Question

Below are the jersey numbers of $11$ players randomly selected from a football team. Find the​ range, variance, and standard deviation for the given sample data. What do the results tell​ us? $1475312377638840681160$ Range=nothing ​(Round to one decimal place as​ needed.)

Answer 1

Step 1: What is the​ range, variance, and standard deviation for the given sample data:

The given sample data is,

$X:\hspace{0.17em}14,75,31,23,77,63,88,40,68,11,60$.

Let $n=11$.

Arrange the data into ascending order.

$X:11,14,23,31,40,60,63,68,75,77,88$.

Here the minimum value is $11$ and the maximum value is $88$.

Step 2: Calculate the range of the sample data.

$\begin{array}{rcl}\text{Range}& =& \text{Maximum}-\text{Minimum}\\ & =& 88-11\\ & =& 77\end{array}$

Step 3: Calculate the mean of the sample data.

$\begin{array}{rcl}\overline{X}& =& \frac{1}{n}\sum _{i=1}^n{X}_i\\ & =& \frac{1}{11}\left(11+14+23+31+40+60+63+68+75+77+88\right)\\ & =& \frac{1}{11}\left(550\right)\\ & =& 50\end{array}$

Step 4: Calculate the variance of the sample data.

$$\begin{gathered} \begin{array}{rcl}{S}^2& =& \frac{1}{n-1}\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2\\ & =& \frac{1}{11-1}\left[{\left(X_1-\overline{X}\right)}^2+{\left(X_2-\overline{X}\right)}^2+{\left(X_3-\overline{X}\right)}^2+{\left(X_4-\overline{X}\right)}^2+{\left(X_5-\overline{X}\right)}^2+{\left(X_6-\overline{X}\right)}^2 \\ +{\left(X_7-\overline{X}\right)}^2+{\left(X_8-\overline{X}\right)}^2+{\left(X_9-\overline{X}\right)}^2+{\left(X_{10}-\overline{X}\right)}^2+{\left(X_{11}-\overline{X}\right)}^2\right]\\ & =& \frac{1}{10}\left[{\left(11-50\right)}^2+{\left(14-50\right)}^2+{\left(23-50\right)}^2+{\left(31-50\right)}^2+{\left(40-50\right)}^2+{\left(60-50\right)}^2 \\ +{\left(63-50\right)}^2+{\left(68-50\right)}^2+{\left(75-50\right)}^2+{\left(77-50\right)}^2+{\left(88-50\right)}^2\right]\\ & =& \frac{1}{10}\left[{\left(-39\right)}^2+{\left(-36\right)}^2+{\left(-27\right)}^2+{\left(-19\right)}^2+{\left(-10\right)}^2+{\left(10\right)}^2 \\ +{\left(13\right)}^2+{\left(18\right)}^2+{\left(25\right)}^2+{\left(25\right)}^2+{\left(38\right)}^2\right]\\ & =& \frac{1}{10}\left[1521+1296+729+361+100+100 \\ +169+324+625+729+1444\right]\\ & =& \frac{1}{10}\left[7398\right]\\ & =& 739.8\end{array} \end{gathered}$$

Step 5: Calculate the standard deviation of the sample data.

$\begin{array}{rcl}\sigma & =& \sqrt{S^2}\\ & =& \sqrt{739.8}\\ & =& 27.2\end{array}$

From the data the result can tell us that jersey numbers are nominal data and are just replacements for names, so the resulting statistics are meaningless.

Therefore, the range will be $\mathbf{77}$, the variance will be $\mathbf{739}\mathbf{.}\mathbf{8}$ and the standard deviation will be $\mathbf{27}\mathbf{.}\mathbf{2}$.