Question

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

Explanation:

Step1: Find the data items and their frequencies

From the bar - graph, the data items \(x_i\) and their frequencies \(f_i\) are:
When \(x_1 = 6\), \(f_1=3\); when \(x_2 = 9\), \(f_2 = 5\); when \(x_3=12\), \(f_3 = 3\)
The total number of data items \(n=\sum_{i = 1}^{k}f_i=f_1 + f_2+f_3=3 + 5+3=11\)

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

The formula for the mean of a frequency - distribution is \(\bar{x}=\frac{\sum_{i = 1}^{k}f_ix_i}{n}\)
\(\sum_{i = 1}^{k}f_ix_i=f_1x_1 + f_2x_2+f_3x_3=3\times6 + 5\times9+3\times12\)
\(=18+45 + 36=99\)
\(\bar{x}=\frac{99}{11}=9\)

Step3: Calculate \((x_i-\bar{x})^2f_i\)

For \(i = 1\): \((x_1-\bar{x})^2f_i=(6 - 9)^2\times3=(-3)^2\times3=9\times3 = 27\)
For \(i = 2\): \((x_2-\bar{x})^2f_i=(9 - 9)^2\times5=0^2\times5=0\)
For \(i = 3\): \((x_3-\bar{x})^2f_i=(12 - 9)^2\times3=3^2\times3=9\times3=27\)
\(\sum_{i = 1}^{k}(x_i-\bar{x})^2f_i=27+0 + 27=54\)

Step4: Calculate the standard deviation \(s\)

The formula for the standard deviation of a frequency - distribution is \(s=\sqrt{\frac{\sum_{i = 1}^{k}(x_i-\bar{x})^2f_i}{n-1}}\)
Since \(n = 11\), then \(s=\sqrt{\frac{54}{11 - 1}}=\sqrt{\frac{54}{10}}=\sqrt{5.4}\approx2.32\)

Answer:

\(2.32\)