Question

Calculate the mean, median and standard deviation of the following distribution:

Class-interval:	31-35	36-40	41-45	46-50	51-55	56-60	61-65	66-70
Frequency:	2	3	8	12	16	5	2	3

Answer 1

Class Interval	\[f_i\]	Midpoint  \[x_i\]	\[u_i = \frac{x_i - 53}{4}\]	ui 2	\[f_i u_i\]	\[f_i {u_i}^2\]
31−35	2	33	-5	25	- 10	50
36−40	3	38	-3.75	14.06	- 11.25	42.18
41−45	8	43	-2.5	6.25	- 20	50
46−50	12	48	-1.25	1.56	- 15	18.72
51−55	16	53	0	0	0	0
56−60	5	58	1.25	1.56	6.25	7.8
61−65	2	63	2.5	6.25	5	12.5
66−70	3	68	3.75	14.06	11.25	42.18
	N = 51					\[\sum^n_{i = 1} f_i {u_i}^2 = 223 . 38\]

\[X^{} = a + h\left( \frac{\sum^n_{i = 1} f_i u_i}{N} \right)\]
\[ = 53 + 4\left( \frac{- 33 . 75}{51} \right)\]
\[ = 50 . 36\]

\[\sigma^2 = h^2 \left( \frac{\sum^n_{i = 1} f_i {u_i}^2}{N} - \left( \frac{\sum^n_{i = 1} f_i u_i}{N} \right)^2 \right)\]

\[ = 16\left( \frac{223 . 38}{51} - \frac{1139 . 06}{2601} \right)\]

\[ = 63 . 07\]

\[\sigma = \sqrt{63 . 07}\]

\[ = 7 . 94\]

\[f_i\]	\[CF\] (Cumulative frequency)
2	2
3	5
8	13
12	25
16	41
5	46
2	48
3	51

\[\sum f_i = 51 = N\]
\[\frac{N}{2} = 25 . 5\]

Median class interval is 51−55.

\[L = 51\]
\[F = 25\]
\[f = 16\]
\[h = 4\]

\[Median = L + \frac{\frac{N}{2} - F}{f} \times h\]

\[= 51 + \frac{25 . 5 - 25}{16} \times 4\]
\[ = 51 + \frac{0 . 5}{4}\]
\[ = 51 . 125\]