Question

There are 10 observations arranged in ascending order as given below.

45, 47 ,50 ,52, x, x+2, 60,62,63,74. The median of these observations is 53.

Find the value of x. Also find the mean and the mode of the data.

Answer 1

We have,

The 10 observations are arranged in ascending order as given below.

45, 47, 50, 52, x, x + 2, 60 ,62 ,63 ,74.

Since, the number of observation is 10, which is an even number and,

The median of the given observations is 53.

$$\begin{gathered} \Rightarrow \mathrm{Median}=53 \\ \Rightarrow \frac{1}{2}\left[\left(\frac{10}{2}\right)\mathrm{th}\mathrm{observation}+\left(\frac{10}{2}+1\right)\mathrm{th}\mathrm{observation}\right]=53 \\ \Rightarrow \frac{1}{2}\left[5\mathrm{th}\mathrm{observation}+6\mathrm{th}\mathrm{observation}\right]=53 \\ \Rightarrow \frac{1}{2}\left[x+x+2\right]=53 \\ \Rightarrow 2x+2=53\times 2 \\ \Rightarrow 2x=106-2 \\ \Rightarrow x=\frac{104}{2} \\ \therefore x=52 \end{gathered}$$

$\mathrm{Mean}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{data}=\frac{45+47+50+52+52+54+60+62+63+74}{10}=\frac{559}{10}=55.9$

Since, the observation that is occurring the maximum is 52.

So, the mode of the data is 52.