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A bowler's scores for a sample of six games were 172, 168, 188, 190, 172, 182, 174. The sample standard deviation is closest to:

A. 8.00
B. 8.64
C. 64.00
D. 74.67

Answer :

To find the sample standard deviation of the given scores, you can follow these steps:

  1. Find the Mean

    First, add up all the scores and divide by the number of scores.

    [tex]\text{Mean} = \frac{172 + 168 + 188 + 190 + 172 + 182 + 174}{6} = \frac{1046}{6} \approx 174.33[/tex]

  2. Calculate Each Score's Deviation from the Mean

    Subtract the mean from each score and square the result:

    • [tex](172 - 174.33)^2 = 5.43[/tex]
    • [tex](168 - 174.33)^2 = 40.13[/tex]
    • [tex](188 - 174.33)^2 = 187.43[/tex]
    • [tex](190 - 174.33)^2 = 246.43[/tex]
    • [tex](172 - 174.33)^2 = 5.43[/tex]
    • [tex](182 - 174.33)^2 = 58.13[/tex]
    • [tex](174 - 174.33)^2 = 0.11[/tex]

  3. Find the Average of These Squared Deviations

    Sum all the squared deviations:

    [tex]5.43 + 40.13 + 187.43 + 246.43 + 5.43 + 58.13 + 0.11 = 541.09[/tex]

    Divide by the number of scores minus 1 (since this is a sample standard deviation, not a population standard deviation):

    [tex]\text{Variance} = \frac{541.09}{5} = 108.22[/tex]

  4. Take the Square Root of the Variance

    This gives you the sample standard deviation:

    [tex]\text{Standard Deviation} = \sqrt{108.22} \approx 8.64[/tex]

Therefore, the sample standard deviation is closest to 8.64. The correct choice from the provided options is 8.64.

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