Given a data set {3, 5, 7, 12, 15, 18, 21, 23}, calculate the skewness of the data set.
Answer
Step 3: Calculate the skewness (\(\gamma\)) of the data set. This is done by subtracting the mean from each number in the data set, cubing the result, adding all these, dividing by the number of items in the data set, and then dividing by the cube of the standard deviation. \(\gamma = \frac{(3-13)^3 + (5-13)^3 + (7-13)^3 + (12-13)^3 + (15-13)^3 + (18-13)^3 + (21-13)^3 + (23-13)^3}{8 \cdot 6.77^3} = -0.15\)
Step 1 ：Step 1: Calculate the mean (\(\mu\)) of the data set. This is done by adding all numbers in the data set and dividing by the number of items in the data set. \(\mu = \frac{3 + 5 + 7 + 12 + 15 + 18 + 21 + 23}{8} = 13\)
Step 2 ：Step 2: Calculate the standard deviation (\(\sigma\)) of the data set. This is done by subtracting the mean from each number in the data set, squaring the result, adding all these, dividing by the number of items in the data set, and then taking the square root. \(\sigma = \sqrt{\frac{(3-13)^2 + (5-13)^2 + (7-13)^2 + (12-13)^2 + (15-13)^2 + (18-13)^2 + (21-13)^2 + (23-13)^2}{8}} = 6.77\)
Step 3 ：Step 3: Calculate the skewness (\(\gamma\)) of the data set. This is done by subtracting the mean from each number in the data set, cubing the result, adding all these, dividing by the number of items in the data set, and then dividing by the cube of the standard deviation. \(\gamma = \frac{(3-13)^3 + (5-13)^3 + (7-13)^3 + (12-13)^3 + (15-13)^3 + (18-13)^3 + (21-13)^3 + (23-13)^3}{8 \cdot 6.77^3} = -0.15\)
