Question

The mean and variance of 7 observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12, 14, find the remaining 2 observations.

Or

Calculate the mean deviation about the mean of the following data.

$\begin{array}{cccccccc}\text{Classes}& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& 70-80\\ \text{Frequency}& 2& 3& 8& 14& 8& 3& 2\end{array}$

Answer 1

Let x and y be the remaining 2 observations. Then, mean = 8

$\Rightarrow \frac{2+4+10+12+14+x+y}{7}=8$

$\Rightarrow 42+x+y=56\Rightarrow x+y=14$ ...(i)

and variannce = 16

$\Rightarrow \frac{1}{7}(2^2+4^2+{10}^2+{12}^2+{14}^2+x^2+y^2)-(\text{Mean}{)}^2=16$

$\Rightarrow \frac{1}{7}(4+16+100+144+196+x^2+y^2)-(8{)}^2=16$

$\Rightarrow \frac{1}{7}(460+x^2+y^2)=16+64\Rightarrow 460+x^2+y^2=80\times 7$

$\Rightarrow x^2+y^2=100$

Now, $(x+y{)}^2+(x-y{)}^2=2(x^2=Y^2)$

$\Rightarrow (14{)}^2+(x-y{)}^2=2\left(100\right)$

$\Rightarrow (x-y{)}^2=200-196=4$

$\Rightarrow x-y=\pm 2$

On solving Eqs. (i) and (iii), we get

x=6, 8 and y= 8, 6

Hence, the remaining 2 observations are 6 and 8.

Firstly, make a table for computation of mean deviation about mean.

$\begin{array}{cccccc}\text{Classes}& \text{Mid-level(x\_i)}& \text{Frequency(f\_i)}& f_i{x}_i& |x_i-\overline{x}|-|x_i-45|& f_i|x_i-x|\\ 10-20& 15& 2& 30& 30& 60\\ 20-30& 25& 3& 75& 20& 60\\ 30-40& 35& 8& 280& 10& 80\\ 40-50& 45& 14& 630& 0& 0\\ 50-60& 55& 8& 440& 10& 80\\ 60-70& 65& 3& 195& 20& 60\\ 70-80& 75& 2& 150& 30& 60\\ & & \sum f_i=40& \sum f_i{x}_i=1800& & \sum f_i|x_i-\overline{x}|=400\end{array}$

We have $\sum f_i=40$ and $\sum f_i{x}_i=1800$

$\therefore \overline{X\left(mean\right)}=\frac{\sum f_i{x}_i}{\sum f_i}=\frac{1800}{40}=45$

Now, $\sum f_i=|x_i-\overline{x}|=400$ and $\sum f_i=40$
$\therefore$ Mean deviation about mean $=\frac{\sum f_i|x_i-\overline{x}|}{\sum f_i}$

$=\frac{400}{40}=10$

Hence, the mean deviation about mean is 10.