Answer
Range $=7.7$
Standard Deviation $=2.6$
Work Step by Step
$$\text{Solution}$$
Given Information:
Data Set ⟹ $(8.2, 10.1, 2.6, 4.8, 2.4, 5.6, 7.0, 3.3) $
Total Number of Items in Data Set $n=8$
Formula:
Range = Highest Value - Lowest Value
Standard Deviation $=\sigma =$ $\frac{\sqrt {\Sigma (|x_i- \mu|)^2}}{n}$
Mean = $\frac {\text{Sum of All Values}}{\text{Total Number of Values}}$
To Find:
a)Range
b)Standard Deviation
Answer:
(a) To Find Range:
Re-Arrange the Given Data Set in Ascending Order
$$2.4, 2.6, 3.3, 4.8, 5.6, 7.0, 8.2, 10.1$$
$$\text{Range = Highest Value - Lowest Value}$$
$$ \text{Range} = 10.1-2.4$$
$$\text{Range} =7.7$$
(b) To Find Standard Deviation:
Re-Arrange the Given Data Set in Ascending Order
$$2.4, 2.6, 3.3, 4.8, 5.6, 7.0, 8.2, 10.1 $$
Mean = $\frac{2.4+ 2.6+ 3.3+ 4.8+ 5.6+ 7.0+ 8.2+ 10.1 }{8}$
Mean = $\frac{44}{8}$
Mean= $\mu$ = $5.5 $
$x_i=2.4, 2.6, 3.3, 4.8, 5.6, 7.0, 8.2, 10.1 $
$|x_i-\mu|= 3.1, 2.9, 2.2, 0.7, 0.1 ,1.5 ,2.7 ,4.6 $
$|x_i-\mu|^2$= $9.61, 8.41, 4.84, 0.09, 0.001 ,2.25 ,7.29 ,21.16 $
Standard Deviation $=\sigma =$ $\frac{\sqrt {\Sigma (|x_i- \mu|)^2}}{n}$
$=\sigma $ = $\sqrt{\frac{9.61+ 8.41+ 4.84+ 0.09+ 0.001 +2.25 +7.29 +21.16}{8}}$
$=\sigma $= $\sqrt{\frac{54.06}{8}}$
$=\sigma $ = $\sqrt {6.76}$
$=\sigma $ = $2.6$
