Activity- statistics problems

Problem 1

If n =10, ${\bar {x}}$ = 12 and
$\sum {x^{2}}=1530findthestandarddeviation$

Solution:

${\frac {\sum x}{n}}={\bar {x}}$

$\sum {x}={{\bar {x}}\times {n}}$

$\sum {x}={12\times {10}}$

$\sum {x}=120$
σ=${\sqrt {{\frac {\sum {x^{2}}}{n}}-({{\frac {\sum x}{n}})^{2}}}}$
σ=${\sqrt {{\frac {1530}{10}}-{144}}}$
σ=${\sqrt {{\frac {1530}{10}}-{144}}}$
σ=${\sqrt {153-144}}$
σ=${\sqrt {9}}$
σ=3

Problem 2

Problem No.1 of excercise No.6.1
Find the Standard deviation for the following data.

x	03	08	13	18	23
f	07	10	15	10	08

Solution:

x	f	fx	x²	fx²
03	07	021	009	0063
08	10	080	064	0640
13	15	195	169	2535
18	10	180	324	3240
23	08	184	529	4232
	n=50	Σfx=660		Σfx²=10710

Standard deviation σ=${\sqrt {{\frac {\sum {fx^{2}}}{n}}-({{\frac {\sum fx}{n}})^{2}}}}$
σ=${\sqrt {{\frac {10700}{50}}-({{\frac {660}{50}})^{2}}}}$
σ=${\sqrt {214-174.24}}$
σ=${\sqrt {39.96}}$
σ=6.3

Problem 3

Problem No.5 of excercise No.6.3
Find the varience and Standard deviation for the following data.

class intervals(CI)	30-34	34-38	38-42	42-46	46-50	50-54
freequency(f)	04	07	09	11	06	03

Solution:

C.I.	f	x	fx	d=${\frac {x-A}{c}}$	fd	d²	fd²
30-34	4	32	128	-2	-8	4	16
34-38	7	36	252	-1	-7	1	7
38-42	9	40	360	0	0	0	0
42-46	11	44	484	1	11	1	11
46-50	6	48	288	2	12	4	24
50-54	3	52	156	3	9	9	27
	n=40		Σfx=1668		Σfd=17		Σfd²=85

A=assumed average.
c=4
d=${\frac {x-A}{c}}$=${\frac {32-40}{4}}$=${\frac {-8}{4}}=-2$

assumed mean A=${\frac {\sum fx}{n}}$=${\frac {1668}{40}}=41.7$

Varience σ²=[${\frac {\sum {fd^{2}}}{n}}-({{\frac {\sum fx}{n}})^{2}}]c^{2}$

σ²=[${\frac {85}{40}}-({{\frac {17}{40}})^{2}}]4^{2}$

σ²=[2.125-0.180]16

σ²=[1.945]16

σ²=31.12
standard deviation, σ=${\sqrt {varience}}$
σ=${\sqrt {31.12}}$
σ=5.58