Question

Find the mean, mode, S.D. and coefficient of skewness for the following data: 

Year render:	10	20	30	40	50	60
No. of persons (cumulative):	15	32	51	78	97	109

Answer 1

Year render	No. of persons (cumulative)	\[f_i\]	\[u_i = \frac{x_i - 35}{10}\]	\[f_i u_i\]	\[{u_i}^2\]	\[f_i {u_i}^2\]
10	15	15	- 2.5	- 37.5	6.25	93.75
20	32	17	- 1.5	- 25.5	2.25	38.25
30	51	19	- 0.5	- 9.5	0.25	4.75
40	78	27	0.5	13.5	0.25	6.75
50	97	19	1.5	28.5	2.25	42.75
60	109	12	2.5	30	6.25	75
		\[\sum f_i = N = 109\]		\[\sum f_i u_i = - 0 . 5\]		\[\sum f_i {u_i}^2 = 261 . 25\]

\[\bar{X} = A + h\left( \frac{\sum f_i u_i}{N} \right)\]

\[ = 35 + 10\left( \frac{- 0 . 5}{109} \right)\]

\[ = 34 . 96 \text{ years } \]

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

\[ = 100\left[ \frac{261 . 25}{109} - \frac{0 . 25}{11881} \right]\]

\[ = 100 \times 2 . 396\]

\[ = 239 . 6\]

\[\sigma = \sqrt{239 . 6}\]

\[ = 15 . 47 \text{ years } \]

Coefficient of skewness = Mean - Mode

                                    = 34.96 - 40

 

                                    = -5.04