Consider the following sample data:
39 41 31 50 37 40
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 sample variance. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
d. Calculate the sample 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 minimum value from the maximum value. In this case, the minimum value is 31 and the maximum value is 50. Therefore, the range is 50 - 31 = 19.
b. To calculate the mean absolute deviation (MAD), first we need to find the mean of the data set.
Mean = (39 + 41 + 31 + 50 + 37 + 40) / 6 = 238 / 6 = 39.67
Next, we find the absolute difference between each data point and the mean:
|39 - 39.67| = 0.67
|41 - 39.67| = 1.33
|31 - 39.67| = 8.67
|50 - 39.67| = 10.33
|37 - 39.67| = 2.67
|40 - 39.67| = 0.33
The sum of these absolute differences is 24.
MAD = 24 / 6 = 4
c. To calculate the sample variance, we first need to find the squared difference between each data point and the mean:
(39 - 39.67)^2 = 0.4489
(41 - 39.67)^2 = 1.7689
(31 - 39.67)^2 = 74.5089
(50 - 39.67)^2 = 109.3489
(37 - 39.67)^2 = 7.1689
(40 - 39.67)^2 = 0.1089
The sum of these squared differences is 193.3464.
Sample variance = 193.3464 / (6 - 1) = 38.66928
d. The sample standard deviation is the square root of the sample variance:
Sample standard deviation = √38.66928 = 6.22
a. To calculate the range, subtract the minimum value from the maximum value:
Minimum value: 31
Maximum value: 50
Range = Maximum value - Minimum value
= 50 - 31
= 19
Therefore, the range is 19.
b. To calculate the mean absolute deviation (MAD), follow these steps:
Step 1: Calculate the mean (average) of the data.
Sum of the data = 39 + 41 + 31 + 50 + 37 + 40 = 238
Number of data points = 6
Mean = Sum of data / Number of data points
= 238 / 6
= 39.67 (rounded to two decimal places)
Step 2: For each data point, calculate the absolute difference between the data point and the mean.
Absolute difference for each data point:
|39.67 - 39| = 0.67
|39.67 - 41| = 1.67
|39.67 - 31| = 8.67
|39.67 - 50| = 10.33
|39.67 - 37| = 2.67
|39.67 - 40| = 0.33
Step 3: Calculate the mean of the absolute differences.
Sum of the absolute differences = 0.67 + 1.67 + 8.67 + 10.33 + 2.67 + 0.33 = 24.34
MAD = Sum of the absolute differences / Number of data points
= 24.34 / 6
= 4.06 (rounded to two decimal places)
Therefore, the MAD is 4.06.
c. To calculate the sample variance, follow these steps:
Step 1: Calculate the mean (average) of the data (which we found to be 39.67 in step b).
Step 2: For each data point, subtract the mean and square the result.
Squared differences for each data point:
(39 - 39.67)^2 = 0.4489
(41 - 39.67)^2 = 1.7689
(31 - 39.67)^2 = 74.0889
(50 - 39.67)^2 = 108.9889
(37 - 39.67)^2 = 7.3489
(40 - 39.67)^2 = 0.1089
Step 3: Calculate the sum of the squared differences.
Sum of the squared differences = 0.4489 + 1.7689 + 74.0889 + 108.9889 + 7.3489 + 0.1089 = 192.7464
Step 4: Divide the sum of squared differences by the number of data points minus 1 (in this case, 6 - 1 = 5).
Sample variance = Sum of squared differences / (Number of data points - 1)
= 192.7464 / 5
= 38.54928 (rounded to two decimal places)
Therefore, the sample variance is 38.54928.
d. To calculate the sample standard deviation, take the square root of the sample variance:
Sample standard deviation = √(sample variance)
= √(38.54928)
= 6.21 (rounded to two decimal places)
Therefore, the sample standard deviation is 6.21.