Question

Using only integers between 0 and 10, construct two data sets with at least 10 observations each so that the two sets have the same mean but different variances. Construct dot plots for each of your data sets and mark the mean of each data set on its dot diagram.

Short answer

Set A: 1, 0, 3, 2, 4, 2, 2, 3, 3, 0

Set B: 0, 4, 4, 4, 4, 3, 2, 0, 0, 0

Step-by-step solution

Creating 2 data sets only by using integers between 0 and 10

Set A: 1, 0, 3, 2, 4, 2, 2, 3, 3, 0

Mean = 2

Variance = 1.77

$$\begin{gathered} \mathbf{Variance}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{{\left(1-2\right)}^{\mathbf{2}}\mathbf{+}\mathbf{(}\mathbf{0}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\left(3-2\right)}^{\mathbf{2}}\mathbf{+}{\left(2-2\right)}^{\mathbf{2}}\mathbf{+}{\left(4-2\right)}^{\mathbf{2}}\mathbf{+} \\ {\left(2-2\right)}^{\mathbf{2}}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\left(3-2\right)}^{\mathbf{2}}\mathbf{+}{\left(3-2\right)}^{\mathbf{2}}\mathbf{+}{\left(0-2\right)}^{\mathbf{2}} \\ }{\mathbf{10}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{1}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{4}}{\mathbf{9}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{16}}{\mathbf{9}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{77} \end{gathered}$$

Set B: 0, 4, 4, 4, 4, 3, 2, 0, 0, 0

Mean = 2

Variance = 3.66

$$\begin{gathered} \mathbf{Variance}\mathbf{=}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{{\left(0-2\right)}^{\mathbf{2}}\mathbf{+}\mathbf{(}\mathbf{4}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\left(4-2\right)}^{\mathbf{2}}\mathbf{+}{\left(4-2\right)}^{\mathbf{2}}\mathbf{+}{\left(4-2\right)}^{\mathbf{2}}\mathbf{+} \\ {\left(3-2\right)}^{\mathbf{2}}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\left(0-2\right)}^{\mathbf{2}}\mathbf{+}{\left(0-2\right)}^{\mathbf{2}}\mathbf{+}{\left(0-2\right)}^{\mathbf{2}}}{\mathbf{10}\mathbf{-}\mathbf{1}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{4}}{\mathbf{9}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{33}}{\mathbf{9}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{66} \end{gathered}$$

Constructing a dot plot for Set A

Creating a dot plot for Set B