Question 3.19: Comparison of Outdoor Paint Find the variance and standard d......

Comparison of Outdoor Paint

Find the variance and standard deviation for brand B paint data in Example 3–15. The months brand B lasted before fading were

35, 45, 30, 35, 40, 25

Step-by-Step
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Step 1 Find the mean.

$\mu = \frac{ΣX}{N} = \frac{35 + 45 + 30 + 35 + 40 + 25}{6} = \frac{210}{6} = 35$

Step 2 Subtract the mean from each value, and place the result in column B of the table.

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

Step 3 Square each result and place the squares in column C of the table.

Step 4 Find the sum of the squares in column C.

Σ(X − μ)² = 0 + 100 + 25 + 0 + 25 + 100 = 250

Step 5 Divide the sum by N to get the variance.

$\sigma^2=\frac{\Sigma(X-\mu)^2}{N}=\frac{250}{6}=41.7$

Step 6 Take the square root to get the standard deviation.

$\sigma=\sqrt{\frac{\Sigma(X-\mu)^2}{N}}=\sqrt{41.7} \approx 6.5$

Hence, the standard deviation is 6.5.

 A B C X X − μ (X − μ)² 35 0 0 45 10 100 30 -5 25 35 0 0 40 5 25 25 -10 100

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