Question

The following table gives distribution of marks for 50 students of a class. Calculate mean deviation from the mean and median respectively from the data:

Marks Obtained	140−150	150−160	160−170	170−180	180−190	190−200
Frequency	4	6	10	18	9	3

Answer 1

Calculation of M.D from Median

Marks	Mid value m	Frequency (f)	Cumulative frequency	|dM| = |m − M| M = 172.78	f|dM|
140 − 150 150 − 160 160 − 170 170 − 180 180 − 190 190 − 200	145 155 165 175 185 195	4 6 10 18(f) 9 3	4 4 + 6 = 10 10 + 10 = 20(c.f) 20 + 18 = 38 38 + 9 = 47 47 + 3 = 50	27.78 17.78 7.78 2.22 12.22 22.22	111.12 106.68 77.8 39.96 109.98 66.66
		Σf = N = 50			Σf|dM| = 512.2

Median = Size of $\left(\frac{\mathrm{N}}{2}\right)\mathrm{th}$ item
= Size of $\left(\frac{50}{2}\right)\mathrm{th}$ item
= Size of 25th item

25th item lies under the 38th cumulative frequency therefore 170 − 180 is the median class interval.

Thus, median value will be as follows:

$$\begin{gathered} \mathrm{M}=l_1+\frac{{\displaystyle \frac{N}{2}}-c.f}{f}\times i \\ \mathrm{or},\mathrm{M}=170+\frac{25-20}{18}\times 10 \\ \mathrm{or},\mathrm{M}=170+\frac{50}{18} \\ \mathrm{or},\mathrm{M}=170+2.78 \\ \Rightarrow \mathrm{M}=172.78 \\ Hence,\mathrm{Mean}\mathrm{deviation}{\mathrm{MD}}_{\mathrm{M}}=\frac{\mathrm{\Sigma }f\left|d_M\right|}{\mathrm{\Sigma }f}=\frac{512.2}{50}=10.24 \end{gathered}$$

Calculation of M.D from Mean

Marks	Mid value m	Frequency (f)	fm	$$\begin{gathered} \left|{\mathbf{d}}_{\overline{\mathbf{X}}}\right|\mathbf{=}\left|\mathbf{m}\mathbf{-}\overline{\mathbf{X}}\right| \\ \overline{\mathbf{X}}\mathbf{=}\mathbf{171}\mathbf{.}\mathbf{2} \end{gathered}$$	$\mathit{f}\left|{\mathbf{d}}_{\overline{\mathbf{X}}}\right|$
140 − 150 150 − 160 160 − 170 170 − 180 180 − 190 190 − 200	145 155 165 175 185 195	4 6 10 18(f) 9 3	580 930 1650 3150 1665 585	26.2 16.2 6.2 3.8 13.8 23.8	104.8 97.2 62 68.4 124.2 71.4
		Σf = 50	Σfm = 8560		$\mathrm{\Sigma }f\left|d_{\overline{\mathrm{X}}}\right|=528$

$\mathrm{Mean}\left(\overline{\mathrm{X}}\right)=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{8560}{50}=171.2$

Hence, Mean deviation from the arithmetic mean is:

${\mathrm{MD}}_{\overline{\mathrm{X}}}=\frac{\mathrm{\Sigma }f\left|d_{\overline{X}}\right|}{\mathrm{\Sigma }f}=\frac{528}{50}=10.56$