Question

A distribution consists of three components with frequencies 200,250 and 300 having means 25,10,and 15 and standard deviations 3,4, and 5 respectively.Calculate the mean and standard deviation.​

Answer 1

Answer:

The mean and standard deviation of the combined distribution is 16 and 7.192 respectively.

Step-by-step explanation:

We have given that a distribution consists of three components with frequencies 200, 250, and 300 having means 25, 10, and 15 and standard deviations 3, 4, and 5 respectively.

And we have to find the mean and standard deviation of the combined distribution.

Firstly let us represent some symbols;

$n_1$ = 200           $\bar X_1$ = 25                 $\sigma_1$ = 3

$n_2$ = 250            $\bar X_2$ = 10                  $\sigma_2$ = 4

$n_3$ = 300            $\bar X_3$ = 15                  $\sigma_3$ = 5  

Here, $\bar X_1, \bar X_2 , \bar X_3$ represent the means and $\sigma_1,\sigma_2,\sigma_3$ represent the standard deviations.

Now, as we know that Mean of the combined distribution is given by;

                         $\bar X = \frac{n_1 \times \bar X_1+n_2 \times \bar X_2+n_3 \times \bar X_3}{n_1+n_2+n_3}$

Putting the above values in the formula we get;

                         $\bar X = \frac{200 \times 25+250 \times 10+300 \times 15}{200+250+300}$

                         $\bar X = \frac{5000+2500 +4500}{750}$

                         $\bar X = \frac{12000}{750}$  =  16

Similarly, the formula for combined standard deviation is given by;

       $\sigma = \sqrt{\frac{n_1\sigma_1^{2} + n_1(\bar X_1-\bar X)^{2}+n_2\sigma_2^{2} + n_2(\bar X_2-\bar X)^{2}+n_3\sigma_3^{2} + n_3(\bar X_3-\bar X)^{2} }{n_1+n_2+n_3} }$

       $\sigma = \sqrt{\frac{(200 \times 3^{2}) + 200 \times (25-16)^{2}+(250 \times 4^{2}) + 250 \times (10-16)^{2}+(300 \times 5^{2}) + 300 \times (15-16)^{2} }{200+250+300} }$

       $\sigma = \sqrt{\frac{1800 + (200 \times 81)+4000 + (250 \times 36)+7500 +( 300 \times 1) }{750} }$        

       $\sigma = \sqrt{\frac{1800 + 16200+4000 + 9000+7500 +300 }{750} }$  

       $\sigma = \sqrt{\frac{38800 }{750} }$  =  7.192

Hence, the mean and standard deviation of the combined distribution is 16 and 7.192 respectively.