Question

Find the standard deviation of the data $3,5,6,7$. Then add $4$ to each item and find the standard deviation of the new data.

Answer 1

Given data $3,5,6,7$
Take $A=6$

$x$ $d=x-10$ $d^2$ $3$ $-3$ $9$ $5$ $-1$ $1$ $6$ $0$ $0$ $7$ $1$ $1$
	$\sum d=-3$	$\sum d^2=11$

Standard deviation, $\sigma =\sqrt{\frac{\sum d^2}{n}-{\left(\frac{\sum d}{n}\right)}^2}$
$=\sqrt{\frac{11}{4}-{\left(\frac{-3}{4}\right)}^2}$
$\sigma =\sqrt{\frac{11}{4}-\frac{9}{16}}=\frac{\sqrt{35}}{4}$
Let us add $4$ to each term of the given data to get the new data $7,9,10,11$
Take A$=10$

$x$	$d=x-10$	$d^2$
$7$	$-3$	$9$
$9$	$-1$	$1$
$10$	$0$	$0$
$11$	$1$	$1$
	$\sum d=-3$	$\sum d^2=11$

Standard deviation, ${\sigma }_1=\sqrt{\frac{\sum d^2}{n}-{\left(\frac{\sum d}{n}\right)}^2}$
$=\sqrt{\frac{11}{4}-{\left(\frac{-3}{4}\right)}^2}$
${\sigma }_1=\sqrt{\frac{11}{4}-\frac{9}{16}}=\frac{\sqrt{35}}{4}$
In the example, the standard deviation remains unchanged even when each item is added by the constant $4$.