Maharashtra Board Class 11 Economics Chapter 3 Partition Values PDF Download

Chapter 3: Partition Values

Let's Recall

Are you familiar with the word 'averages'?

Can you tell the meaning of individual series, discrete data and continuous data.

Name the positional averages that you have previously studied.

Introduction

The procedure of dividing the data into equal parts is called 'partitioning'. Values dividing the data into a required number of equal parts are called 'Partition Values'.

In Class X, you have already studied about the measures of central tendency i.e. averages such as Arithmetic Mean, Median and Mode. Median is the value of the middlemost observation in the data when the observations are arranged in increasing or decreasing order of values. 'Median', is a special type of partition value because there are equal number of observations above as well as below it. Like Median, Quartiles, Deciles and Percentiles are also partition values, since they divide the given set of observations into equal number of parts.

In general, they are referred to as 'fractiles'. Partition values form a part of descriptive statistics.

In the forthcoming chapters such as Population, Unemployment and Poverty, students will get acquainted with the use of partition values in economic data analysis.

Do You Know?

Statistics Day: Prof. Prasanta Chandra Mahalanobis, an Indian Statistician was instrumental in formulating India's strategy for industrialization in the Second Five Year Plan (1956-61) which later came to be known as Mahalanobis Model.

Mahalanobis devised a measure of comparison between two data sets that is known as the Mahalanobis distance. He also devised a statistical method called 'fractile graphical analysis' which could be used to compare the socio-economic conditions of different groups of people. In recognition of the notable contributions made by P. C. Mahalanobis in the field of economic planning and statistical development, the Government of India has designated 29th of June every year, coinciding with his birth anniversary as 'Statistics Day', in the category of Special day to be celebrated at the national level.

Teacher's Note

Partition values help us understand data better by dividing it into equal parts. For example, schools use quartiles to compare student scores in exams.

Exam Trick

Remember: Q2 = D5 = P50 = Median. All three give you the middle value. They are just different names for the same thing!

Points to Remember

Partitioning means dividing data into equal parts.
Quartiles divide data into 4 parts.
Deciles divide data into 10 parts.
Percentiles divide data into 100 parts.
Median is the same as the second quartile.

Need For Partition Values

The data consists of extreme values on the lower side and also on the higher side in magnitude. Such values are known as 'outliers'. The average used for such data often misinterprets its representative value. To overcome this misinterpretation, generally partition values like median, quartiles, deciles and percentiles are used.

Always Remember

\(Q_2 = D_5 = P_{50} = \text{Median}\)

You Should Know

Application of Quartiles, Deciles and Percentiles in Economics:

Quartiles are used in the study of all types of financial information concerning economic data, income data, stock data, sales and survey data etc.

Income quartiles is the most objective method of comparing changes among individual income groups caused by economic changes such as wage fluctuations and inflation.

Deciles too have wide application in finance and economics. Government uses deciles to study the level of economic inequality, measurement of poverty line, drought conditions etc.

Deciles are used in investments, particularly to assess the performance of a portfolio investment such as a group of mutual funds.

Percentiles are used in the measurement of test scores, health indicators, household income, household wealth, percentile wages.

Percentiles can be used for benchmarking and baseline purposes.

Teacher's Note

Quartiles help us compare income groups and understand economic changes. For example, India uses quartiles to study if rich and poor groups are getting more equal or unequal.

Exam Trick

Remember the numbers: Quartiles divide into 4, Deciles into 10, and Percentiles into 100. The bigger the denominator, the more parts!

Points to Remember

Quartiles are useful for financial data like income and stock prices.
Deciles help measure poverty and economic inequality.
Percentiles measure test scores and health data.
Q2 is always the median value.
Outliers are extreme values that can mislead averages.

Let's Learn: Quartiles

Meaning

'Quartiles' are values of data which divide the whole set of observations into four equal parts. There are three Quartiles which divide the data into 4 equal parts, when data is arranged in ascending or descending order. They are known as Q₁, Q₂ and Q₃ respectively. Second quartile is nothing but the median.

It is explained in the following example:

(1) Q₁ (2) Q₂ (3) Q₃ (4)

a) In general, for individual and ungrouped data we get the formula for Q₁, Q₂ and Q₃ as given below:

\(Q_i = \text{size of } i\left(\frac{n+1}{4}\right)^{\text{th}} \text{Observation}\)

i = 1, 2, 3

b) For grouped data or continuous data,

\(Q_i = l + \left(\frac{in}{4} - cf\right) \times \frac{h}{f}\) i = 1, 2, 3

Where

l = Lower limit of quartile class.

f = Frequency of the quartile class

cf = Cumulative frequency of the class preceding the quartile class.

n = Total of frequency.

h = Upper limit - lower limit of the quartile class.

Calculation of Quartiles

Solved Examples

A) Individual Data

1) Calculate Q₁ and Q₃ of the first semester examination marks scored by the students as given: 40, 85, 84, 83, 82, 69, 68, 65, 64, 55, 45

Solution: Arrange the series in ascending order i.e. 40, 45, 55, 64, 65, 68, 69, 82, 83, 84, 85

n = Total number of observations

n = 11

\(Q_1 = \text{size of } \left(\frac{n+1}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_1 = \text{size of } \left(\frac{11+1}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_1 = \text{size of } \left(\frac{12}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_1 = \text{size of } 3^{\text{rd}} \text{Observation}\)

Q₁ = size of 3rd Observation is 55

∴ Q₁ = 55

Third Quartile

\(Q_3 = \text{size of } 3\left(\frac{n+1}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } 3\left(\frac{11+1}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } 3\left(\frac{12}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } (3 \times 3)^{\text{th}} \text{Observation}\)

Q₃ = size of 9th Observation is 83

∴ Q₃ = 83

Ans: Q₁ = 55, Q₃ = 83

2) Calculate Q₃ for the given distribution.

20, 28, 31, 18, 19, 17, 32, 33, 22, 21

Solution: Arrange the data in ascending order.

17, 18, 19, 20, 21, 22, 28, 31, 32, 33

n = 10

\(Q_3 = \text{size of } 3\left(\frac{n+1}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } 3\left(\frac{10+1}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } \left(3 \times \frac{11}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } \left(\frac{33}{4}\right)^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } 8.25^{\text{th}} \text{Observation}\)

\(Q_3 = \text{size of } 8^{\text{th}} \text{observation} + 0.25(9^{\text{th}} \text{observation} - 8^{\text{th}} \text{observation})\)

Q₃ = 31 + 0.25(32 - 31)

Q₃ = 31 + 0.25 × 1

∴ Q₃ = 31.25

Ans: Q₃ = 31.25

Teacher's Note

Quartiles help arrange students into groups based on exam marks. For example, Q₁ shows the lowest 25% of students, and Q₃ shows the lowest 75% of students.

Exam Trick

To find quartile position, use the formula with (n+1)/4 for individual data. Always arrange data first from smallest to largest number!

Points to Remember

Q₁ is at position (n+1)/4 in the arranged data.
Q₂ is always the median value.
Q₃ is at position 3(n+1)/4 in the arranged data.
Always arrange data in order before finding quartiles.
For fractional positions, use interpolation method.

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