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? 51 26 29 23 55 39 63 79 16 65 53 range = 63 (round to one decimal place as needed.) sample standard deviation = (round to one decimal place as needed.)

Explanation:

Step1: Calculate the mean

First, find the sum of the data values: \(51 + 26+29 + 23+55+39+63+79+16+65+53=509\).
The number of data points \(n = 11\).
The mean \(\bar{x}=\frac{509}{11}\approx46.3\).

Step2: Calculate the squared - differences from the mean

For \(x = 51\): \((51 - 46.3)^2=(4.7)^2 = 22.09\)
For \(x = 26\): \((26-46.3)^2=(-20.3)^2 = 412.09\)
For \(x = 29\): \((29 - 46.3)^2=(-17.3)^2=299.29\)
For \(x = 23\): \((23-46.3)^2=(-23.3)^2 = 542.89\)
For \(x = 55\): \((55 - 46.3)^2=(8.7)^2 = 75.69\)
For \(x = 39\): \((39-46.3)^2=(-7.3)^2 = 53.29\)
For \(x = 63\): \((63 - 46.3)^2=(16.7)^2 = 278.89\)
For \(x = 79\): \((79-46.3)^2=(32.7)^2 = 1069.29\)
For \(x = 16\): \((16 - 46.3)^2=(-30.3)^2 = 918.09\)
For \(x = 65\): \((65 - 46.3)^2=(18.7)^2 = 349.69\)
For \(x = 53\): \((53 - 46.3)^2=(6.7)^2 = 44.89\)

Step3: Calculate the sample variance

The sum of the squared - differences \(\sum_{i = 1}^{n}(x_i-\bar{x})^2=22.09 + 412.09+299.29+542.89+75.69+53.29+278.89+1069.29+918.09+349.69+44.89 = 4066.1\)
The sample variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n - 1}=\frac{4066.1}{10}=406.61\)

Step4: Calculate the sample standard deviation

The sample standard deviation \(s=\sqrt{s^2}=\sqrt{406.61}\approx20.2\)

Answer:

Sample standard deviation \(=20.2\)