Question

the data is xi:0,1,2,3,4,5,6,7 fi:3,5,10,15,21,17,4,1​

Answer 1

Answer:

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 meas-

ures 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) RangeThe 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 n observation x1

, x2

, ..., xn

, the mean deviation about their mean x is

given by

M.D ( x ) =

| | i

x x

n

−

(1)

Mean deviation about their median M is given by

M.D (M) =

| M | i

x

n

−

(2)

(ii) Mean deviation for discrete frequency distribution

Let the given data consist of discrete observations x1

, x2

, ... , xn

occurring with

frequencies f

1

, f

2

, ... , f

n

, respectively. In this case

Chapter 15

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M.D ( x ) =

| | | |

N

i i i i

i

f x x f x x

f

− −

=

(3)

M.D (M) =

| M |

N

i i f x −

(4)

where N = i

f .

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

M.D ( x ) =

| |

N

i i f x x −

(5)

M.D (M) =

| M |

N

i i f x −

(6)

where xi

are the midpoints of the classes, x and M are, respectively, the mean

and median of the distribution.

(c) Variance : Let x1

, x2

, ..., xn

be n observations with x as the mean. The variance,

denoted by σ

2

, is given by

σ

2

=

1 2

( ) i

x x

n

− (7)

(d) Standard Deviation: If σ

2

is the variance, then σ, is called the standard deviation,

is given by

σ =

1 2

( ) i

x x

n

−

(8)

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

σ =

1 2

( )

N

i i f x x − (9)

where f

i

’s are the frequencies of xi

’ s and N =

1

n

i

i

f

=

.

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

is given by

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