Question

use the display of data items to find the standard deviation
the standard deviation is 1.84
(do not round until the final answer then round to two decimal places as needed.)

Explanation:

Step1: List out the data items

From the histogram, the data items are: \(6,6,9,9,9,9,9,9,9,9,11\) (since frequency of \(6\) is \(2\), frequency of \(9\) is \(7\) and frequency of \(11\) is \(2\)).

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n=2 + 7+2=11\), \(\sum_{i=1}^{n}x_{i}=2\times6 + 7\times9+2\times11=12 + 63+22 = 97\). So \(\bar{x}=\frac{97}{11}\approx8.82\).

Step3: Calculate \((x_{i}-\bar{x})^{2}\) for each \(x_{i}\)

• For \(x = 6\): \((6 - 8.82)^{2}=(- 2.82)^{2}=7.9524\)
• For \(x = 9\): \((9 - 8.82)^{2}=(0.18)^{2}=0.0324\)
• For \(x = 11\): \((11 - 8.82)^{2}=(2.18)^{2}=4.7524\)

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

The formula for the variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=2\times7.9524+7\times0.0324 + 2\times4.7524\)
\(=15.9048+0.2268+9.5048=25.6364\)
\(s^{2}=\frac{25.6364}{10}=2.56364\)

Step5: Calculate the standard deviation \(s\)

The standard deviation \(s=\sqrt{s^{2}}\). So \(s=\sqrt{2.56364}\approx1.60\)

Answer:

\(1.60\)