Measures of Central Tendency

CJ 526 Statistical Analysis in Criminal Justice

Introduction

Central Tendency

Characteristics of a Measure of Central Tendency1. 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 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

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

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

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

N

XM

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

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