Measures of Central Tendency CJ 526 Statistical Analysis in Criminal Justice

Introduction Central Tendency Single number that represents the entire set of data (example: the average score)

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 X: 8, 6, 7, 9, 10, 6 Mode = 6 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

Example Subject # Test Score Mode = ?

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

Finding the Median 1. Arrange the scores in ascending or descending numerical order 2. If there is an odd number of scores, the Md is the middle score

Finding the Median -- continued 3. 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 X: 6, 6, 7, 8, 9, 10, 11 Median = 8 Y: 1, 3, 5, 6, 8, 12 Median = 5.5

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

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 X: 8, 6, 7, 11, 3 Sum = 35 N = 5 M = 7

Selecting a Measure of Central Tendency Choice depends on Measurement level of data If the data is nominal, the mode must be used The mode can also be used for other levels of measurement

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

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