# STANDARD DEVIATION OF GROUPED DATA IN INTERVAL

Standard Deviation of grouped Data in Interval :

Here we are going to see, some example problems of finding standard deviation for the grouped data.

## Calculation of Standard deviation for grouped data

Mean method

Standard deviation

(ii)  Assumed Mean method

Let x1, x2, x3, ...........xn be the given data with frequencies f1, f2,f3, .......fn respectively. Let x bar be their mean and A be the assumed mean.

## Standard Deviation for Continuous Frequency Distribution

(i) Mean method

Standard deviation

where xi =Middle value of the i th class.

fi = Frequency of the ith class

(ii) Shortcut method (or) Step deviation method :

To make the calculation simple, we provide the following formula. Let A be the assumed mean, xi be the middle value of the ith class and c is the width of the class interval.

## Standard Deviation of grouped Data in Interval - Examples

Question 1 :

The measurements of the diameters (in cms) of the plates prepared in a factory are given below. Find its standard deviation.

Solution :

 20.5 - 24.515 24.5 - 28.518 28.5 - 32.520 32.5 - 36.516 36.5 - 40.58 40.5 - 44.57

x is the mid value of the given set.

d = (x - 34.5)/2

 x22.526.530.534.538.542.5 d -6-4-2024 f1518201687 d2361640416 fd254028880032112 fd-90-72-4001628

Σf  =  N  =  84

Σfd =  1052

(Σfd2/N)  =  (1052/84)  =  263/21

Σfd  =  -158

(Σfd/N)2  =  (-158/84)=  (79/42)

σ  =  √(22092-6241)/1764

σ  =  (√15851/1764) ⋅ 2

σ  =  √8.98 ⋅ 2

σ  =  2.99 ⋅ 2

σ  =  5.99

Hence the required standard deviation is 6.

Question 2 :

The time taken by 50 students to complete a 100 meter race are given below. Find its standard deviation

Solution :

x is the mid value of the given set.

d = (x - 22)/2

 x1820222426 d -2-1012 f6817109 d241014 fd224801036 fd-12-801018

Σf  =  N  =  50

Σfd =  78

(Σfd2/N)  =  (78/50)  =  39/25

Σfd  =  8

(Σfd/N)2  =  (8/50)=  (4/25)2   =  16/625

σ  =  √(975 - 16)/625

σ  =  √959/625

σ  =  (√1.533) ⋅ 2

σ  =  1.23 ⋅ 2

σ  =  2.47

Hence the required standard deviation is 2.47.

After having gone through the stuff given above, we hope that the students would have understood, "Standard Deviation of grouped Data in Interval".

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