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?
85 42 11 2 61 12 46 10 47 29 8
range = 83 (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: \(85 + 42+11 + 2+61 + 12+46 + 10+47 + 29+8=353\).
The number of data points \(n = 11\).
The mean \(\bar{x}=\frac{353}{11}\approx32.1\)

Step2: Calculate the squared differences from the mean

For each data value \(x_i\):
\((85 - 32.1)^2=(52.9)^2 = 2798.41\)
\((42-32.1)^2=(9.9)^2 = 98.01\)
\((11 - 32.1)^2=(-21.1)^2=445.21\)
\((2-32.1)^2=(-30.1)^2 = 906.01\)
\((61-32.1)^2=(28.9)^2=835.21\)
\((12 - 32.1)^2=(-20.1)^2=404.01\)
\((46-32.1)^2=(13.9)^2 = 193.21\)
\((10-32.1)^2=(-22.1)^2=488.41\)
\((47-32.1)^2=(14.9)^2=222.01\)
\((29-32.1)^2=(-3.1)^2 = 9.61\)
\((8-32.1)^2=(-24.1)^2=580.81\)

Step3: Calculate the variance

The variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^{2}}{n - 1}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^{2}=2798.41+98.01 + 445.21+906.01+835.21+404.01+193.21+488.41+222.01+9.61+580.81=7080.1\)
\(s^{2}=\frac{7080.1}{11 - 1}=\frac{7080.1}{10}=708.01\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}\)
\(s=\sqrt{708.01}\approx26.6\)

Answer:

Sample standard deviation = \(26.6\)