Question
Table below shows the frequency $f$ with which $'x'$ alpha particles 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

Mean, $\overline{\text{x}}=\frac{\sum\text{f}_{\text{i}}\text{x}_{\text{i}}}{\sum\text{f}_{\text{i}}}=\frac{10078}{2600}=3.88$
$x_i$
$f_i$
$f_ix_i$
$\text{x}_{\text{i}}-\overline{\text{X}}$
$\big(\text{x}_{\text{i}}-\overline{\text{X}}\big)^2$
$\text{f}_{\text{i}}\big(\text{x}_{\text{i}}-\overline{\text{X}}\big)^2$
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
 
$\sum\text{f}_{\text{i}}=\text{N}=2600$
$\sum\text{f}_{\text{i}}\text{x}_{\text{i}}=10078$
 
 
$\sum\text{f}_{\text{i}}\big(\text{x}_{\text{i}}-\overline{\text{X}}\big)^2=9448.848$
Variance, $\sigma^2=\frac{\sum\text{f}_{\text{i}}\big(\text{x}_{\text{i}}-\overline{\text{X}}\big)^2​​}{{\text{N}}}=\frac{9448.848}{2600}=3.63$

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