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Measures of Central Tendency CJ 526 Statistical Analysis in Criminal Justice

Introduction n Central Tendency

Characteristics of a Measure of Central Tendency 1. Single number that represents the entire set of data (average)

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

Three Measures of Central Tendency Mode n Median n Mean n

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

Mode Example of modal category: n Sample of 25 married, 30 single, 22 divorced n Married is the modal category n n 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 n Mode = 6 n Y: 1, 8, 12, 3, 8, 5, 6 n Mode = 8 n Can have more than one mode n 1, 2, 2, 8, 10, 5, 5, 6 n Mode = 2 and 5 n

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

n 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 n Median = 8 n Y: 1, 3, 5, 6, 8, 12 n Median = 5. 5 n

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

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

Characteristics of the Mean n 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 nx = X - M n

Example of Finding the Mean X: 8, 6, 7, 11, 3 n Sum = 35 n. N = 5 n. M = 7 n

Selecting a Measure of Central Tendency n Choice depends on

Nature of the Variable n Nominal -- Mode

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

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

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

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