Question

Heights of students of Class X are given in the foloowing frequency distribution:

Height (in cm)	150−155	155−160	160−165	165−170	170−175
Number of students	15	8	20	12	5

Find the modal height.
Also, find the mean height. Compare and interpret the two measures of central tendency. [CBSE 2014]

Answer 1

Here the maximum class frequency is 20, and the class corresponding to this frequency is 160−165. So, the modal class is 160−165.

Now,

Modal class = 160−165, lower limit (l) of modal class = 160, class size (h) = 5,

frequency (f₁) of the modal class = 20,

frequency (f₀) of class preceding the modal class = 8,

frequency (f₂) of class succeeding the modal class = 12.

Now, let us substitute these values in the formula:

$$\begin{gathered} \mathrm{Mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h \\ =160+\left(\frac{20-8}{40-8-12}\right)\times 5 \\ =160+\left(\frac{12}{20}\right)\times 5 \\ =160+3 \\ =163 \end{gathered}$$

Hence, the mode is 163.

It represents that the height of maximum number of students is 163 cm.

Now, to find the mean let us put the data in the table given below:

Height (in cm)	Number of students (fi)	Class mark (xi)	fixi
150−155	15	152.5	2287.5
155−160	8	157.5	1260
160−165	20	162.5	3250
165−170	12	167.5	2010
170−175	5	172.5	862.5
Total	∑fi = 60		∑fixi = 9670

$$\begin{gathered} \mathrm{Mean}=\frac{{\displaystyle \sum _i}{f}_i{x}_i}{{\displaystyle \sum _i}{f}_i} \\ =\frac{9670}{60} \\ =161.17 \end{gathered}$$

Thus, mean of the given data is 161.17.

It represents that on an average, the height of a student is 161.17 cm.