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? 12 37 31 6 65 54 33 82 38 86 91
range = 86 (round to one decimal place as needed.)
sample standard deviation = (round to one decimal place as needed.)

Explanation:

Step1: Identify the data set

The data set is \(37,31,6,65,94,33,82,38,86,91,0\)

Step2: Calculate the range

The range is the difference between the maximum and minimum values. Maximum value \(= 94\), minimum value \(= 0\). Range \(=94 - 0=94\)

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

\(\bar{x}=\frac{37 + 31+6+65+94+33+82+38+86+91+0}{11}=\frac{563}{11}\approx51.2\)

Step4: Calculate the squared - differences \((x_i-\bar{x})^2\) for each data point \(x_i\)

For \(x_1 = 37\): \((37 - 51.2)^2=(-14.2)^2 = 201.64\)
For \(x_2 = 31\): \((31 - 51.2)^2=(-20.2)^2 = 408.04\)
For \(x_3 = 6\): \((6 - 51.2)^2=(-45.2)^2 = 2043.04\)
For \(x_4 = 65\): \((65 - 51.2)^2=(13.8)^2 = 190.44\)
For \(x_5 = 94\): \((94 - 51.2)^2=(42.8)^2 = 1831.84\)
For \(x_6 = 33\): \((33 - 51.2)^2=(-18.2)^2 = 331.24\)
For \(x_7 = 82\): \((82 - 51.2)^2=(30.8)^2 = 948.64\)
For \(x_8 = 38\): \((38 - 51.2)^2=(-13.2)^2 = 174.24\)
For \(x_9 = 86\): \((86 - 51.2)^2=(34.8)^2 = 1211.04\)
For \(x_{10}=91\): \((91 - 51.2)^2=(39.8)^2 = 1584.04\)
For \(x_{11}=0\): \((0 - 51.2)^2=(-51.2)^2 = 2621.44\)

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

\(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\)
\(\sum_{i = 1}^{11}(x_i-\bar{x})^2=201.64+408.04 + 2043.04+190.44+1831.84+331.24+948.64+174.24+1211.04+1584.04+2621.44 = 11345.6\)
\(s^2=\frac{11345.6}{10}=1134.6\)

Step6: Calculate the standard deviation \(s\)

\(s=\sqrt{s^2}=\sqrt{1134.6}\approx33.7\)

Answer:

33.7