## Q. 3.18

Comparison of Outdoor Paint

Find the variance and standard deviation for the data set for brand A paint in Example 3–15. The number of months brand A lasted before fading was

10, 60, 50, 30, 40, 20

## Verified Solution

Step 1 Find the mean for the data.

$\mu = \frac{ΣX}{N} = \frac{10 + 60 + 50 + 30 + 40 + 20}{6} = \frac{210}{6} = 35$

Step 2 Subtract the mean from each data value (X − μ).

10 − 35 = −25     50 − 35 = 15      40 − 35 = 5
60 − 35 = 25       30 − 35 = −5      20 − 35 = −15

Step 3 Square each result (X − μ)².

(−25)² = 625      (15)² = 225          (5)² = 25
(25)² = 625         (−5)²= 25            (−15)² = 225

Step 4 Find the sum of the squares Σ(X − μ)².

625 + 625 + 225 + 25 + 25 + 225 = 1750

Step 5 Divide the sum by N to get the variance $\frac{Σ(X − μ)^2}{N}$

Variance = 1750 ÷ 6 ≈ 291.7

Step 6 Take the square root of the variance to get the standard deviation. Hence, the
standard deviation equals $\sqrt{291.7}$ , or 17.1. It is helpful to make a table.

Column A contains the raw data X. Column B contains the differences X − μ obtained in step 2. Column C contains the squares of the differences obtained in step 3.

 A Values X B X − 𝛍 C (X − 𝛍)² 10 −25 625 60 25 625 50 15 225 30 −5 25 40 5 25 20 −15 225 1750