Question

Calculate the mean, median and mode of the following frequency

distribution:Classintervals:15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 40 45 45 - 50 50 55Frequency: 3 8 9 10 4 3 2 1Draw the less than and more than O give (Cumulative Frequency Curve) for theabove data and hence find the median.

Answer 1

C I	fi	xi	fi xi
15 - 20	3	17.5	87.5
20 - 25	8	22.5	180
25 - 30	9	27.5	247.5
30 - 35	10	32.5	325
35 - 40	4	37.5	150
40 - 45	3	42.5	127.5
45 - 50	2	47.5	95
50 - 55	1	52.5	52.5
	Ʃfi = 40​		Ʃfi xi = 1265

$$\begin{gathered} \mathrm{Mean}=\frac{\sum f_i{x}_i}{\sum f_i} \\ =\frac{1265}{40} \\ =31.625 \end{gathered}$$

$$\begin{gathered} \mathrm{Now},\mathrm{Modal}\mathrm{class}=30-35 \\ \mathrm{Lower}\mathrm{limit}\mathrm{of}\mathrm{modal}\mathrm{class}=l=30 \\ \mathrm{Frequency}\mathrm{of}\mathrm{modal}\mathrm{class},f_1=10 \\ \mathrm{Frequency}\mathrm{of}\mathrm{class}\mathrm{preceeding}\mathrm{modal}\mathrm{class},f_0=9 \\ \mathrm{Frequency}\mathrm{of}\mathrm{class}\mathrm{succeeding}\mathrm{modal}\mathrm{class},\mathit{}{f}_2=4 \\ \mathrm{Class}\mathrm{size},h\mathit{}=5 \\ \mathrm{Now},\mathrm{mode}=\mathit{}l+\left(\frac{f_1-\mathit{}{f}_0}{2f_1-\mathit{}{f}_0-\mathit{}{f}_2}\right)\times \mathit{}h \\ =30+\left[\frac{10-9}{20-9-4}\right]\times 5 \\ =30+\frac{1}{7}\times 5 \\ =\frac{210+5}{7} \\ =\frac{215}{7} \\ =30.71 \\ \mathrm{Median}=\frac{\mathrm{Mode}+2\mathrm{Mean}}{3} \\ =\frac{30.71+63.25}{3} \\ =\frac{93.96}{3}=31.32 \end{gathered}$$