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: \(18,17,17,18,16,18,23,23,23,20\)

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 10\) and \(\sum_{i=1}^{n}x_{i}=18 + 17+17+18+16+18+23+23+23+20=193\)
\(\bar{x}=\frac{193}{10}=19.3\)

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

For \(x = 18\): \((18 - 19.3)^{2}=(-1.3)^{2}=1.69\) (occurs \(3\) times)
For \(x = 17\): \((17 - 19.3)^{2}=(-2.3)^{2}=5.29\) (occurs \(2\) times)
For \(x = 16\): \((16 - 19.3)^{2}=(-3.3)^{2}=10.89\)
For \(x = 23\): \((23 - 19.3)^{2}=(3.7)^{2}=13.69\) (occurs \(3\) times)
For \(x = 20\): \((20 - 19.3)^{2}=(0.7)^{2}=0.49\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=3\times1.69+2\times5.29 + 10.89+3\times13.69+0.49\)
\(=5.07+10.58+10.89+41.07+0.49=68.1\)

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

The formula for the variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\) (sample variance, assuming this is a sample). Here \(n = 10\), so \(s^{2}=\frac{68.1}{9}\approx7.57\)

Step5: Calculate the standard deviation \(s\)

The standard deviation \(s=\sqrt{s^{2}}\), so \(s=\sqrt{\frac{68.1}{9}}\approx2.75\)

Answer:

\(2.75\)