Question

yun computes the mean and standard deviation for the sample data set 12, 14, 9, and 21. he finds the mean is 14.
yuri’s work
$s = \sqrt{\frac{(12 - 14)^2 + (14 - 14)^2 + (9 - 14)^2 + (21 - 14)^2}{4}}$
$= \sqrt{\frac{(-2)^2 + (0)^2 + (-5)^2 + (7)^2}{4}}$
$= \sqrt{\frac{4 + 0 + 25 + 49}{4}}$
$= \sqrt{\frac{78}{4}}$
$= \sqrt{19.5}$
what is the first error he made in computing the standard deviation?
yuri did not subtract 9 − 14 correctly.
yuri failed to find the difference between each data point and the mean.
yuri failed to square −2 correctly.
yuri divided by n instead of n − 1.

Explanation:

To find the sample standard deviation, the formula is \( s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}} \), where \( n \) is the number of data points. Here, the data set has \( n=4 \) (data points: 12, 14, 9, 21). Yuri used \( n = 4 \) in the denominator (divided by \( n \)) instead of \( n-1=3 \). Let's check other options:

• For "Yuri did not subtract \( 9 - 14 \) correctly", \( 9-14=-5 \), and he used \( - 5 \) (as \( (-5)^2 \) is part of his calculation), so this is incorrect.
• For "Yuri failed to find the difference between each data point and the mean", he did find \( 12 - 14 \), \( 14 - 14 \), \( 9 - 14 \), \( 21 - 14 \), so this is incorrect.
• For "Yuri failed to square \( - 2 \) correctly", \( (-2)^2 = 4 \), and he used 4, so this is incorrect.

So the first error is dividing by \( n \) instead of \( n - 1 \).

Answer:

Yuri divided by \( n \) instead of \( n - 1 \).