Question

Find the mean and variance for the data:
$\begin{array}{cccccccc}{x}_i& 92& 93& 97& 98& 102& 104& 109\\ f_i& 3& 2& 3& 2& 6& 3& 3\end{array}$

Answer 1

Finding mean
$\begin{array}{ccc}{x}_i& f_i& f_i{x}_i\\ 92& 3& 276\\ 93& 2& 186\\ 97& 3& 291\\ 98& 2& 196\\ 102& 6& 612\\ 104& 3& 312\\ 109& 3& 327\\ & \sum f_i=22& \sum x_i{f}_i=2200\end{array}$
Mean $\overline{x}=\frac{\sum x_i{f}_i}{\sum f_i}$
$\Rightarrow \overline{x}=\frac{2200}{22}$
$\Rightarrow \overline{x}=100$

Finding variance and standard deviation
For $x_i=92,f_i=3,\overline{x}=100,x_i-\overline{x}=92-100=-8,(x_i-\overline{x}{)}^2=(-8{)}^2=64,f_i(x_i-\overline{x}{)}^2=3\times 64=192$
$\begin{array}{ccccc}{x}_i& f_i& x_i-\overline{x}& (x_i-\overline{x}{)}^2& f_i(x_i-\overline{x}{)}^2\\ 92& 3& -8& 64& 192\\ 93& 2& -7& 49& 98\\ 97& 3& -3& 9& 27\\ 98& 2& -2& 4& 8\\ 102& 6& 2& 4& 24\\ 104& 3& 4& 16& 48\\ 109& 3& 9& 81& 243\\ & \sum f_i=22& & & \sum f_i(x_i-\overline{x}{)}^2=640\end{array}$
Variance $\left({\sigma }^2\right)=\frac{\sum f_i(x_i-\overline{x}{)}^2}{\sum f_i}$
$(\sigma {)}^2=\frac{640}{22}$
$(\sigma {)}^2=29.09$
Hence, the mean is $100$ and the variance is $\mathrm{29.09.}$