Question

for the following set of data, find the sample standard deviation, to the nearest hundredth.

data	frequency
31	6
33	2
34	4
38	9
39	8
43	4

Explanation:

Step1: Calculate the sample mean $\bar{x}$

First, find the sum of the products of data - value and frequency ($\sum_{i = 1}^{n}x_if_i$) and the total frequency ($n=\sum_{i = 1}^{n}f_i$).
$\sum_{i = 1}^{n}x_if_i=30\times5 + 31\times6+33\times2+34\times4+38\times9+39\times8+43\times4$
$=150 + 186+66+136+342+312+172$
$=1364$
$n=5 + 6+2+4+9+8+4=38$
$\bar{x}=\frac{\sum_{i = 1}^{n}x_if_i}{n}=\frac{1364}{38}\approx35.89$

Step2: Calculate the sum of squared - deviations $\sum_{i = 1}^{n}f_i(x_i-\bar{x})^2$

$(30 - 35.89)^2\times5+(31 - 35.89)^2\times6+(33 - 35.89)^2\times2+(34 - 35.89)^2\times4+(38 - 35.89)^2\times9+(39 - 35.89)^2\times8+(43 - 35.89)^2\times4$
$=(-5.89)^2\times5+(-4.89)^2\times6+(-2.89)^2\times2+(-1.89)^2\times4+(2.11)^2\times9+(3.11)^2\times8+(7.11)^2\times4$
$=34.6921\times5 + 23.9121\times6+8.3521\times2+3.5721\times4+4.4521\times9+9.6721\times8+50.5521\times4$
$=173.4605+143.4726 + 16.7042+14.2884+40.0689+77.3768+202.2084$
$=667.5808$

Step3: Calculate the sample variance $s^2$

The formula for the sample variance of grouped data is $s^2=\frac{\sum_{i = 1}^{n}f_i(x_i-\bar{x})^2}{n - 1}$.
$s^2=\frac{667.5808}{38 - 1}=\frac{667.5808}{37}\approx18.0427$

Step4: Calculate the sample standard deviation $s$

$s=\sqrt{s^2}=\sqrt{18.0427}\approx4.25$

Answer:

$4.25$