WP.2.5: PEARSON'S COEFFICIENT OF SKEWNESS

[WP.2.5]

Skewness can be measured in a number of different ways but one very simple way to start evaluating skewness is to calculate Pearson’s Coefficient of Skewness.  While not the most precise way to quantify skewness, Pearson’s method provides a clear picture of the primary variables that help to establish symmetry or asymmetry: the relative position of the Mean, Median (in some variants the Mode), and amount of overall variability as measured by the standard deviation.

I. PEARSON’S COEFFICIENT OF SKEWNESS

Pearson’s Coefficient can be mathematically summarized as follows:

Skewness = (3(x_bar-Median))/s

Note: Pearson’s Coefficent can only exist over an interval from -3 to +3 and can only be used for interval or ratio scale data.

Demonstration 1:  A small sample of students in a section of BA254 were asked how many hours of sleep they got the night before.  The results were as follows:

Calculate Pearson’s Coefficient of Skewness.

Answer: [Note: xbar = 7 ; Median = 8 s = 2] sk = -1.50

Problem 1:  Suppose the mean is 11.2, the median is 12.8, the mode is 13, the standard deviation is 2.0 and the variance is 4.0.

a.     Calculate Pearson’s coefficient of skewness.

b.     Comment on this result.

Problem 2: Suppose Pearson’s coefficient of skewness for a particular data set is calculated to be 2.30.  Based on this result, we could say that this distribution is

a.     symmetrical

b.     negatively skewed

c.      normal

d.     positively skewed

e.     bimodal

Problem 3:  What is the value of Pearson’s coefficient of skewness for the standard normal distribution?

Problem 4:  The mean GPA for a group of students is 3.25 with a standard deviation of 0.75.  The median GPA is 2.75. 

a.     Calculate Pearson’s coefficient of skewness.

b.     Comment on this result.

Problem 5:  The mean age of a listener of a particular radio station is 40 years with a standard deviation of 10 years.  The median age is 40 years. 

a.     Calculate Pearson’s coefficient of skewness.

b.     Comment on this result.

Problem 6:  If a distribution is negatively skewed, is more of the data above or below the mean?

Problem 7: True or False: The empirical rule can be used on skewed data

Answers:

Problem 1: a. -2.4  b. Negatively skewed

Problem 2: D

Problem 3: 0 (Zero)

Problem 4: a. 2.0  b. Positively skewed

Problem 5: a. 0  b. Symmetrical

Problem 6: Above the mean

Problem 7: False, must be normally distributed