Question

toby took a sample of 4 employees at his company and looked at how many years they had each worked for the company. here is what he found:
employee jim pam dwight angela
experience (years) 6 5 9 8
toby found their mean was \\(\bar{x} = 7\\) years. he thinks the standard deviation is
\\(s_x = \sqrt{\frac{(6 - 7) + (5 - 7) + (9 - 7) + (8 - 7)}{3}}\\)
what is the error in tobys standard deviation calculation?
choose 1 answer:

Explanation:

Step1: Recall standard deviation formula

The formula for the sample standard deviation \( s_x \) is \( s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}} \), where \( (x_i-\bar{x})^2 \) means we square the deviations from the mean, not just take the deviations.

Step2: Analyze Toby's formula

In Toby's formula \( s_x=\sqrt{\frac{(6 - 7)+(5 - 7)+(9 - 7)+(8 - 7)}{3}} \), he used \( (x_i-\bar{x}) \) (the deviations) instead of \( (x_i-\bar{x})^2 \) (the squared deviations) in the numerator. The correct formula requires squaring each deviation from the mean before summing them up.

Answer:

Toby did not square the deviations from the mean (\( (x_i - \bar{x}) \)) in the numerator. The correct formula for sample standard deviation requires squaring each deviation (\( (x_i - \bar{x})^2 \)) before summing them, then dividing by \( n - 1 \) (where \( n = 4 \), so \( 4-1 = 3 \)) and taking the square root.