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: \(x = 6,7,8,9,10,11,12\) and \(f=11,0,8,0,0,0,11\) (assuming the frequencies from left - to - right).
\(\sum(x\times f)=6\times11 + 8\times8+12\times11=66 + 64+132 = 262\)
\(\sum f=11 + 8+11=30\)
The mean \(\bar{x}=\frac{\sum(x\times f)}{\sum f}=\frac{262}{30}\approx8.73\)

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

For \(x = 6\): \((6 - 8.73)^2\times11=( - 2.73)^2\times11 = 7.4529\times11 = 81.9819\)
For \(x = 8\): \((8 - 8.73)^2\times8=( - 0.73)^2\times8=0.5329\times8 = 4.2632\)
For \(x = 12\): \((12 - 8.73)^2\times11=(3.27)^2\times11 = 10.6929\times11=117.6219\)
\(\sum((x - \bar{x})^2\times f)=81.9819+4.2632 + 117.6219=203.867\)

Step3: Calculate the variance

The variance \(s^{2}=\frac{\sum((x - \bar{x})^2\times f)}{\sum f-1}=\frac{203.867}{30 - 1}=\frac{203.867}{29}\approx6.9954\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}=\sqrt{6.9954}\approx2.64\)

Answer:

\(2.64\)