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? 54 4 22 57 16 89 85 64 17 12 80 range = (round to one decimal place as needed.)

Explanation:

Step1: Find the maximum and minimum values

The data set is \(54,4,22,57,16,89,85,64,17,12,80\). The maximum value \(x_{max}=89\) and the minimum value \(x_{min} = 4\).

Step2: Calculate the range

The formula for the range \(R\) of a data - set is \(R=x_{max}-x_{min}\). So, \(R = 89 - 4=85\).

Step3: Calculate the mean \(\bar{x}\)

\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 11\) and \(x_{i}\) are the data - points. \(\sum_{i=1}^{11}x_{i}=54 + 4+22+57+16+89+85+64+17+12+80=500\). Then \(\bar{x}=\frac{500}{11}\approx45.45\).

Step4: Calculate the squared differences \((x_{i}-\bar{x})^2\)

For \(x_1 = 54\), \((54 - 45.45)^2=(8.55)^2 = 73.1025\)
For \(x_2 = 4\), \((4 - 45.45)^2=(-41.45)^2 = 1718.1025\)
...
For \(x_{11}=80\), \((80 - 45.45)^2=(34.55)^2 = 1193.7025\)

Step5: Calculate the variance \(s^{2}\)

The formula for the sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}{n - 1}\).
\(\sum_{i=1}^{11}(x_{i}-\bar{x})^2=73.1025+1718.1025+\cdots+1193.7025 = 7797.2725\)
\(s^{2}=\frac{7797.2725}{10}=779.72725\approx779.7\)

Step6: Calculate the standard deviation \(s\)

The formula for the sample standard deviation \(s=\sqrt{s^{2}}\). So \(s=\sqrt{779.72725}\approx27.9\)

Answer:

Range = 85.0
Variance \(\approx779.7\)
Standard Deviation \(\approx27.9\)
The range tells us the spread between the highest and lowest jersey numbers. The variance and standard - deviation measure the average amount by which each number in the data set differs from the mean. A relatively large variance and standard deviation indicate that the jersey numbers are quite spread out from the mean value.