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Statistics — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsStatistics3 Marks
Question
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.
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Answer

Given that $\text{n}_1=20,\sigma_1=5,\bar{\text{x}}_1=17$
and $\text{n}_2=20,\sigma_2=5,\bar{\text{x}}_2=22$
Now we know for combined two series that
$\sigma=\sqrt{\frac{\text{n}_1\text{s}^2_1+\text{n}_2\text{s}^2_2}{\text{n}_1+\text{n}_2}+\frac{\text{n}_1\text{n}_2(\bar{\text{x}}_1-\bar{\text{x}}_2)^2}{(\text{n}_1+\text{n}_2)^2}}$
$=\sqrt{\frac{20\times(5)^2+20\times(5)^2}{20+20}+\frac{20\times20(17-22)^2}{(20+20)^2}}$
$=\sqrt{\frac{1000}{40}+\frac{400\times25}{1600}}$
$=\sqrt{25+\frac{25}{4}}=\sqrt{\frac{125}{4}}$
$=\sqrt{31.25}=5.59$
Hence, the required SD = 5.59

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