Consider the following sample of five measurements: 2, 1, 1, 0, 3.

a.Calculate the range, s², and s.

b.Add 3 to each measurement and repeat part a.

c.Subtract 4 from each measurement and repeat part a.

d.Considering your answers to parts a, b, and c, what seems to be the effect on the variability of a data set by adding the same number to or subtracting the same number from each measurement?

Range = 3 – 0 = 3

s² = 1.3

s = 1.14

$$\begin{gathered} \overline{\mathbf{x}}\mathbf{=}\frac{\mathbf{2}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{3}}{\mathbf{5}} \\ \mathbf{=}\frac{\mathbf{7}}{\mathbf{5}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{4} \\ {\mathbf{s}}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{=}\frac{{\left(2-1.4\right)}^{\mathbf{2}}\mathbf{+}{\left(1-1.4\right)}^{\mathbf{2}}\mathbf{+}{\left(1-1.4\right)}^{\mathbf{2}}\mathbf{+}{\left(0-1.4\right)}^{\mathbf{2}}\mathbf{+}{\left(3-1.4\right)}^{\mathbf{2}}}{\mathbf{5}\mathbf{-}\mathbf{1}} \\ \mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{36}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{16}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{16}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{96}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{56}}{\mathbf{4}} \\ \mathbf{=}\frac{\mathbf{5}\mathbf{.}\mathbf{2}}{\mathbf{4}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{3} \\ \mathbf{s}\mathbf{=}\sqrt{{\mathbf{s}}^{\mathbf{2}}} \\ \mathbf{=}\sqrt{\mathbf{1}\mathbf{.}\mathbf{3}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{14} \end{gathered}$$

After adding 3, the new measurements are as follows:

5, 4, 4, 3, 6

$$\begin{gathered} \overline{\mathbf{x}}\mathbf{=}\frac{\mathbf{5}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{3}\mathbf{+}\mathbf{6}}{\mathbf{5}} \\ \mathbf{=}\frac{\mathbf{22}}{\mathbf{5}} \\ \mathbf{=}\mathbf{4}\mathbf{.}\mathbf{4} \\ {\mathbf{s}}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{=}\frac{{\left(5-4.4\right)}^{\mathbf{2}}\mathbf{+}{\left(4-4.4\right)}^{\mathbf{2}}\mathbf{+}{\left(4-4.4\right)}^{\mathbf{2}}\mathbf{+}{\left(3-4.4\right)}^{\mathbf{2}}\mathbf{+}{\left(6-4.4\right)}^{\mathbf{2}}}{\mathbf{5}\mathbf{-}\mathbf{1}} \\ \mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{36}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{16}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{16}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{96}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{56}}{\mathbf{4}} \\ \mathbf{=}\frac{\mathbf{5}\mathbf{.}\mathbf{2}}{\mathbf{4}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{3} \\ \mathbf{s}\mathbf{=}\sqrt{{\mathbf{s}}^{\mathbf{2}}} \\ \mathbf{=}\sqrt{\mathbf{1}\mathbf{.}\mathbf{3}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{14} \end{gathered}$$

Range = 6 – 3 = 3

s² = 1.3

s = 1.14

After subtracting 4, we get,

-2, -3, -3, -4, -1

$$\begin{gathered} \overline{\mathbf{x}}\mathbf{=}\frac{\left(-2\right)\mathbf{+}\left(-3\right)\mathbf{+}\left(-3\right)\mathbf{+}\left(-4\right)\mathbf{+}\left(-1\right)}{\mathbf{5}} \\ \mathbf{=}\frac{\mathbf{-}\mathbf{13}}{\mathbf{5}} \\ \mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{6} \\ {\mathbf{s}}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{=}\frac{{\left(\left(-2\right)-\left(-2.6\right)\right)}^{\mathbf{2}}\mathbf{+}{\left(\left(-3\right)-\left(-2.6\right)\right)}^{\mathbf{2}}\mathbf{+}{\left(\left(-3\right)-\left(-2.6\right)\right)}^{\mathbf{2}}\mathbf{+}{\left(\left(-4\right)-\left(-2.6\right)\right)}^{\mathbf{2}}\mathbf{+}{\left(\left(-1\right)-\left(-2.6\right)\right)}^{\mathbf{2}}}{\mathbf{5}\mathbf{-}\mathbf{1}} \\ \mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{36}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{16}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{16}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{96}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{56}}{\mathbf{4}} \\ \mathbf{=}\frac{\mathbf{5}\mathbf{.}\mathbf{2}}{\mathbf{4}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{3} \\ \mathbf{s}\mathbf{=}\sqrt{{\mathbf{s}}^{\mathbf{2}}} \\ \mathbf{=}\sqrt{\mathbf{1}\mathbf{.}\mathbf{3}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{14} \end{gathered}$$

Range = 6 – 3 = 3

s² = 1.3

s = 1.14

There is no effect on the variability of a data set by adding the same number to or subtracting the same number from each measurement. All values remain constant.