Consider the following population data:
38 40 15 12 24
a. Calculate the range.
b. Calculate MAD. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
c. Calculate the population variance. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
d. Calculate the population standard deviation. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
a. To calculate the range, subtract the smallest value from the largest value in the population data:
Range = Largest value - Smallest value
= 40 - 12
= 28
b. To calculate the MAD (Mean Absolute Deviation), first find the mean of the population data:
Mean = (38 + 40 + 15 + 12 + 24) / 5
= 17.8 (rounded to 1 decimal place)
Next, calculate the absolute deviation of each value from the mean:
|38 - 17.8| = 20.2
|40 - 17.8| = 22.2
|15 - 17.8| = 2.8
|12 - 17.8| = 5.8
|24 - 17.8| = 6.2
Now, find the average of these absolute deviations:
MAD = (20.2 + 22.2 + 2.8 + 5.8 + 6.2) / 5
= 11.24 (rounded to 2 decimal places)
c. To calculate the population variance, first find the mean of the population data (as done in part b). Then, subtract the mean from each value, square the result, and find the average of these squared differences:
(38 - 17.8)^2 = 328.84
(40 - 17.8)^2 = 487.84
(15 - 17.8)^2 = 7.84
(12 - 17.8)^2 = 34.84
(24 - 17.8)^2 = 38.44
Population Variance = (328.84 + 487.84 + 7.84 + 34.84 + 38.44) / 5
= 179.08 (rounded to 2 decimal places)
d. To calculate the population standard deviation, take the square root of the variance calculated in part c:
Population Standard Deviation = √(179.08)
= 13.38 (rounded to 2 decimal places)
a. To calculate the range, subtract the minimum value from the maximum value:
Maximum value = 40
Minimum value = 12
Range = Maximum value - Minimum value
Range = 40 - 12
Range = 28
Therefore, the range is 28.
b. To calculate the mean absolute deviation (MAD), follow these steps:
1. Calculate the mean (average) of the data set.
2. Subtract the mean from each data point and take the absolute value.
3. Calculate the average of those absolute differences.
Step 1: Calculate the mean.
Mean = (38 + 40 + 15 + 12 + 24) / 5
Mean = 129 / 5
Mean = 25.8
Step 2: Calculate the absolute differences between each data point and the mean.
|38 - 25.8| = 12.2
|40 - 25.8| = 14.2
|15 - 25.8| = 10.8
|12 - 25.8| = 13.8
|24 - 25.8| = 1.8
Step 3: Calculate the average of those absolute differences.
MAD = (12.2 + 14.2 + 10.8 + 13.8 + 1.8) / 5
MAD = 52.8 / 5
MAD = 10.56
Therefore, the mean absolute deviation (MAD) is 10.56.
c. To calculate the population variance, follow these steps:
1. Calculate the mean (average) of the data set.
2. Subtract the mean from each data point and square the result.
3. Calculate the average of those squared differences.
Using the mean calculated in step 1 above (25.8), follow the remaining steps:
(38 - 25.8)^2 = 144.04
(40 - 25.8)^2 = 206.44
(15 - 25.8)^2 = 115.24
(12 - 25.8)^2 = 190.44
(24 - 25.8)^2 = 3.24
Population variance = (144.04 + 206.44 + 115.24 + 190.44 + 3.24) / 5
Population variance = 659.4 / 5
Population variance = 131.88
Therefore, the population variance is 131.88.
d. The population standard deviation is the square root of the population variance.
Population standard deviation = √(131.88)
Population standard deviation ≈ 11.49 (rounded to 2 decimal places)
Therefore, the population standard deviation is approximately 11.49.