Question

Calculate the mean deviation about the mean for the following frequency distribution:

Class interval:	0–4	4–8	8–12	12–16	16–20
Frequency	4	6	8	5	2

Answer 1

Let the assumed mean A = 10 and h = 4.

Class Interval	Mid-Value(xi)	Frequency(fi)	$d_i=\frac{x_i-10}{4}$	$f_i{d}_i$	$$\begin{gathered} \left|x_i-\overline{X}\right| \\ =\left|x_i-9.2\right| \end{gathered}$$	$f_i\left|x_i-\overline{X}\right|$
0–4	2	4	−2	−8	7.2	28.8
4–8	6	6	−1	−6	3.2	19.2
8–12	10	8	0	0	0.8	6.4
12–16	14	5	1	5	4.8	24
16–20	18	2	2	4	8.8	17.6
		N = 25		$\sum _{}{f}_i{d}_i$ = −5		$\sum _{}{f}_i\left|x_i-\overline{X}\right|=$96

Here, N = 25 and $\sum _{}{f}_i{d}_i$ = −5

Mean,
$$\begin{gathered} \overline{X}=A+h\left(\frac{1}{N}\sum _{}{f}_i{d}_i\right) \\ =10+4\left(\frac{1}{25}\times \left(-5\right)\right) \\ =10-\frac{20}{25} \\ =10-0.8 \\ =9.2 \end{gathered}$$

∴ Mean deviation about mean$=\frac{1}{N}\sum _{}{f}_i\left|x_i-\overline{X}\right|=\frac{1}{25}\times 96=3.84$