CENTRAL TENDENCY I. Measures of central tendency summarize a distribution by describing a typical or average score. A. Measures of central tendency are efficient, describing an entire distribution with a single value or score. B. In numerical distributions, measures of central tendency describe where on a number line the distribution is located and can give some idea of the general shape of the distribution. II. Measures of central tendency. A. The mode. 1. The mode is simply the most commonly occurring value or score. 2. Distributions may have more than one mode. 3. The mode may be determined for variables measured at any level. B. The median. 1. The median is the score or value in a distribution at which 1/2 of the observations fall below it and 1/2 fall above it. It is the mid-point of a distribution. 2. The median can be calculated for variables measured at the ordinal or interval level. However, unless an ordinal variable 7 or more attributes, the median is usually not meaningful and is seldom used. 3. Determining the median. a. Order and number the observations in the distribution from high to low. b. If the total number of observations in the distribution is an odd number, add 1 to the total number of observations and divide the result by 2. The value of the observation with that number is the median. c. If the total number of observations in the distribution is an even number, divide the total number of observations by 2, add the value of the observation with that number to the value of the next observation then divide by 2. The result is the median. d. The median can be a fractional value even if all the observations in the data set are whole numbers. C. The mean. 1. The mean is the sum of all observations in a distribution divided by the total number ofobservations. It is the average score. 2. Technically, the mean is only appropriate for continuous numerical variables, however, it is frequently used with discrete numerical variables as well. In fact, it is sometimes calculated for ordinal data, although this is not as widely accepted. 3. The mean is calculated by adding all the observations in a data set together, then dividing by the total number of observations. 4. While the mean is usually considered the best measure of central tendency for numerical variables, the median, and the mode, can be informative. In some cases, they may even be better measures than the mean. a. Skewed distributions. b. Bimodal distributions. III. Reporting measures of central tendency. A. When reporting central tendency, generally, you report the measure that uses as much of the available information as possible. This means report the mean for numerical variables, report the median for ordinal variables, and report the mode for nominal variables. B. However, there are exceptions. 1. If a numerical distribution is highly skewed, the median should be reported in addition to or instead of the mean. 2. If a numerical distribution is bimodal, the modes should be reported in addtion to or instead of the mean. 3. For variables with three or fewer attributes, report the mode regardless of level of measurement.