Question

A group of 10 observations has mean 5 and S.D. $2\sqrt 6 $. Another group of 20 observations has mean 5 and S.D. $3\sqrt 2 $ , then the S.D. of combined group of 30 observations is:
A) $\sqrt 5 $
B) $2\sqrt 5 $
C) $3\sqrt 5 $
D) None of these

Answer 1

Hint:We will first write down the formulas of combined mean and combined standard deviation, which we are going to use. Now, after then we will first find the combined mean which is required in the formula for combined S.D. and thus putting in values, we will get the required answer.

Formula used:
Complete step-by-step answer:
Let us first write the formula for Standard Deviation of the combined data and the arithmetic mean of the combined data which we will require.
S.D. for the combined data is given by $\sqrt {\dfrac{{{n_1}({\sigma _1}^2 + {d_1}^2) + {n_2}({\sigma _2}^2 + {d_2}^2)}}{{{n_1} + {n_2}}}} $ …….(1); where ${n_1}$ and ${n_2}$ stands for the number of observations in first group and second group respectively; ${\sigma _1}$ and ${\sigma _2}$ stand for the standard deviations of group 1 and group 2 respectively; ${d_1} = {\bar x_1} - \overline {{x_g}} $ and ${d_2} = {\bar x_2} - \overline {{x_g}} $ where ${\bar x_1},{\bar x_2},\overline {{x_g}} $ are the arithmetic mean of group 1, group 2 and combined data respectively.
Now, combined mean is given by ${\bar x_g} = \dfrac{{{n_1}{{\bar x}_1} + {n_2}{{\bar x}_2}}}{{{n_1} + {n_2}}}$ ………..(2)
Now, we are already given that ${n_1} = 10,{\sigma _1} = 2\sqrt 6 ,{\bar x_1} = 5$ and ${n_1} = 20,{\sigma _1} = 3\sqrt 2 ,{\bar x_2} = 5$. ……..(3)
Putting these values in (2), we will get:-
$ \Rightarrow {\bar x_g} = \dfrac{{10 \times 5 + 20 \times 5}}{{10 + 20}}$
On simplifying the RHS, we will get:-
$ \Rightarrow {\bar x_g} = 5$ ……….(4)
Now, finding ${d_1}$: ${d_1} = {\bar x_1} - \overline {{x_g}} = 5 - 5 = 0$. …….(5)
Now, finding ${d_2}$: ${d_2} = {\bar x_2} - \overline {{x_g}} = 5 - 5 = 0$ ………(6)
Now, putting in (3), (4), (5) and (6) in (1), we will get:-
Combined S.D. will be $\sqrt {\dfrac{{10\left\{ {{{(2\sqrt 6 )}^2} + {0^2}} \right\} + 20\left\{ {{{(3\sqrt 2 )}^2} + {0^2}} \right\}}}{{30}}} $
On simplifying it, we will get:-
Combined S.D. = $\sqrt {\dfrac{{10 \times 24 + 20 \times 18}}{{30}}} $
On simplifying it further, we will get:-
Combined S.D. = $2\sqrt 5 $.

So, the correct answer is “Option B”.

Note:The students must note that they can be provided with the data instead of all this, they will first require to find the mean and standard deviation separately and then use the formula.Students must know that standard deviation actually represents how much the members of a group differ from the mean value. This implies that the more the standard deviation, the more scattered our data is and the less the standard deviation and more close our data is.Mean in simpler words is the sum divided by the count.