Consider the following sample data:



40 46 28 48 38 44



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.)

1 answer

a. To calculate the range, subtract the smallest value from the largest value:

Range = Largest value - Smallest value
= 48 - 28
= 20

b. To calculate the MAD (mean absolute deviation), first calculate the mean of the data. The mean is the sum of all the values divided by the total number of values:

Mean = (40 + 46 + 28 + 48 + 38 + 44) / 6
= 244 / 6
= 40.67

Next, subtract the mean from each value, take the absolute value of each difference, and calculate the mean of these absolute values:

MAD = (|40 - 40.67| + |46 - 40.67| + |28 - 40.67| + |48 - 40.67| + |38 - 40.67| + |44 - 40.67|) / 6
= (0.67 + 5.33 + 12.67 + 7.33 + 2.33 + 3.33) / 6
= 31.66 / 6
= 5.28

c. To calculate the sample variance, first calculate the squared difference of each value from the mean, sum up these squared differences, and divide by the total number of values minus 1:

Variance = ((40 - 40.67)^2 + (46 - 40.67)^2 + (28 - 40.67)^2 + (48 - 40.67)^2 + (38 - 40.67)^2 + (44 - 40.67)^2) / (6 - 1)
= (0.4489 + 34.2289 + 152.2289 + 54.5089 + 4.3489 + 13.1089) / 5
= 259.87 / 5
= 51.97

d. To calculate the sample standard deviation, take the square root of the sample variance:

Standard Deviation = √(51.97)
= 7.21