Question

The following table shows the frequency distribution of milk(in litres) produced per week by 50 cows.

Milk(in litres)	24-30	30-36	36-42	42-48	48-54	54-60	60-66	66-72	72-78	78-84	84-90
No. of cows	1	3	8	5	5	5	8	4	6	2	3

Using the shift of origin method, find the average (mean) amount of milk produced by the cows.

Answer 1

Milk (in litres) Class Intervals	Class Marks $\left(x_i\right)$	$$\begin{gathered} d_i=x_i-A \\ {d}_i=x_i-57 \end{gathered}$$	$u_i=\frac{d_i}{h}=\frac{d_i}{6}$	No. of cows $\left(f_i\right)$	$f_i{u}_i$
24-30	27	$-$30	$-$5	1	$-$5
30-36	33	$-$24	$-$4	3	$-$12
36-42	39	$-$18	$-$3	8	$-$24
42-48	45	$-$12	$-$2	5	$-$10
48-54	51	$-$6	$-$1	5	$-$5
54-60	$57\to A$	0	0	5	0
60-66	63	6	1	8	8
66-72	69	12	2	4	8
72-78	75	18	3	6	18
78-84	81	24	4	2	8
84-90	87	30	5	3	15
Total	-	-	-	$\sum _{}^{}{f}_i=50$	$\sum _{}^{}{f}_i{u}_i=1$

$$\begin{gathered} \overline{u}=\frac{{\displaystyle \sum _{}^{}}{f}_i{u}_i}{{\displaystyle \sum _{}^{}}{f}_i}=\frac{1}{50}=0.02 \\ \mathrm{Mean}=\overline{x}=A+h\overline{u} \\ =57+6(0.02) \\ =57+0.12=57.12 \end{gathered}$$

The mean amount of milk produced by the cows is 57.12 litres.