Question

Find the standard deviation for the following data:

$x_i$	4	8	11	17	20	24	32
$f_i$	3	5	9	5	4	3	1

Answer 1

Answer: 6.77

$x_i$	$f_i$	$f_i{x}_i$	$(x_i-\bar{x})$	$(x_i-\bar{x}{)}^2$	$f_i(x_i-\bar{x}{)}^2$
4	3	12	-10	100	300
8	5	40	-6	36	180
11	9	99	-3	9	81
17	5	85	3	9	45
20	4	80	6	36	144
24	3	72	10	100	300
32	1	32	18	324	324
Total	30	420			1374

$N=30,\sum _{i=1}^7{f}_i{x}_i=420$,
$\sum _{i=1}^7{f}_i{\left(x_i-\bar{x}\right)}^2=1374$
Therefore, $\bar{x}=\frac{\sum _{i=1}^7{f}_i{x}_i}{N}=\frac{1}{30}\times 420=14$
Hence, variance $\left({\sigma }^2\right)$
$=\frac{1}{N}\sum _{i=1}^7{f}_i{\left(x_i-\bar{x}\right)}^2$
$=\frac{1}{30}\times 1374=45.8$
and Standard deviation $(\sigma )=\sqrt{45.8}=6.77$