Question

Calculate mean deviation from the following data, using mean and median, respectively.

Size	4	6	8	10	12	14	16
Frequency	2	4	5	3	2	1	4

Answer 1

Calculation of mean deviation from median

Size (X)	Frequency (f)	Cumulative frequency (c.f.)	Deviation from median |dM| = |X − M| M = 8	f|dM|
4 6 8 10 12 14 16	2 4 5 3 2 1 4	2 2 + 4 = 6 6 + 5 = 11 11 + 3 = 14 14 + 2 = 16 16 + 1 = 17 17 = 4 = 21	4 2 0 2 4 6 8	8 8 0 6 8 6 32
	Σf=N= 21			Σf|dM| = 68

(a) Median or (M) = Size of $\left(\frac{N+1}{2}\right)\mathrm{th}$ item
= Size of $\left(\frac{21+1}{2}\right)\mathrm{th}$ item
= Size of 11th item
= 8

(b) Mean Deviation from median

${\mathrm{MD}}_{\mathrm{M}}=\frac{\mathrm{\Sigma }f\left|d_M\right|}{\mathrm{N}}=\frac{68}{21}=3.24$

Calculation of mean deviation from mean

Size (X)	Frequency (f)	fX	From mean deviation from mean $$\begin{gathered} \left|{\mathit{d}}_{\overline{\mathbf{X}}}\right|\mathbf{=}\left|\mathit{X}\mathbf{-}\overline{\mathbf{X}}\right| \\ \left(\overline{\mathbf{X}}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{71}\right) \end{gathered}$$	$f\left|d_{\overline{X}}\right|$
4 6 8 10 12 14 16	2 4 5 3 2 1 4	8 24 40 30 24 14 64	5.71 3.71 1.71 0.29 2.29 4.29 6.29	11.42 14.84 8.55 0.87 4.58 4.29 25.16
	Σf = 21	ΣfX = 204		$\Sigma f\left|d\overline{{}_X}\right|$= 69.71

(a) $\mathrm{Mean}\left(\overline{X}\right)=\frac{\mathrm{\Sigma }fX}{\mathrm{\Sigma }f}=\frac{204}{21}=9.71$

(b) Mean deviation from A.M.

${\mathrm{MD}}_{\overline{\mathrm{X}}}=\frac{\mathrm{\Sigma }f\left|d_{\overline{X}}\right|}{\mathrm{\Sigma }f}=\frac{69.71}{21}=3.32$