Question

Calculate the mean, median and mode of the number of persons per house in a village with the help of the following information:

Number of Persons per House	1	2	3	4	5	6	7	8	9	10
Number of Houses	26	113	120	95	60	42	21	14	5	4

Answer 1

No. of Persons per House (X)	No. of Houses (f)	fx	Cumulative Frequency (c.f.)
1 2 3 4 5 6 7 8 9 10	26 113 120 95 60 42 21 14 5 4	26 226 360 380 300 252 147 112 45 40	26 139 259 354 414 456 477 491 496 500
	N = ∑f = 500	∑fx = 1888

$$\begin{gathered} \underline{\mathbf{Mean}} \\ \mathrm{Mean}\left(\overline{X}\right)=\frac{\Sigma fx}{\Sigma f}=\frac{1888}{500}=3.78 \end{gathered}$$

$$\begin{gathered} \underline{\mathbf{Median}} \\ \mathrm{Median}=\mathrm{size}\mathrm{of}{\left(\frac{N+1}{2}\right)}^{th}\mathrm{item} \\ =\mathrm{size}\mathrm{of}{\left(\frac{500+1}{2}\right)}^{th}\mathrm{item} \\ =\mathrm{size}\mathrm{of}250.5^{th}\mathrm{item} \end{gathered}$$

Now, we need to locate this item in the column of Cumulative Frequency. The item just exceeding 250.5^th is 259 (in the c.f. column), which is corresponding to 3. ​ Hence, median is 3.

Mode

The data given in the question is a discrete series. Therefore, using the inspection method, we can say that 3 is mode of the given series. This is becasue 3 has the highest frequency of 120 times.
Therefore, Mode (Z) is 3.