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? 74 70 94 97 29 38 52 59 77 36 37 range = 68 (round to one decimal place as needed.) sample standard deviation = (round to one decimal place as needed.)

Explanation:

Step1: List the data

The data set is \( 74, 70, 94, 97, 29, 38, 52, 59, 77, 36, 37 \) (11 values).

Step2: Calculate the mean (\(\bar{x}\))

First, sum the data:
\( 74 + 70 + 94 + 97 + 29 + 38 + 52 + 59 + 77 + 36 + 37 = 663 \)
Mean: \( \bar{x} = \frac{663}{11} \approx 60.2727 \)

Step3: Calculate squared deviations

For each data point \( x_i \), compute \( (x_i - \bar{x})^2 \):

• \( (74 - 60.2727)^2 \approx 189.67 \)
• \( (70 - 60.2727)^2 \approx 94.59 \)
• \( (94 - 60.2727)^2 \approx 1137.59 \)
• \( (97 - 60.2727)^2 \approx 1351.09 \)
• \( (29 - 60.2727)^2 \approx 977.97 \)
• \( (38 - 60.2727)^2 \approx 496.09 \)
• \( (52 - 60.2727)^2 \approx 68.47 \)
• \( (59 - 60.2727)^2 \approx 1.62 \)
• \( (77 - 60.2727)^2 \approx 279.67 \)
• \( (36 - 60.2727)^2 \approx 589.17 \)
• \( (37 - 60.2727)^2 \approx 541.09 \)

Step4: Sum squared deviations

Sum these values:
\( 189.67 + 94.59 + 1137.59 + 1351.09 + 977.97 + 496.09 + 68.47 + 1.62 + 279.67 + 589.17 + 541.09 \approx 5726.18 \)

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

Sample variance: \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{5726.18}{10} \approx 572.62 \)

Step6: Calculate sample standard deviation (\(s\))

Sample standard deviation: \( s = \sqrt{572.62} \approx 23.9 \)

Answer:

\( 23.9 \)