Measures of Central Tendency CJ 526 Statistical Analysis in Criminal Justice

Introduction Central Tendency

Characteristics of a Measure of Central Tendency Single number that represents the entire set of data (average)

Alternate Names Also known as _____ value Average Typical Usual Representative Normal Expected

Three Measures of Central Tendency Mode Median Mean

The Mode Score or qualitative category that occurs with the greatest frequency Always used with nominal data, we find the most frequently occurring category

Mode Example of modal category: Sample of 25 married, 30 single, 22 divorced Married is the modal category Determined by inspection, not by computation, counting up the number of times a value occurs

Example of Finding the Mode Y: 1, 8, 12, 3, 8, 5, 6 Mode = 8 Can have more than one mode 1, 2, 2, 8, 10, 5, 5, 6 Mode = 2 and 5

The Median The point in a distribution that divides it into two equal halves

Symbolized by Md

Finding the Median Arrange the scores in ascending or descending numerical order

Finding the Median -- continued If there is an even number of scores, the median corresponds to a value halfway between the two middle scores

Example of Finding the Median Y: 1, 3, 5, 6, 8, 12 Median = 5.5

The Mean The sum of the scores divided by the number of scores

Formula for finding the Mean Symbolized by M or “X-bar”

Characteristics of the Mean The mean may not necessarily be an actual score in a distribution

Deviation Score Measure of how far away a given score is from the mean x = X - M

Example of Finding the Mean Sum = 35 N = 5 M = 7

Selecting a Measure of Central Tendency Choice depends on

Nature of the Variable Nominal -- Mode

Shape of the Distribution Symmetrical – Mean Not symmetrical—the median will be better Any time there are extreme scores the median will be better

Example Median income: if someone loses their job, an income of 0—this would pull the average down Median housing values: an unusually nice house or poor house would affect the average Better to use the median

Intended Use of Statistic Descriptive -- Mode, Median, or Mean

Central Tendency and the Shape of a Distribution Symmetrical Unimodal: Mo = Md = M