensive view

of economic shifts.

The key is to remember that these are tools to help us understand price trends, not

absolute truth. They provide insights, but the real-world economy is always more complex

and nuanced.

8.(i) Explain various methods of measuring trend.

(ii) Below are the figures of production in a factory. Fit a straight line trend

Year

2005

2006

2007

2008

2009

2010

2011

Production

Ans: Methods of Measuring Trend in Time Series Analysis

Introduction

In business, economics, and statistics, analyzing trends is crucial for understanding long-

term movements in data. A trend refers to the general direction in which something moves

over time. For example, the increasing use of mobile phones, rising fuel prices, or the

decline in landline phones are all examples of trends.

To make informed decisions, businesses and analysts need to measure trends accurately.

There are four main methods to measure trends in time series data:

Easy2Siksha

1. Graphical Method

2. Moving Average Method

3. Method of Least Squares (Linear Trend Equation)

4. Semi-Average Method

Each method has its own advantages and applications, and we will discuss them in detail

with examples.

1. Graphical Method

This is the simplest and most visual way to measure a trend. It involves plotting the data

points on a graph and observing the pattern formed over time.

How It Works:

• The x-axis (horizontal axis) represents time (e.g., years, months, days).

• The y-axis (vertical axis) represents the variable being measured (e.g., sales,

temperature, population).

• By plotting the data points and connecting them with a line, we can see whether the

trend is increasing, decreasing, or stable.

Example:

Imagine a small business tracking its monthly sales revenue over the past five years. If the

plotted points show an upward movement, it indicates a growth trend. If the points move

downward, it signals a decline in sales.

Advantages:

✔ Easy to understand and interpret.

✔ Provides a quick idea of the trend direction.

✔ Suitable for non-mathematical users.

Disadvantages:

✖ Not precise; relies on visual judgment.

✖ Not useful for predicting future values accurately.

2. Moving Average Method

The moving average method smooths out short-term fluctuations and highlights the long-

term trend by averaging a fixed number of past observations.

How It Works:

• Choose a period for averaging (e.g., 3-month, 5-year, etc.).

• Calculate the average for that period and continue shifting forward.

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• The resulting trend line will be smoother and clearer.

Example:

A supermarket wants to analyze the monthly demand for a product (e.g., milk). Demand

fluctuates due to factors like holidays and weather. If the supermarket takes a 3-month

moving average, it will get a better idea of the general trend, reducing the impact of

temporary highs and lows.

Advantages:

✔ Reduces short-term fluctuations, making the trend clearer.

✔ Simple to calculate and understand.

✔ Helps in seasonal adjustments.

Disadvantages:

✖ Cannot predict long-term trends accurately.

✖ Less useful when there is a sudden change in trend direction.

3. Method of Least Squares (Linear Trend Equation)

This is the most mathematically accurate method. It uses a mathematical formula to fit a

straight-line trend to the data.

How It Works:

• It assumes that the trend follows a linear pattern (i.e., a straight-line increase or

decrease).

• A formula is used to find the best-fitting straight line through the data points.

• The equation of the trend line is:

Where:

o Y = predicted value of the trend

o X = time period (e.g., year, month)

o a = constant (starting value)

o b = rate of increase or decrease (slope)

Example:

A company’s profit over the last 10 years is recorded, and the trend is found to be

increasing. By using the least squares method, they calculate the trend equation and use it

to predict profits for the next few years.

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Advantages:

✔ Provides a precise mathematical trend.

✔ Can be used for forecasting future values.

✔ Works well when the data follows a straight-line pattern.

Disadvantages:

✖ Assumes a linear trend, which may not always be the case.

✖ Requires calculations, which may be difficult for non-mathematical users.

4. Semi-Average Method

This method divides the data into two equal parts and calculates the average of each half.

These averages are then used to identify the trend.

How It Works:

• The given data is split into two equal parts.

• The average of the first half is calculated.

• The average of the second half is calculated.

• A straight line is drawn between these two average points to represent the trend.

Example:

A school wants to analyze student enrollment over the last 10 years. They divide the data

into two halves (first 5 years and last 5 years), calculate the averages, and plot them. If the

second average is higher than the first, it shows an increasing trend in student enrollment.

Advantages:

✔ Simple to calculate and understand.

✔ Provides a rough idea of the trend direction.

Disadvantages:

✖ Less accurate than the least squares method.

✖ Assumes the trend follows a straight line, which may not always be true.

Comparison of Methods

Method

Best For

Main Drawback

Graphical Method

Quick visualization of trends

Not precise

Moving Average

Smoothing short-term fluctuations

Cannot predict

Least Squares

Accurate trend analysis and forecasting

Assumes linear trend

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Method

Best For

Main Drawback

Semi-Average

Simple trend identification

Less accurate

Conclusion

Each method of measuring trends has its own strengths and weaknesses. The choice of

method depends on the purpose of analysis, the nature of data, and the required accuracy.

• If a quick, visual representation is needed, the graphical method is best.

• If short-term fluctuations need to be removed, the moving average method is useful.

• For accurate trend analysis and future predictions, the least squares method is most

reliable.

• If a simple yet effective method is required, the semi-average method can be used.

(ii) Below are the figures of production in a factory. Fit a straight line trend

Year

2005

2006

2007

2008

2009

2010

2011

Production

Ans: Understanding the Concept of Trend in Data

Imagine you own a factory that produces goods every year, and you want to understand

how your production is changing over time. Is it increasing, decreasing, or fluctuating

randomly? One way to analyze this is by fitting a straight-line trend, which helps you see the

overall direction of change in production.

A straight-line trend is like drawing a best-fit straight line through your data points. This line

helps to predict future values and understand whether your production is improving or

declining over time.

To fit a straight-line trend, we use a mathematical equation of the form

Where:

• Y = Predicted production for a given year

• a = Intercept (starting point of the trend line when X = 0)

• b = Slope (rate of increase or decrease per year)

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• X = Year (expressed in a simple numerical form)

Step-by-Step Explanation of Fitting the Trend

Let's go step by step to fit a straight-line trend to the given production data.

Step 1: Convert the Years into Simple Numbers

When working with years like 2005, 2006, etc., the numbers are too large, making

calculations difficult. So, we convert the years into simpler values by assigning:

• X=−3 for 2005

• X=−2 for 2006

• X=−1 for 2007

• X=0 for 2008 (taking the middle year as reference)

• X=1X for 2009

• X=2X for 2010

• X=3X for 2011

This transformation helps in easier calculations.

Year

2005

2006

2007

2008

2009

2010

2011

-3

-2

-1

Production (Y)

Step 2: Calculate the Required Values

Now, we calculate:

1. Sum of X values:

Since the sum of X values is 0, it simplifies our calculations.

2. Sum of Y values (Total Production over the years):

3. Sum of X×Y (Multiplying each X by its respective Y value and summing them up):

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4. Sum of X

(Squaring each X value and summing them up):

Step 3: Calculate the Values of a and b

We use the formulas:

Where N is the number of years (7 in this case).

1. Find a (Average production over the years):

So, the intercept a=90, meaning that in the middle year (2008), the average production level

is 90.

2. Find b (Slope of the trend line):

This means the production is increasing by 2 units per year on average.

Step 4: Write the Trend Equation

Now that we have a=90 and b=2b , we can write the straight-line equation as:

Y=90+2X

This equation helps predict production for any given year.

Step 5: Use the Trend Line for Predictions

We can now use this equation to estimate production for any year:

1. For 2005 (X = -3):

The estimated production is 84, which is close to the actual production of 80.

2. For 2006 (X = -2):

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Estimated production is 86, close to the actual value 90.

3. For 2009 (X = 1):

The actual production was 94, so the estimate is reasonable.

This trend line gives a general idea of how production is changing, even if actual values

fluctuate.

Understanding the Importance of Trend Analysis

1. Predicting the Future:

If the trend continues, we can estimate production for 2012, 2013, etc.

o For 2012 (X = 4):

Y=90+2(4)=98 So, the estimated production for 2012 is 98 units.

2. Identifying Patterns:

o If production is increasing, managers can prepare for higher demand and

expansion.

o If decreasing, they can take corrective actions like improving efficiency or

marketing.

3. Comparing with Actual Data:

o If the actual production differs a lot from the trend line, something unusual

(like a factory shutdown or a big order) might have happened.

Real-Life Example:

Think of this trend line like tracking your weight over time. If you record your weight every

month and fit a straight-line trend, you can see if you’re gaining, losing, or maintaining

weight over time. Small fluctuations happen, but the overall trend tells the big picture.

Conclusion

Fitting a straight-line trend to data helps us analyze past performance and predict the

future. In this example, we found that production is increasing by 2 units per year on

average. This information can help businesses make smart decisions for the future.

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