Question

Consider the following data
$6,8,10,12,14,16,18,20,22,24$
I. The variance of the data is 33.
II. The standard deviation of the data is $4.74$.

• A. Only Statement I is true
• B. Only Statement II is true
• C. Both statements are true
• D. Both statements are false

Answer 1

Answer: (A)

From the given data we can form the following table. The mean is calculated by step-deviation method taking 14 as assumed mean. The number of observations is $n=10$.

$x_i$	$d_i=\frac{x_i-14}{2}$	Deviations from mean$(|x_i-\bar{x}|$	$(x_i-\bar{x}{)}^2$
6	-4	-9	81
8	-3	-7	49
10	-2	-5	25
12	-1	-3	9
14	0	-1	1
18	2	3	9
20	3	5	25
22	4	7	49
24	5	9	81
	5		330

$\therefore$ Mean $(\bar{x})=$ Assumed mean $+\frac{\sum _{i=1}^n{d}_i}{n}\times h$
$=14+\frac{5}{10}\times 2$
$=15$
Variance $\left({\sigma }^2\right)=\frac{1}{n}\sum _{i=1}^{10}{\left(x_i-\bar{x}\right)}^2=\frac{1}{10}\times 330=33$
$\therefore$ Standard deviation $(\sigma )=\sqrt{33}$
$=5.74$