Question

The distribution of the heights of 50 children are given. If the mean height for the distribution is 117.8 cm, then complete the following table.
$\begin{array}{ccccccc}Height& 110& 115& x_1& 120& 121& 125\\ Numberofstudents& 6& 8& 14& f_1& 4& 3\end{array}$

• A. $x_1=118,f_1=15$
• B. $x_1=121,f_1=15$
• C. $x_1=118,f_1=8$
• D. $x_1=221,f_1=45$

Answer 1

The correct option is A

$x_1=118,f_1=15$

$\begin{array}{ccc}{x}_i& f_i& f_i{x}_i\\ 110& 6& 660\\ 115& 8& 920\\ x_1& 14& 14x_1\\ 120& f_1& 120f_1\\ 121& 4& 484\\ 125& 3& 375\\ Total& \sum f_i=35+f_1& \sum f_1{x}_1=2439+14x_1+120f_1\end{array}$

$\text{Given},\sum f_i=50$
$\Rightarrow 35+f_1=50\Rightarrow f_1=15$
$\text{We know that,}\overline{x}=\frac{\sum x_1{f}_i}{\sum f_i}$
$\Rightarrow 117.8=\frac{2439+14x_1+120f_1}{50}\Rightarrow 5890=2439+14x_1+120\times 15[\because f_1=15]\Rightarrow 5890-4239=14x_1\Rightarrow 1651=14x_1\therefore x_1=117.93\approx 118$