Question

find the standard deviation for the group of data items.
17, 13, 17, 13, 17, 13, 17, 13
the standard deviation is
(round to two decimal places as needed.)

Explanation:

Step1: Calculate the mean

The data set is \(x = \{17,13,17,13,17,13,17,13\}\).
The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 8\), \(\sum_{i=1}^{n}x_{i}=(17 + 13+17 + 13+17 + 13+17 + 13)=120\).
So, \(\bar{x}=\frac{120}{8}=15\).

Step2: Calculate the squared - deviation from the mean

The formula for the squared - deviation \((x_{i}-\bar{x})^{2}\).
For \(x = 17\): \((17 - 15)^{2}=4\).
For \(x = 13\): \((13 - 15)^{2}=4\).
Since \(17\) appears \(4\) times and \(13\) appears \(4\) times, \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=4\times4+4\times4=32\).

Step3: Calculate the variance

The formula for the variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\) (for sample variance, if it's a population variance, the formula is \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\)). Here, assuming it's a sample, \(n = 8\), so \(s^{2}=\frac{32}{8 - 1}=\frac{32}{7}\approx4.57\).

Step4: Calculate the standard deviation

The formula for the standard deviation \(s=\sqrt{s^{2}}\).
\(s=\sqrt{\frac{32}{7}}\approx2.14\).

Answer:

\(2.14\)