What do you want to do? How many variables? What level of data? Central tendency Describe Median Make inferences Univariate Bivariate Multivariate Nominal Ordinal Interval/ Ratio Central tendency Mode Central tendency Mean Form Skew Form Skew Kurtosis Kurtosis Kurtosis Form Skew Dispersion Dispersion Dispersion Variance, Standard deviation Average Absolute deviation Range, Index of dispersion 16304_CH06_Walker.indd 146 7/12/12 10:02:10 AM © Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION. F U L M E R , M A R J O R I E E L I Z A B E T H 1 3 1 4 B U Learning Objectives Understand the number of modes, skewness, and kurtosis as they relate to ■■ explaining a data set. Explain the difference between the mode and the number of modes.■■ Interpret the values of skewness and kurtosis as they relate to univariate ■■ analysis. Discuss the importance of the normal curve in statistics.■■ Describe the properties of the normal curve.■■ The final univariate descriptive statistic, the form of the distribution, ties together the central tendency and dispersion of the data. Three characteristics make up the form of the distribution: the number of modes, the symmetry, and the kurtosis. In addressing the form of a distribution, a polygon can generally be used to represent these charac- teristics visually. 1 Moments of a Distribution6- In some statistics books and other places, distributions and the form of distributions are referred to in terms of the moments of the distribution. There are four moments that are considered important to a distribution. Moments are calculated as follows: S1X 2 X2 i N where 1X 2 X2 represents the deviations from the mean (as has been the case in Chap- ters 4 and 5), N is the total number of cases in the distribution, and i is the moment being calculated. Since the sum of the deviations around the mean is always zero, the first moment is always zero. If X is taken to the second power in the formula above, you can see that this is the formula for the variance—thus, the second moment is the variance. The third moment is usually associated with the skew of the distribution, although the exact formula is to divide the formula for the third moment by the variance to the power of 1.5. Similarly, the kurtosis of a distribution is associated with the fourth Chapter 6 The Form of a Distribution 147 16304_CH06_Walker.indd 147 7/12/12 10:02:10 AM © Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION. F U L M E R , M A R J O R I E E L I Z A B E T H 1 3 1 4 B U moment, although the exact formula is to divide the formula for the fourth moment by the variance squared. The mean and variance were discussed in “Measures of Central Tendency” and “Measures of Dispersion” (Chapters 4 and 5). The skew and kurtosis of a distribution are discussed in this chapter, together with the third measure of the form of a distribution: the number of modes..