# NCERT Class 11-Math՚s: Exemplar Chapter – 15 Statistics Part 1 (For CBSE, ICSE, IAS, NET, NRA 2023)

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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

## 15.2 Solved Examples

### Short Answer Type

**Question 1**:

Find the mean deviation about the mean of the following data:

Size : | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 |

Frequency (f) : | 3 | 3 | 4 | 14 | 7 | 4 | 3 | 4 |

**Answer**: