Question

Find the value of p, when the mean of the following distribution is 20.

x	15	17	19	20 + p	23
f	2	3	4	5p	6

Answer 1

We know that,
Mean = $\frac{{\displaystyle \sum _{}{x}_i{f}_i}}{\sum _{}{f}_i}$

For the following data:

x	15	17	19	20 + p	23
f	2	3	4	5p	6

Mean = $\frac{\left(15\times 2\right)+\left(17\times 3\right)+\left(19\times 4\right)+\left(\left(20+p\right)\times 5p\right)+\left(23\times 6\right)}{2+3+4+5p+6}$
$$\begin{gathered} \Rightarrow 20=\frac{30+51+76+100p+5p^2+138}{15+5p} \\ \Rightarrow 20\left(15+5p\right)=5p^2+100p+295 \\ \Rightarrow 300+100p=5p^2+100p+295 \\ \Rightarrow 5p^2=300-295 \\ \Rightarrow 5p^2=5 \\ \Rightarrow p^2=\frac{5}{5} \\ \Rightarrow p^2=1 \\ \Rightarrow p=\pm 1 \end{gathered}$$

Hence, the value of p is ±1.