Question

yuri computes the mean and standard deviation for the sample data set 12, 14, 9, and 21. he finds the mean is 14. his steps for finding the standard deviation are below. 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 failed to find the difference between each data point and the mean.

Explanation:

Step1: Recall sample - standard - deviation formula

The formula for the sample standard deviation $s$ is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $x_{i}$ are the data - points, $\bar{x}$ is the mean, and $n$ is the number of data - points. Here, $n = 4$, and $\bar{x}=14$.

Step2: Identify Yuri's error

Yuri used the formula for the population standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$ instead of the sample standard deviation formula. He divided by $n = 4$ instead of $n-1=3$.

Answer:

Yuri divided by $n = 4$ instead of $n - 1=3$ in the formula for the sample standard deviation.