NCERT Class 11-Math՚s: Exemplar Chapter – 15 Statistics Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

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15.1 Overview

In earlier classes, you have studied measures of central tendency such as mean, mode, median of ungrouped and grouped data. In addition to these measures, we often need to calculate a second type of measure called a measure of dispersion which measures the variation in the observations about the middle value – mean or median etc.

This chapter is concerned with some important measures of dispersion such as mean deviation, variance, standard deviation etc. , and finally analysis of frequency distributions.

15.1. 1 Measures of dispersion

(a) Range The measure of dispersion which is easiest to understand and easiest to calculate is the range. Range is defined as: Range Largest observation Smallest observation

(b) Mean Deviation

(i) Mean deviation for ungrouped data:

For observation the mean deviation about their mean is given by

Mean deviation about their median M is given by

(ii) Mean deviation for discrete frequency distribution

Let the given data consist of discrete observations occurring with frequencies , respectively. In this case

Where .

(iii) Mean deviation for continuous frequency distribution (Grouped data) .

Where are the midpoints of the classes, and are, respectively, the mean and median of the distribution.

(c) Variance: Let be n observations with as the mean. The variance, denoted by , is given by

(d) Standard Deviation: If is the variance, then σ, is called the standard deviation, is given by

(e) Standard deviation for a discrete frequency distribution is given by

Where s are the frequencies of s and .

(f) Standard deviation of a continuous frequency distribution (grouped data) is given by

Where xi are the midpoints of the classes and fi their respective frequencies. Formula (10) is same as

(g) Another formula for standard deviation:

Where h is the width of class intervals and and A is the assumed mean.

15.1. 2 Coefficient of variation:

It is sometimes useful to describe variability by expressing the standard deviation as a proportion of mean, usually a percentage. The formula for it as a percentage is