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
		$$\sum f_i=N=109$$		$$\sum f_i{u}_i=-0.5$$		$$\sum f_i{u_i}^2=261.25$$

$$\overline{X}=A+h\left(\frac{\sum f_i{u}_i}{N}\right)$$

$$=35+10\left(\frac{-0.5}{109}\right)$$

$$=34.96\text{ years }$$

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

$$=100[\frac{261.25}{109}-\frac{0.25}{11881}]$$

$$=100\times 2.396$$

$$=239.6$$

$$\sigma =\sqrt{239.6}$$

$$=15.47\text{ years }$$

Coefficient of skewness = Mean - Mode

                                    = 34.96 - 40

 

                                    = -5.04