Question

For the following distribution, calculate mean using all suitable methods:

Size of item:	1−4	4−9	9−16	16−27
Frequency:	6	12	26	20

Answer 1

Direct Method:

We may prepare the table as shown:

We know that mean,

Hence, the mean is 13.25.

Short-Cut Method:

We may prepare the table as shown:

Size of Item	Mid value(xi)	$$\begin{gathered} d_i=x_i-A \\ =x_i-12.5 \end{gathered}$$	Frequency(fi)	$f_i{d}_i$
1–4	2.5	−10	6	−60
4–9	6.5	−6	12	−72
9–16	12.5 = A	0	26	0
16–27	21.5	9	20	180
			N = $\sum _{}{f}_i=64$	$\sum _{}{f}_i{d}_i=48$

Let the assumed mean be A = 12.5.

We know that mean, $\overline{X}=A+\frac{\sum _{}{f}_i{d}_i}{\sum _{}{f}_i}$
$$\begin{gathered} =12.5+\frac{48}{64} \\ =12.5+0.75 \\ =13.25 \end{gathered}$$

Hence, the mean is 13.25.

Step-deviation method cannot be used to evaluate the mean of the distribution as the width of the class intervals are not equal. Here, h is not fixed.