Question

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying numbers of mangoes. The following was the distribution of mangoes according to the number of boxes. Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

Number of mangoes	50-52	53-55	56-58	59-61	62-64
Number of boxes	15	110	135	115	25

Answer 1

Find the mean:

We can observe that the class intervals are not continuous. So, to make them a continuous set of data we add $0.5$ to the upper limit and subtract $0.5$ from the lower limit since the gap between the two intervals is $1$.

Now, to determine the Mid Point, we use the formula:

Class Mark $=\frac{(\mathrm{Upper}\mathrm{class}\mathrm{Limit}+\mathrm{Lower}\mathrm{Class}\mathrm{Limit})}{2}$

In this case, let us assume the mean value, $A=57$ and class size is $h=3$.

Class Interval	Number of boxes $\left(f_i\right)$	Mid-point $\left(x_i\right)$	$d_i=x_i–A$	$u_i=\frac{d_i}{h}=\frac{d_i}{3}$	$f_i{u}_i$
$49.5-52.5$	$15$	$51$	$-6$	$-2$	$-30$
$52.5-55.5$	$110$	$54$	$-3$	$-1$	$-110$
$55.5-58.5$	$135$	$57=A$	$0$	$0$	$0$
$58.5-61.5$	$115$	$60$	$3$	$1$	$115$
$61.5-64.5$	$25$	$63$	$6$	$2$	$50$
	Sum $f_i=400$				Sum $f_i{u}_i=25$

The mean formula is:

Mean $=\mathrm{x̄}=A+h\frac{\sum f_i{u}_i}{\sum f_i}$

$$\begin{gathered} =57+3\left[\frac{25}{400}\right] \\ =57+\frac{3}{16} \\ =57.19 \end{gathered}$$

Hence, the mean number of mangoes are $57.19$ and the method used is Step Deviation Method.