Question

The mean of a numerical data set is $\overline{x}$ and the standard deviation is S. if a number P is added to each term in the data set then the mean and standard deviation become.

• A. $\overline{x},S$
• B. $\overline{x}+P,S$
• C. $\overline{x},S+P$
• D. $\overline{x}+P,S+P$

Answer 1

The correct option is B $\overline{x}+P,S$

Let's take an example:

We have to find mean $\overline{x}$ and standard deviation(s) the given numbers : 1, 2 and 3

Mean, $\overline{x}=\frac{1+2+3}{3}=2$

Standard deviation $S=\sqrt{\frac{\sum (x-\overline{x}{)}^2}{n-1}}$

$=\sqrt{\frac{(1-2{)}^2+(2-2{)}^2+(3-2{)}^2}{3-1}}$

$=\sqrt{\frac{1+0+1}{2}}$

S = 1

Now, add a number 1 (here P = 1) in each numbers, i.e.
1+1, 2+1, 3+1

i.e. 2, 3, 4

Now, mean $\left({\overline{X}}^{\prime }\right)=\frac{2+3+4}{3}=3=\overline{x}+1$

So we can say that new mean ${\overline{X}}^{\prime }=\overline{X}+P$

Now, standard deviation,

$S^{\prime }=\sqrt{\frac{(2-3{)}^2+(3-3{)}^2+(4-3{)}^2}{3-1}}$

$S=\sqrt{\frac{1+0+1}{2}}$

S' = 1 = S (remain same)

On the above discussion, we can say that, by adding a number P to each term in the data set mean $\left(\overline{x}\right)$ and standard deviation (S), the new mean and standard deviation will be $\overline{x}+P$ and S.