STANDARD DEVIATION FOR CONTINUOUS DATA

Mean Method :

where x_i = Middle value of the i^th class.

f_i = Frequency of the i_th class

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 i^th class and c is the width of the class interval.

Example 1 :

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

Solution :

x is the mid value of the given set.

d = (x - 34.5)/2

Σf  =  N  =  84

Σfd²  =  1052

(Σfd²/N)  =  (1052/84)  =  263/21

Σfd  =  -158

(Σfd/N)²  =  (-158/84)² =  (79/42)² 

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.

Example 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

Σf  =  N  =  50

Σfd²  =  78

(Σfd²/N)  =  (78/50)  =  39/25

Σfd  =  8

(Σfd/N)²  =  (8/50)² =  (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.

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