A testing lab wishes to test two experimental brands of outdoor paint to see how long each will last before fading. The testing lab makes 6 gallons of each paint to test. Since different chemical agents are added to each group and only six cans are involved, these two groups constitute two small populations from a previous example, the data for Brand B is given here

Brand B

35

45

30

35

40

25

We will now find the standard deviation for Brand B's data set.

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Step 1: Find the mean

\[\mu=\frac{\sum X}{N}=\frac{35+45+30+35+40+25}{6}=\frac{210}{6}=35\]

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Step 2: Subtract the mean from each value \((X-\mu)\)

35-35=0	45-35=10	30-35=-5
35-35=0	40-35=5	25-35=-10

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Step 3: Square each result \((X-\mu)^2\)

\(0^2=0\)	\(10^2=100\)	\((-5)^2=25\)
\(0^2=0\)	\(5^2=25\)	\((-10)^2=100\)

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Step 4: Find the sum of the squares \(\sum (X-\mu)^2\)

\[\sum (X-\mu)^2=0+100+25+0+25+100=250\]

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Step 5: Divide the sum by \(N\) to get the variance

\[\sigma^2=\frac{\sum (X-\mu)^2}{N}=\frac{250}{6}=41.7\]

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Step 6: Take the square root of the variance to get the standard deviation

\[\sigma=\sqrt{\frac{\sum (X-\mu)^2}{N}}=\sqrt{41.7}\approx 6.5\]