Question

The weight of coffee in 70 jars is shown in the following table:

Weight (in grams):	200–201	201–202	202–203	203–204	204–205	205–206
Frequency:	13	27	18	10	1	1

Determine the variance and standard deviation of the above distribution. [NCERT EXEMPLAR]

Answer 1

Weight (in grams)	Mid-Values$\left(x_i\right)$	Frequency$\left(f_i\right)$	$d_i=x_i-202.5$	$d_i^2$	$f_i{d}_i$	$f_i{d}_i^2$
200–201	200.5	13	−2	4	−26	52
201–202	201.5	27	−1	1	−27	27
202–203	202.5	18	0	0	0	0
203–204	203.5	10	1	1	10	10
204–205	204.5	1	2	4	2	4
205–206	205.5	1	3	9	3	9
		N = $\sum _{}{f}_i=70$			$\sum _{}{f}_i{d}_i=-38$	$\sum _{}{f}_i{d}_i^2=102$

Now,

Variance, ${\sigma }^2$
$$\begin{gathered} =\left(\frac{1}{N}\sum _{}{f}_i{d}_i^2\right)-{\left(\frac{1}{N}\sum _{}{f}_i{d}_i\right)}^2 \\ =\left(\frac{1}{70}\times 102\right)-{\left(\frac{1}{70}\times \left(-38\right)\right)}^2 \\ =1.457-0.295 \\ =1.162\mathrm{gm} \end{gathered}$$

Standard deviation, $\sigma$ = $\sqrt{\mathrm{Variance}}=\sqrt{1.162}=1.08\mathrm{gm}$