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.)
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
a. To calculate the range, subtract the minimum value from the maximum value.
Minimum value: 28
Maximum value: 48
Range = Maximum value - Minimum value
Range = 48 - 28
Range = 20
The range of the data is 20.
b. To calculate the Mean Absolute Deviation (MAD), follow these steps:
1. Calculate the mean (average) of the data.
Mean = (40 + 46 + 28 + 48 + 38 + 44) / 6
Mean = 244 / 6
Mean = 40.67 (rounded to two decimal places)
2. Subtract the mean from each data point and take the absolute value of the difference.
Absolute deviations:
|40 - 40.67| = 0.67
|46 - 40.67| = 5.33
|28 - 40.67| = 12.67
|48 - 40.67| = 7.33
|38 - 40.67| = 2.67
|44 - 40.67| = 3.33
3. Calculate the mean of the absolute deviations.
MAD = (0.67 + 5.33 + 12.67 + 7.33 + 2.67 + 3.33) / 6
MAD = 32 / 6
MAD = 5.33 (rounded to two decimal places)
The Mean Absolute Deviation (MAD) of the data is 5.33.
c. To calculate the sample variance, follow these steps:
1. Calculate the mean (average) of the data.
Mean = 40.67 (from the previous calculation)
2. Subtract the mean from each data point, square the difference, and sum the squared differences.
Squared differences:
(40 - 40.67)^2 = 0.4489
(46 - 40.67)^2 = 28.2249
(28 - 40.67)^2 = 163.4881
(48 - 40.67)^2 = 53.4769
(38 - 40.67)^2 = 7.1689
(44 - 40.67)^2 = 11.1329
Sum of the squared differences:
0.4489 + 28.2249 + 163.4881 + 53.4769 + 7.1689 + 11.1329 = 264.9396
3. Divide the sum of the squared differences by (n-1), where n is the number of data points.
Sample variance = Sum of the squared differences / (n-1)
Sample variance = 264.9396 / (6-1)
Sample variance = 264.9396 / 5
Sample variance = 52.98792 (rounded to two decimal places)
The sample variance of the data is 52.98.
d. To calculate the sample standard deviation, take the square root of the sample variance.
Sample standard deviation = √(Sample variance)
Sample standard deviation = √(52.98)
Sample standard deviation ≈ 7.28 (rounded to two decimal places)
The sample standard deviation of the data is approximately 7.28.