The following data sets represent the highway fuel efficiency in miles per gallon (mpg) of randomly selected cars from two car companies. Which company has cars that are more consistent in highway fuel efficiency?
Car Company A: 35, 28, 35, 30, 31, 36, 35, 30
Car Company B: 29, 33, 40, 27, 34, 34, 34, 25
To determine which company has cars that are more consistent in highway fuel efficiency, we can calculate the standard deviation for each data set.
For Car Company A:
1. Calculate the mean:
(35 + 28 + 35 + 30 + 31 + 36 + 35 + 30) / 8 = 32.5
2. Subtract the mean from each data point and square the result:
(35 - 32.5)^2 = 6.25
(28 - 32.5)^2 = 18.25
(35 - 32.5)^2 = 6.25
(30 - 32.5)^2 = 6.25
(31 - 32.5)^2 = 2.25
(36 - 32.5)^2 = 12.25
(35 - 32.5)^2 = 6.25
(30 - 32.5)^2 = 6.25
3. Find the sum of the squared differences:
6.25 + 18.25 + 6.25 + 6.25 + 2.25 + 12.25 + 6.25 + 6.25 = 54.75
4. Divide by the number of data points minus one (n-1):
54.75 / 7 = 7.82
5. Take the square root of the result to get the standard deviation:
√7.82 = 2.79
For Car Company B:
1. Calculate the mean:
(29 + 33 + 40 + 27 + 34 + 34 + 34 + 25) / 8 = 31.5
2. Subtract the mean from each data point and square the result:
(29 - 31.5)^2 = 6.25
(33 - 31.5)^2 = 2.25
(40 - 31.5)^2 = 72.25
(27 - 31.5)^2 = 18.25
(34 - 31.5)^2 = 6.25
(34 - 31.5)^2 = 6.25
(34 - 31.5)^2 = 6.25
(25 - 31.5)^2 = 42.25
3. Find the sum of the squared differences:
6.25 + 2.25 + 72.25 + 18.25 + 6.25 + 6.25 + 6.25 + 42.25 = 159.75
4. Divide by the number of data points minus one (n-1):
159.75 / 7 = 22.82
5. Take the square root of the result to get the standard deviation:
√22.82 = 4.78
Based on these calculations, Car Company A has a standard deviation of 2.79 mpg, while Car Company B has a standard deviation of 4.78 mpg. This means that Car Company A's cars are more consistent in highway fuel efficiency, as their data points are closer together and have less variability.
To determine which company has cars that are more consistent in highway fuel efficiency, we can calculate the standard deviation for each data set.
For Company A:
Step 1: Calculate the mean (average) of the data set.
Mean = (35 + 28 + 35 + 30 + 31 + 36 + 35 + 30) / 8 = 32.5
Step 2: Calculate the difference between each data point and the mean, and square these differences.
(35 - 32.5)^2 = 6.25
(28 - 32.5)^2 = 16.25
(35 - 32.5)^2 = 6.25
(30 - 32.5)^2 = 6.25
(31 - 32.5)^2 = 2.25
(36 - 32.5)^2 = 12.25
(35 - 32.5)^2 = 6.25
(30 - 32.5)^2 = 6.25
Step 3: Calculate the average of these squared differences.
Mean squared difference = (6.25 + 16.25 + 6.25 + 6.25 + 2.25 + 12.25 + 6.25 + 6.25) / 8 = 7.125
Step 4: Take the square root of the mean squared difference to get the standard deviation.
Standard deviation = √7.125 ≈ 2.67
For Company B:
Step 1: Calculate the mean (average) of the data set.
Mean = (29 + 33 + 40 + 27 + 34 + 34 + 34 + 25) / 8 = 31.75
Step 2: Calculate the difference between each data point and the mean, and square these differences.
(29 - 31.75)^2 = 7.5625
(33 - 31.75)^2 = 1.5625
(40 - 31.75)^2 = 67.5625
(27 - 31.75)^2 = 18.0625
(34 - 31.75)^2 = 5.0625
(34 - 31.75)^2 = 5.0625
(34 - 31.75)^2 = 5.0625
(25 - 31.75)^2 = 45.5625
Step 3: Calculate the average of these squared differences.
Mean squared difference = (7.5625 + 1.5625 + 67.5625 + 18.0625 + 5.0625 + 5.0625 + 5.0625 + 45.5625) / 8 = 18.03125
Step 4: Take the square root of the mean squared difference to get the standard deviation.
Standard deviation = √18.03125 ≈ 4.25
Comparing the standard deviations, we can see that:
- Company A has a standard deviation of approximately 2.67.
- Company B has a standard deviation of approximately 4.25.
The lower the standard deviation, the more consistent the data set is. Therefore, Company A has cars that are more consistent in highway fuel efficiency.