Answer
Range $=2.0$
Standard Deviation $=0.65$
Work Step by Step
$$\text{Solution}$$
Given Information:
Data Set ⟹ $(0.5, 2.0, 2.5, 1.5, 1.0, 1.5) $
Total Number of Items in Data Set $n=6$
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
$$0.5, 1.0, 1.5, 1.5, 2.0, 2.5$$
$$\text{Range = Highest Value - Lowest Value}$$
$$ \text{Range} = 2.5-0.5$$
$$\text{Range} =2.0$$
(b) To Find Standard Deviation:
Re-Arrange the Given Data Set in Ascending Order
$$0.5, 1.0, 1.5, 1.5, 2.0, 2.5 $$
Mean = $\frac{0.5+ 1.0+ 1.5+ 1.5+ 2.0+ 2.5 }{6}$
Mean = $\frac{9}{6}$
Mean= $\mu$ = $1.5 $
$x_i=0.5, 1.0, 1.5, 1.5, 2.0, 2.5$
$|x_i-\mu|= 1.0, 0.5 ,0, 0 ,0.5, 0.25, 1 $
$|x_i-\mu|^2$= $1, 0.25, 0, 0, 0.25 ,1 $
Standard Deviation $=\sigma =$ $\frac{\sqrt {\Sigma (|x_i- \mu|)^2}}{n}$
$=\sigma $ = $\sqrt{\frac{1+ 0.25+ 0+ 0+ 0.25 +1}{6}}$
$=\sigma $= $\sqrt{\frac{2.5}{6}}$
$=\sigma $ = $\sqrt {0.42}$
$=\sigma $ = $0.65$
