Question

if the sum and sum of squares of 10 observations are 12 and 18 respectively, then the S.D of the observation is,
(a)$\dfrac{1}{5}$
(b)$\dfrac{2}{5}$
(c)$\dfrac{3}{5}$
(d)$\dfrac{4}{5}$

Answer 1

Hint: To solve this question, we will first determine the values of n which represents the number of observation, mean a sum of the square of observation, and then, we will substitute the values in the formula of Standard deviation and on solving, we will have a value of standard deviation.

Complete step by step answer:
Before we solve this question, let us see what does mean and S.D of observation means.
The mean of observation in statistics is the sum of a collection of numbers divided by the count of numbers in the collection, where the collection is set of an observational study or survey.
Let us consider n – items say, ${{n}_{1}},{{n}_{2}},{{n}_{3}},........,{{n}_{n}}$ , then the average or mean of these n – items will be $\bar{x}=\dfrac{{{n}_{1}}+{{n}_{2}}+........+{{n}_{n}}}{n}$ , where $\bar{x}$ denotes average or mean value.
Standard Deviation in statistics is a measure of the amount of variation or dispersion of a set of values, where a low standard deviation indicates that the values tend to be close to the mean of the set, while high standard deviation indicates that the values are spread out over a wider range.
Let us consider n – items say, ${{n}_{1}},{{n}_{2}},{{n}_{3}},........,{{n}_{n}}$ , then the standard deviation of these n items will be $S.D=\sqrt{\dfrac{\sum\limits_{1}^{n}{{{x}_{i}}^{2}}}{n}-{{\left( \dfrac{\sum\limits_{1}^{n}{{{x}_{i}}}}{n} \right)}^{2}}}$ , where $\sum\limits_{i=1}^{n}{{{x}_{i}}^{2}}$ denotes the sum of squares of values of n items and $\sum\limits_{i=1}^{n}{{{x}_{i}}}$ denotes the sum of values of n items.
Now in question, we are provided that the sum of 10 items is 12 and sum of squares of 10 items is 18.
Let, 10 observations be ${{x}_{1}},{{x}_{2}},....,{{x}_{10}}$, then
So, we have n = 10, $\sum\limits_{i=1}^{10}{{{x}_{i}}}={{x}_{1}}+{{x}_{2}}+....+{{x}_{10}}=12$, and $\sum\limits_{i=1}^{10}{{{x}_{i}}^{2}}={{x}_{1}}^{2}+{{x}_{2}}^{2}+....+{{x}_{10}}^{2}=18$
Substituting, values of n, $\sum\limits_{i=1}^{10}{{{x}_{i}}}={{x}_{1}}+{{x}_{2}}+....+{{x}_{10}}=12$ and $\sum\limits_{i=1}^{10}{{{x}_{i}}^{2}}={{x}_{1}}^{2}+{{x}_{2}}^{2}+....+{{x}_{10}}^{2}=18$ in formula of Standard Deviation, we get
$S.D=\sqrt{\dfrac{18}{10}-{{\left( \dfrac{12}{10} \right)}^{2}}}$
On simplifying, we get
$=\sqrt{\dfrac{9}{5}-\dfrac{36}{25}}$
On solving, we get
$=\sqrt{\dfrac{45-36}{25}}$
\[=\dfrac{3}{5}\]
Hence, option ( c ) is correct.

Note:
 To solve such a question, one must know the concept and formulas of mean and standard deviation as without formula, it is not possible to solve these problems. The calculation part is a bit harder so, try not to make any calculation mistakes.