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Home » Two sets each of 20 observations have the same standard derivation 5 . The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.
Two sets each of 20 observations have the same standard derivation 5 . The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.
CBSE | Class 11 | Maths | NCERT Exemplar | Statistics
Two sets each of 20 observations have the same standard derivation 5 . The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

Solution:

Provided: Two sets each of 20 observations, have the same standard derivation 5 . The first set has a mean 17 and the second a mean 22 .

We now need to show that the standard deviation of the set obtained by combining the given two sets

As per the criteria given, for the first set

No. of observations, n_{1}=20

Standard deviation, s_{1}=5

And mean, \bar{x}_{1}=17

For the second set, no. of observations, n_{2}=20

Standard deviation, s_{2}=5

And mean, \overline{\mathrm{x}}_{2}=22

It is known to us that the standard deviation for combined two series is

SD(\sigma)=\sqrt{\frac{n_{1} s_{1}^{2}+n_{2} s_{2}^{2}}{n_{1}+n_{2}}+\frac{n_{1} n_{2}\left(\bar{x}_{1}-\bar{x}_{2}\right)^{2}}{\left(n_{1}+n_{2}\right)^{2}}}

Substitute the corresponding values,

SD(\sigma)=\sqrt{\frac{(20)(5)^{2}+(20)(5)^{2}}{20+20}+\frac{(20 \times 20)(17-22)^{2}}{(20+20)^{2}}}

On simplifying we obtain

SD(\sigma)=\sqrt{\frac{(20) 25+(20) 25}{40}+\frac{(400)(-5)^{2}}{(40)^{2}}}

SD(\sigma)=\sqrt{\frac{2(500)}{40}+\frac{(400)(25)}{1600}}

SD(\sigma)=\sqrt{\frac{1000}{40}+\frac{10000}{1600}}

\mathrm{SD}(\sigma)=\sqrt{25+\frac{25}{4}}

Take LCM and simplify,

SD(\sigma)=\sqrt{\frac{100+25}{4}}

SD(\sigma)=\sqrt{\frac{125}{4}}

Or, \sigma=5.59

As a result, 5.59 is the standard deviation of the set obtained by combining the given two sets.

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