Consider the following population data:
27 51 8 19 13
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. The range is calculated by subtracting the smallest value from the largest value:
Range = 51 - 8 = 43
b. To calculate the MAD (Mean Absolute Deviation), we first need to find the mean of the data. The mean is calculated by summing all the values and dividing by the number of values:
Mean = (27 + 51 + 8 + 19 + 13) / 5 = 23.6
Next, we find the absolute deviation of each data point by subtracting the mean from each data point:
|27 - 23.6| = 3.4
|51 - 23.6| = 27.4
|8 - 23.6| = 15.6
|19 - 23.6| = 4.6
|13 - 23.6| = 10.6
Then, we calculate the sum of the absolute deviations:
Sum of Absolute Deviations = 3.4 + 27.4 + 15.6 + 4.6 + 10.6 = 61.6
Finally, we divide the sum of absolute deviations by the number of values to find the MAD:
MAD = 61.6 / 5 = 12.32
c. To calculate the population variance, we need to find the squared deviations from the mean for each data point. Then, we calculate the mean of these squared deviations:
(27 - 23.6)^2 = 13.44
(51 - 23.6)^2 = 755.84
(8 - 23.6)^2 = 239.36
(19 - 23.6)^2 = 21.16
(13 - 23.6)^2 = 112.36
Sum of Squared Deviations = 13.44 + 755.84 + 239.36 + 21.16 + 112.36 = 1142.16
Population Variance = Sum of Squared Deviations / Number of Values = 1142.16 / 5 = 228.43
d. The population standard deviation is the square root of the population variance:
Population Standard Deviation = √(228.43) ≈ 15.10
a. To calculate the range, we need to subtract the smallest value from the largest value.
The smallest value in the data set is 8, and the largest value is 51.
Range = 51 - 8 = 43
b. To calculate the Mean Absolute Deviation (MAD), we need to find the absolute difference between each value and the mean, then calculate the average of these differences.
First, let's find the mean of the data set.
Mean = (27 + 51 + 8 + 19 + 13) / 5 = 118 / 5 = 23.6
Now, we can find the absolute difference for each value:
|27 - 23.6| = 3.4
|51 - 23.6| = 27.4
|8 - 23.6| = 15.6
|19 - 23.6| = 4.6
|13 - 23.6| = 10.6
Next, we calculate the average of these absolute differences:
MAD = (3.4 + 27.4 + 15.6 + 4.6 + 10.6) / 5 = 61.6 / 5 = 12.32
c. To calculate the population variance, we need to find the squared difference between each value and the mean, then calculate the average of these squared differences.
First, let's find the squared difference for each value:
(27 - 23.6)^2 = 13.44
(51 - 23.6)^2 = 776.16
(8 - 23.6)^2 = 242.56
(19 - 23.6)^2 = 21.16
(13 - 23.6)^2 = 112.36
Next, we calculate the average of these squared differences:
Variance = (13.44 + 776.16 + 242.56 + 21.16 + 112.36) / 5 = 1165.68 / 5 = 233.14
d. To calculate the population standard deviation, we take the square root of the variance.
Standard Deviation = √(233.14) = 15.27