Set A of six numbers has a standard deviation of 3 and set B of four numbers has a standard deviation of 5. Both sets of numbers have an equal mean. If the two sets of numbers a…

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asked Feb 3, 2022 24.6k views

1 Answer

Chinmay Atrawalkar · Answer 1 · 2022-02-08T13:59:19+0000

Given:

$\sigma_A=3$

n_A=6

$\sigma_B=5$

n_B=4

$\overline{x}_A=\overline{x}_B$

To find:

The variance. of combined set.

Solution:

Formula for variance is

$\sigma^2=\frac{\sum (x_i-\overline{x})^2}{n}$ ...(i)

Using (i), we get

$\sigma_A^2=\frac{\sum (x_i-\overline{x}_A)^2}{n_A}$

$(3)^2=\frac{\sum (x_i-\overline{x}_A)^2}{6}$

$9=\frac{\sum (x_i-\overline{x}_A)^2}{6}$

$54=\sum (x_i-\overline{x}_A)^2$

Similarly,

$\sigma_B^2=\frac{\sum (x_i-\overline{x}_B)^2}{n_B}$

$(5)^2=\frac{\sum (x_i-\overline{x}_B)^2}{4}$

$25=\frac{\sum (x_i-\overline{x}_B)^2}{4}$

$100=\sum (x_i-\overline{x}_B)^2$

Now, after combining both sets, we get

$\sigma^2=\frac{\sum (x_i-\overline{x}_A)^2+\sum (x_i-\overline{x}_B)^2}{n_A+n_B}$

$\sigma^2=(54+100)/(6+4)$

$\sigma^2=(154)/(10)$

$\sigma^2=15.4$

Therefore, the variance of combined set is 15.4.