Question

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

Explanation:

Step1: Write out the data items

The stem - and - leaf plot gives the data items: \(6\), \(11\), \(16\), \(21\), \(26\).

Step2: Calculate the mean \(\bar{x}\)

The mean formula is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 5\) and \(x_{1}=6\), \(x_{2}=11\), \(x_{3}=16\), \(x_{4}=21\), \(x_{5}=26\).
\(\bar{x}=\frac{6 + 11+16+21+26}{5}=\frac{80}{5}=16\).

Step3: Calculate the squared differences \((x_{i}-\bar{x})^{2}\)

For \(x = 6\): \((6 - 16)^{2}=(-10)^{2}=100\).
For \(x = 11\): \((11 - 16)^{2}=(-5)^{2}=25\).
For \(x = 16\): \((16 - 16)^{2}=0^{2}=0\).
For \(x = 21\): \((21 - 16)^{2}=5^{2}=25\).
For \(x = 26\): \((26 - 16)^{2}=10^{2}=100\).

Step4: Calculate the variance \(s^{2}\)

The variance formula for a sample (assuming this is a sample, if it's a population the formula is similar with \(n\) instead of \(n - 1\)). Here we'll use the sample formula \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=100 + 25+0+25+100=250\), \(n = 5\), so \(s^{2}=\frac{250}{4}=62.5\).

Step5: Calculate the standard deviation \(s\)

The standard deviation formula is \(s=\sqrt{s^{2}}\).
\(s=\sqrt{62.5}\approx7.91\).

Answer:

\(7.91\)