Sports car or convertible? The following table presents the fuel efficiency, in miles per gallon, for a sample of convertibles and a sample of sports cars.
| Convertible Model | MPG | Sports Model | MPG |
|---|---|---|---|
| Ford Mustang V6 | 20 | Honda Civic Si | 27 |
| Volkswagen Eos | 25 | BMW 135i | 23 |
| Mini Cooper | 25 | Mazda3 Mazdaspeed | 24 |
| Saab 9-3 | 24 | Subaru Impreza WRX STi | 21 |
| BMW 328 i | 21 | Mazda RX-8 | 18 |
| Toyota Camry Solara | 21 | Mitsubishi Lancer Evolution | 21 |
Part 1 of 2
(a) Find the sample standard deviation of the mileage for the sample of convertibles. Round the answer to one decimal place.
The sample standard deviation for the convertibles, in miles per gallon, is $\square$ .
Answer (1)
Calculate the sample mean: x ˉ = 6 136 = 22.67 .
Find the squared differences from the mean and sum them: ∑ ( x i − x ˉ ) 2 = 25.33 .
Calculate the sample variance: s 2 = 5 25.33 = 5.07 .
Determine the sample standard deviation: s = 5.07 ≈ 2.3 .
Explanation
Understand the problem and provided data We are given a sample of convertible MPG values: 20, 25, 25, 24, 21, 21. Our goal is to find the sample standard deviation of these values, rounded to one decimal place.
Calculate the sample mean First, we need to calculate the sample mean, which is the average of the MPG values. We add up all the values and divide by the number of values (which is 6). x ˉ = 6 20 + 25 + 25 + 24 + 21 + 21 = 6 136 = 22.666666666666668
Calculate squared differences from the mean Next, we calculate the squared differences from the mean for each MPG value. This means we subtract the mean from each value, square the result, and list them all out:
( 20 − 22.666666666666668 ) 2 = ( − 2.666666666666668 ) 2 = 7.111111111111118
( 25 − 22.666666666666668 ) 2 = ( 2.333333333333332 ) 2 = 5.444444444444439
( 25 − 22.666666666666668 ) 2 = ( 2.333333333333332 ) 2 = 5.444444444444439
( 24 − 22.666666666666668 ) 2 = ( 1.333333333333332 ) 2 = 1.7777777777777746
( 21 − 22.666666666666668 ) 2 = ( − 1.666666666666668 ) 2 = 2.7777777777777817
( 21 − 22.666666666666668 ) 2 = ( − 1.666666666666668 ) 2 = 2.7777777777777817
Sum the squared differences Now, we sum up all the squared differences we just calculated: 7.111111111111118 + 5.444444444444439 + 5.444444444444439 + 1.7777777777777746 + 2.7777777777777817 + 2.7777777777777817 = 25.333333333333336
Calculate the sample variance To find the sample variance, we divide the sum of the squared differences by n − 1 , where n is the sample size (6 in this case). So, n − 1 = 6 − 1 = 5 .
s 2 = 5 25.333333333333336 = 5.066666666666667
Calculate the sample standard deviation Finally, we take the square root of the sample variance to get the sample standard deviation: s = 5.066666666666667 = 2.250925735484551 Rounding this to one decimal place, we get 2.3.
State the final answer The sample standard deviation for the convertibles, rounded to one decimal place, is 2.3 miles per gallon.
Examples
Understanding standard deviation helps in many real-world scenarios. For example, if you're comparing the fuel efficiency of different car models, a lower standard deviation indicates more consistent performance. This is useful for consumers looking for reliable fuel economy and for manufacturers aiming to improve the consistency of their vehicles' performance. In finance, standard deviation is used to measure the volatility of investments, helping investors assess risk. Similarly, in quality control, it helps ensure that products meet consistent standards.
