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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