Question: From the following data, find the standard deviation and coefficient of variation.


Class Interval 20-30 30-40 40-50 50-60 60-70 70-80
Frequency 10 15 25 30 12 8

Solution:
Using actual mean,
[We have, interval between two classes is 10.
To get the frequency, subtract the give frequency of the respective class from one step higher class.]

Arranging the data in a table,


X f m fm d|x-21.5| d² fd²
20-30 10 25 250 24.3 590.49 5904.9
30-40 15 35 525 14.3 204.49 3067.35
40-50 25 45 1125 4.3 18.49 462.25
30-40 30 55 1650 5.7 32.49 974.7
40-50 12 65 780 15.7 246.49
 2957.88
70-808 75 600 25.7 660.49 5283.92
N=100
$\sum$fm
 = 4930		 $\sum$fd²
=18651
Now,

Mean = $\dfrac{\sum fm}{N}$

$= \dfrac{4930}{100}$
$= 49.3$
 

Standard Deviation ($\sigma$) = $ \sqrt{ \dfrac{\sum fd²}{N} }$

$= \sqrt{ \dfrac{18651}{100} }$

$= \sqrt{186.51}$
$= 13.66$
And,
Coefficient of Variation = $\dfrac{\sigma}{mean} × 100%$

$= \dfrac{13.66}{49.3} × 100%$

$= 0.2770 × 100%$
$= 27.7%$
Hence, the required standard deviation using actual mean of the given data is 13.66 and the coefficient of variation is 27.7%.

Related Notes and Solutions:

Here is the website link to the notes of Statistics of Class 10.

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