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
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 |
B X − 𝛍 |
C |
10 | −25 | 625 |
60 | 25 | 625 |
50 | 15 | 225 |
30 | −5 | 25 |
40 | 5 | 25 |
20 | −15 | 225 |
1750 |