PEP 6305 Measurement in Health & Physical Education

Topic 5: The Normal Distribution

Section 5.4

Click to go to back to the previous section (Section 5.3)

Skewness and Kurtosis

n   Recall that the normal distribution is symmetric, and not "too flat" or "too peaked."

¨  But we also know that sample data will likely not show a perfect distribution.

¨  Then how do we know if variables are approximately normally distributed, i.e., symmetric and NOT "too flat" or "too peaked"?

n   Skewness is a measure of how symmetric the data are distributed on either side of the mean.

¨  Evaluating skewness: the measure of skewness uses the cube of the Z scores (Z³) from the data divided by the sample size (N); if this value differs from 0 with a probability of error greater than 5% (0.05), then the data are too asymmetric and cannot be assumed to be normally distributed.

n   Kurtosis is a measure of how peaked or flat the distribution is.

¨  Evaluating kurtosis: the measure of kurtosis uses the Z scores from the data to the fourth power (Z⁴) divided by the sample size (N); if this value differs from 0 with a probability of error greater than 5% (0.05), then the data are too peaked or flat to be assumed to be normal.

n   Evaluation of the skewness and kurtosis measures is one way to evaluate whether a distribution of values is approximately normally distributed or not.

¨  If the skewness and kurtosis statistics are not significantly greater than 0, the distribution is approximately normal.

n   LazStats can be used to compute the skewness and kurtosis measures, and to evaluate normality of a distribution.

¨  Open a data file in LazStats.  Click Analyze>Descriptives>Distribution Statistics… Select one of the variables and move it to the Variables to Analyze field.        

¨  Click OK, and the output (which you've seen before) includes the skewness and kurtosis statistics and their standard errors.    

¨  Divide the kurtosis and skewness measures by their standard errors to obtain their Z scores (as described in the textbook, pp. 88-89).

¨  Z scores for skewness and kurtosis that are < ±2.0 generally indicate that the distribution is approximately normal because (based on what we've learned about Z scores) only <5% of distributions that are normally distributed in the population will have Z scores for skewness and kurtosis that are larger than 2.0 or smaller than -2.0.

Formative Evaluation

n   What is the skewness and kurtosis for our old data set: 525, 505, 507, 654, 631, 281, 771, 575, 485, 626, 780, 626? Are these data approximately normally distributed? (Answers) (remember how to enter data into R Commander?)

n   Work Problems 1 through 4 at the end of Chapter 6 and Problems 1 and 2 at the end of Chapter 7.

You have reached the end of Topic 5.

Make sure to work through the Formative Evaluation above and the textbook problems (end of the chapter).

You must complete the review quiz (in the Quizzes folder on the WebCT course home page) before you can advance to the next topic.