Question

Calculate the range, variance, and standard deviation for the following samples:

a.39, 42, 40, 37, 41

b.100, 4, 7, 96, 80, 3, 1, 10, 2

c.100, 4, 7, 30, 80, 30, 42, 2

Short answer

1. Range = 5, Variance = 3.79, Standard Deviation = 1.95
2. Range = 99, Variance = 1949.25, Standard Deviation = 44.15
3. Range = 98, Variance = 1307.839, Standard Deviation = 36.16

Step-by-step solution

Calculating the range, variance and standard deviation for a

39, 42, 40, 37, 41

Range = 42 – 37 = 5

x		2
39	-0.8	0.64
42	2.2	4.84
40	0.2	0.4
37	-2.8	7.84
41	1.2	1.44
Sum	0	15.16

$$\begin{gathered} \mathbf{Mean}\mathbf{\left(}\overline{\mathbf{x}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{sumofx}}{\mathbf{n}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{199}}{\mathbf{5}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{39}\mathbf{.}\mathbf{8} \end{gathered}$$

$$\begin{gathered} \mathbf{Variance}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{15}\mathbf{.}\mathbf{16}}{\mathbf{5}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{15}\mathbf{.}\mathbf{16}}{\mathbf{4}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{79} \end{gathered}$$

$$\begin{gathered} \mathit{S}\mathit{t}\mathit{a}\mathit{n}\mathit{d}\mathit{a}\mathit{r}\mathit{d}\mathit{D}\mathit{e}\mathit{v}\mathit{i}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{=}\sqrt{\mathit{V}\mathit{a}\mathit{r}\mathit{i}\mathit{a}\mathit{n}\mathit{c}\mathit{e}} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}\sqrt{\mathit{3}\mathit{.}\mathit{79}} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}\mathit{1}\mathit{.}\mathit{95} \end{gathered}$$

Therefore,the variance and standard deviation are 3.79 and 1.95 respectively.

Finding the range, variance and standard deviation for b

100, 4, 7, 96, 80, 3, 1, 10, 2

Range = 100 – 1 = 99

x		2
100	66.33	4400.11
4	-29.67	880.11
7	-26.67	711.11
96	62.33	3885.44
80	46.33	2146.778
3	-30.67	940.44
1	-32.67	1067.11
10	-23.67	560.11
2	-31.67	1002.778
Sum	0	15594

$$\begin{gathered} \mathbf{Mean}\mathbf{\left(}\overline{\mathbf{x}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{sumofx}}{\mathbf{n}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{301}}{\mathbf{9}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{33}\mathbf{.}\mathbf{66} \end{gathered}$$

$$\begin{gathered} \mathbf{Variance}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{15594}}{\mathbf{9}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{15594}}{\mathbf{8}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1949}\mathbf{.}\mathbf{25} \end{gathered}$$

$$\begin{gathered} \mathbf{StandardDeviation}\mathbf{=}\sqrt{\mathbf{Variance}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{1949}\mathbf{.}\mathbf{25}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{44}\mathbf{.}\mathbf{15} \end{gathered}$$

Therefore, Variance = 1949.25 and Standard Deviation = 44.15

Computing the range, variance and standard deviation for c

100, 4, 7, 30, 80, 30, 42, 2

Range = 100 – 2 = 98

x		2
100	63.125	3984.766
4	-32.875	1080.766
7	-29.875	892.5156
30	-6.875	47.26563
80	43.125	1859.766
30	-6.875	47.26563
42	5.125	26.26563
2	-34.875	1216.266
Sum	0	9154.875

$$\begin{gathered} \mathbf{Mean}\mathbf{\left(}\overline{\mathbf{x}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{sumofx}}{\mathbf{n}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{295}}{\mathbf{8}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{36}\mathbf{.}\mathbf{875} \end{gathered}$$

$$\begin{gathered} \mathbf{Variance}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{9154}\mathbf{.}\mathbf{875}}{\mathbf{8}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{9154}\mathbf{.}\mathbf{875}}{\mathbf{7}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1307}\mathbf{.}\mathbf{839} \end{gathered}$$

$$\begin{gathered} \mathbf{StandardDeviation}\mathbf{=}\sqrt{\mathbf{Variance}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{1307}\mathbf{.}\mathbf{839}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{36}\mathbf{.}\mathbf{16} \end{gathered}$$

Hence, thevariance is 1307.839 and standard deviation is 36.16.