Question

A frequency distribution of the of life times 400 T.V. picture tubes tested in a tube company is given below. Find the average life of tube.

Life time ( in hrs)	Frequency
300-399	14
400-499	46
500-599	58
600-699	76
700-68	68
800-899	62
900-999	48
1000-1099	22
1100-1199	6

Answer 1

Here, the class- intervals are formed by exclusive method. If we make the series an inclusive one the mid-values remain same. So, there is no need to convert the series into an inclusive form.
Let the assumed mean be A= 749.5 and h= 100.
Calculation of Mean

Life time (in hrs):	Frequency $f_i$	Mid-Values $x_i$	$d_i=x_i-A=x_i-749.5$	$u_i=\frac{x_i-A}{h}$ $u_i=\frac{x_i-749.5}{100}$	$f_i{u}_i$
300-399	14	349.5	-400	-4	-56
400-499	46	449.5	-300	-3	-138
500-599	58	549.5	-200	-2	-116
600-699	76	649.5	-100	-1	-76
700-799	68	749.5	0	0	0
800-899	62	849.5	100	1	62
900-999	48	949.5	200	2	96
1000-1099	22	1049.5	300	3	66
1100-1199	6	1149.5	400	4	24
	$N=\sum f_i=400$				$\sum f_i{u}_i$=-138

We have,
N= 400, A= 749.5, h= 100 and $\sum f_i{u}_i=-138$
$\therefore \overline{X}=A+h[\frac{1}{N}\sum f_i{u}_i]$
$\Rightarrow \overline{X}=749.5+100\left(\frac{-138}{400}\right)=749.5-34.5=715$.
Hence, the average life time of a tube is 715 hours.