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: Calculate the mean

First, find the sum of \(x\times f\) and the sum of \(f\).
The data: \(x_1 = 6,f_1=4\); \(x_2 = 8,f_2 = 6\); \(x_3=11,f_3 = 4\)
\(\sum(x\times f)=6\times4 + 8\times6+11\times4=24 + 48+44=116\)
\(\sum f=4 + 6+4=14\)
The mean \(\bar{x}=\frac{\sum(x\times f)}{\sum f}=\frac{116}{14}\approx8.29\)

Step2: Calculate \((x - \bar{x})^2\times f\)

For \(x = 6\): \((6 - 8.29)^2\times4=( - 2.29)^2\times4=5.2441\times4 = 20.9764\)
For \(x = 8\): \((8 - 8.29)^2\times6=( - 0.29)^2\times6=0.0841\times6 = 0.5046\)
For \(x = 11\): \((11 - 8.29)^2\times4=(2.71)^2\times4=7.3441\times4 = 29.3764\)
\(\sum((x - \bar{x})^2\times f)=20.9764+0.5046 + 29.3764=50.8574\)

Step3: Calculate the standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum((x - \bar{x})^2\times f)}{n - 1}}\), where \(n=\sum f\)
\(s=\sqrt{\frac{50.8574}{14 - 1}}=\sqrt{\frac{50.8574}{13}}\approx\sqrt{3.9121}\approx1.98\)

Answer:

\(1.98\)