Chapter 3 EDRS 5305 Fall 2005 Gravetter and Wallnau 5 th edition

Central Tendency (defined) ► Definition  A statistical measure to determine a single score that defines the center of a distribution. ► Goal  To find the single score that is most typical or most representative of the entire group (i.e. average).

Central Tendency (cont.) ► Data is easier to understand; ► Problem  No single standard procedure for determining central tendency.  No single measure will always produce a central, representative value in every situation.

Figure. 3.1 The difficulty in defining central tendency Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Page 54

Mean, Median, Mode ► To deal with the problems, statisticians have developed three different methods for measuring central tendency. ► How do you decide which one to use?  Keep in mind – the general purpose of central tendency is to find the single most representative score.

Mean ► Arithmetic average ► Add all the scores and divide by the number of scores. ► For the average of a population use the Greek letter mu,  (myoo) ► For the mean for a sample use X (read as X-bar) or M (read as X-bar) or M

Mean ► The mean for a distribution is the sum of the scores divided by the number of scores. ► Formula    n Population Mean Sample Mean

Why Greek letters? ► Greek letters to identify population values ► Our own alphabet to identify sample values  n Sample n is used for the number of scores in the sample

Example For a population N=4 scores: 3, 7, 4, 6    Page 55

Alternative Definitions for Mean ► The mean can be thought of as an amount each individual would get if the total (  were equally divided among all the individuals (N) in the distribution. ► Example 3.2 pg. 55 n=6 boysn = 4 boys n=6 boysn = 4 boys Buy 180 baseball cardsM = $5 Each gets 30 cards$20 total Do not know how much each boy has

Alternative Definitions for Mean ► Define the mean as a balance point for the distribution. ► Example 3.2 pg. 56

Weighted Mean ► Combining two sets of scores and then finding the overall mean for the combined group. ► Example pg. 57 ► Because the samples are not the same size, one will make a larger contribution to the total group and therefore will carry more weight in determining the overall mean. ► The overall mean is called the weighted mean.

Computing the mean from a frequency distribution table Quiz score (X) f f X f X

Characteristics of the Mean ► Every score in the distribution contributes to the value of the mean.  Every score must be added into the total in order to compute the mean. ► Changing the value of the score will change the mean ► Introducing a new score or removing a score will change the value of the mean

Median ► The score that divides a distribution exactly in half. ► No symbols or notations ► Definition and computations are identical for a sample and for a population ► Goal of a median is to determine the precise midpoint of a distribution.

Example ► When N is an odd number 3, 5, 8, 10, 11 ► When N is an even number 3, 3, 4, 5, 7, 8 Median = 8 Median = 4.5

Median (cont.) ► Used when a researcher wants to divide the sample or population into two groups that are exactly the same size. ► Median split  Where one group is above the median line and the other is below  For example: one of high-scoring subjects and one of low-scoring subjects

Figure 3.5 The median divides the area in the graph in half Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Mode ► The score or category that has the greatest frequency ► No symbols or notation to identify the mode ► The definition is the same for either a population or a sample distribution.

Mode (cont.) ► Can be used to determine the typical or average value for any scale of measurement, including a nominal scale (chapter 1) ► It is possible to have more than one mode

Table 3.4 Favorite restaurants Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning

Figure 3.6 A bimodal distribution Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning vs. multimodal

Selecting a Measure of Central Tendency ► Could be possible to compute two or three measures of central tendency with a set of data. ► Often get similar results.

Mean ► Mean is the most preferred measure.  Usually a good representative value  Goal is to find the single value that best represents the entire distribution. ► Mean has the added advantage of being closely related to variance and standard deviation (the most common measures of variability) ► This relationship makes the mean a valuable measure for purposes of inferential statistics

When to Use the Median ► Three situations in which the median serves as a valuable alternative to the mean.  Extreme scores or skewed distributions  Undetermined values  Open-ended distributions Page 68

When to Use the Mode ► Three situations in which the mode is commonly used as an alternative to the mean, or is used in conjunction with the mean to describe central tendency  Nominal scales  Discrete variables  Describing shape

Figure 3.10 Central tendency and symmetrical distributions Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Normal Bimodal Rectangular Page 73

Figure 3.11 Central tendency and skewed distributions Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning