For the following set of sample data, calculate the range, variance, and standard deviation.
16, 14, 84, 50
To calculate the range, we first need to arrange the data in ascending order: 14, 16, 50, 84
Range = Maximum value - Minimum value
Range = 84 - 14
Range = 70
To calculate the variance, we first need to calculate the mean of the data:
Mean = (14 + 16 + 84 + 50) / 4
Mean = 41
Then, we calculate the variance using the formula:
Variance = [(14-41)^2 + (16-41)^2 + (84-41)^2 + (50-41)^2] / 4
Variance = [(27)^2 + (25)^2 + (43)^2 + (9)^2] / 4
Variance = [729 + 625 + 1849 + 81] / 4
Variance = 3284 / 4
Variance = 821
To calculate the standard deviation, we take the square root of the variance:
Standard Deviation = √821
Standard Deviation ≈ 28.66
Therefore, the range is 70, the variance is 821, and the standard deviation is approximately 28.66.