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?
33 6 80 3 32 75 98 10 71 19 9
range = 95.0 (round to one decimal place as needed.)
sample standard deviation = 34.9 (round to one decimal place as needed.)
sample variance = (round to one decimal place as needed.)

Explanation:

Step1: Recall the formula for sample variance

The formula for sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\), where \(n\) is the sample size, \(x_{i}\) are the data points, and \(\bar{x}\) is the sample mean. Another formula \(s^{2}=\frac{n\sum_{i=1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}{n(n - 1)}\) can also be used.

First, find \(n = 11\).
\(\sum_{i=1}^{n}x_{i}=33 + 6+80 + 3+32+75+98+10+71+19+9=436\)
\(\sum_{i=1}^{n}x_{i}^{2}=33^{2}+6^{2}+80^{2}+3^{2}+32^{2}+75^{2}+98^{2}+10^{2}+71^{2}+19^{2}+9^{2}\)
\(=1089+36 + 6400+9+1024+5625+9604+100+5041+361+81\)
\(=29370\)

Step2: Calculate the sample variance

Using the formula \(s^{2}=\frac{n\sum_{i=1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}{n(n - 1)}\)
Substitute \(n = 11\), \(\sum_{i=1}^{n}x_{i}=436\), \(\sum_{i=1}^{n}x_{i}^{2}=29370\)
\(s^{2}=\frac{11\times29370-(436)^{2}}{11\times(11 - 1)}\)
\(=\frac{323070 - 190096}{110}\)
\(=\frac{132974}{110}\approx1208.9\)

Answer:

\(1208.9\)