Question

Table below shows the frequency f with which 'x' alpha particles were radiated from a diskette 

x :	0	1	2	3	4	5	6	7	8	9	10	11	12
f :	51	203	383	525	532	408	273	139	43	27	10	4	2

Calculate the mean and variance.

 

 

Answer 1

Mean,

\[\bar{X} = \frac{\sum f_i x_i}{\sum f_i} = \frac{10078}{2600} = 3 . 88\]

\[x_i\]	\[f_i\]	\[f_i x_i\]	\[x_i - \bar{X}\]	\[\left( x_i - \bar{X} \right)^2\]	\[f_i \left( x_i - \bar{X} \right)^2\]
0	51	0	−3.88	15.05	767.55
1	203	203	−2.88	8.29	1682.87
2	383	766	−1.88	3.53	1351.99
3	525	1575	−0.88	0.77	404.25
4	532	2128	0.12	0.014	7.448
5	408	2040	1.12	1.25	510
6	273	1638	2.12	4.49	1225.77
7	139	973	3.12	9.73	1352.47
8	43	344	4.12	16.97	729.71
9	27	243	5.12	26.21	707.67
10	10	100	6.12	37.45	374.5
11	4	44	7.12	50.69	202.76
12	2	24	8.12	65.93	131.86
	\[\sum f_i = N = 2600\]	\[\sum f_i x_i = 10078\]			\[\sum f_i \left( x_i - \bar{X} \right)^2 = 9448 . 848\]

Variance,

\[\sigma^2 = \frac{\sum f_i \left( x_i - \bar{X} \right)^2}{N} = \frac{9448 . 848}{2600} = 3 . 63\]