Question

Calculate the value of median and mode from the following distribution.

Mid-point	20	30	40	50	60	70	80	90
Frequency	8	10	15	25	40	20	15	7

Answer 1

For the calculation of median, the given mid-values must be converted into class intervals using the following formula.
$\mathrm{Value}\mathrm{of}\mathrm{adjustment}=\frac{\mathrm{Mid}-\mathrm{point}\mathrm{of}\mathrm{one}\mathrm{class}-\mathrm{Mid}-\mathrm{point}\mathrm{of}\mathrm{preceeding}\mathrm{class}}{2}$
The value obtained is then added to the mid-point to obtain the upper limit and subtracted from the mid-point to obtain the lower limit. In this manner, we obtain the following distribution.

Mid Points	Class Interval	Frequency (f)	Cumulative Frequency (c.f.)
20 30 40 50	15 − 25 25 − 35 35 − 45 45 − 55	8 10 15 25 (f0)	8 18 33 58 (c.f.)
60	55 − 65	40 (f1)	98
70 80 90	65 − 75 75 − 85 85 − 95	20 (f2) 15 7	118 133 140
		N = ∑f =140

Median
Median class is given by the size of ${\left(\frac{N}{2}\right)}^{\mathrm{th}}$ item, i.e.${\left(\frac{140}{2}\right)}^{\mathrm{th}}$ item, which is 70^th item.
This corresponds to the class interval of (55 − 65), so this is the median class.
$$\begin{gathered} \mathrm{Median}=l_1+\frac{{\displaystyle \frac{N}{2}}-c.f.}{f}\times i \\ \mathrm{or},\mathrm{Median}=55+\frac{70-58}{40}\times 10 \\ \mathrm{or},\mathrm{Median}=55+3=58 \\ \mathrm{Thus},\mathrm{Median}=58 \end{gathered}$$

Mode

By inspection, we can say that the modal class is (55 − 65) as it has the highest frequency of 40.
$$\begin{gathered} \mathrm{Mode}\left(Z\right)=l_1+\frac{f_1-f_0}{2f_1-f_0-f_2}\times i \\ \mathrm{or},Z=55+\frac{40-25}{2\left(40\right)-25-20}\times 10 \\ \mathrm{or},Z=55+\frac{15}{35}\times 10 \\ \mathrm{or},Z=55+4.28=59.28 \\ \mathrm{Hence},\mathrm{mode}\left(Z\right)\mathrm{is}59.28. \end{gathered}$$