Chapter 3Chapter 3EDRS 5305EDRS 5305Fall 2005Fall 2005

Gravetter and Wallnau 5Gravetter and Wallnau 5thth edition edition

Central Tendency (defined)Central Tendency (defined)

►DefinitionDefinition A statistical measure to determine a A statistical measure to determine a

single score that defines the center of a single score that defines the center of a distribution.distribution.

►GoalGoal To find the single score that is most To find the single score that is most

typical or most representative of the typical or most representative of the entire group (i.e. average).entire group (i.e. average).

Central Tendency (cont.)Central Tendency (cont.)

►Data is easier to understand;Data is easier to understand;►ProblemProblem

No single standard procedure for No single standard procedure for determining central tendency.determining central tendency.

No single measure will always produce a No single measure will always produce a central, representative value in every central, representative value in every situation.situation.

Figure. 3.1Figure. 3.1

The difficulty in defining central tendencyThe difficulty in defining central tendency

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Mean, Median, ModeMean, Median, Mode

►To deal with the problems, statisticians To deal with the problems, statisticians have developed three different have developed three different methods for measuring central methods for measuring central tendency.tendency.

►How do you decide which one to use?How do you decide which one to use? Keep in mind – the general purpose of Keep in mind – the general purpose of

central tendency is to find the single most central tendency is to find the single most representative score.representative score.

MeanMean

►Arithmetic averageArithmetic average►Add all the scores and divide by the Add all the scores and divide by the

number of scores.number of scores.►For the average of a population use For the average of a population use

the Greek letter mu, the Greek letter mu, (myoo)(myoo)►For the mean for a sample use X For the mean for a sample use X

(read as X-bar) or (read as X-bar) or MM

MeanMean

►The mean for a distribution is the sum The mean for a distribution is the sum of the scores divided by the number of of the scores divided by the number of scores.scores.

►FormulaFormula

nn

Population Mean Sample Mean

Why Greek letters?Why Greek letters?

►Greek letters to identify population Greek letters to identify population valuesvalues

►Our own alphabet to identify sample Our own alphabet to identify sample valuesvalues

nnSample n is used for the number of scores in the sample

ExampleExample

For a population N=4 scores:For a population N=4 scores:

3, 7, 4, 63, 7, 4, 6

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Alternative Definitions for Alternative Definitions for MeanMean

► The mean can be thought of as an amount The mean can be thought of as an amount each individual would get if the total each individual would get if the total ((were equally divided among all the were equally divided among all the individuals (N) in the distribution.individuals (N) in the distribution.

► Example 3.2 pg. 55Example 3.2 pg. 55

n=6 boysn=6 boys n = 4 boysn = 4 boysBuy 180 baseball cardsBuy 180 baseball cards M = $5M = $5Each gets 30 cardsEach gets 30 cards $20 total$20 total

Do not know how much Do not know how much each boy haseach boy has

Alternative Definitions for Alternative Definitions for MeanMean

► Define the mean as Define the mean as a balance point for a balance point for the distribution.the distribution.

► Example 3.2 pg. 56Example 3.2 pg. 56

Weighted MeanWeighted Mean

► Combining two sets of scores and then Combining two sets of scores and then finding the overall mean for the combined finding the overall mean for the combined group.group.

► Example pg. 57Example pg. 57► Because the samples are not the same size, Because the samples are not the same size,

one will make a larger contribution to the one will make a larger contribution to the total group and therefore will carry more total group and therefore will carry more weight in determining the weight in determining the overall meanoverall mean..

► The overall mean is called the weighted The overall mean is called the weighted mean.mean.

Computing the mean from a Computing the mean from a frequency distribution tablefrequency distribution table

Quiz score (X)Quiz score (X) ff f f XX

1010 11 1010

99 22 1818

88 44 3232

77 00 00

66 11 66

Characteristics of the MeanCharacteristics of the Mean

►Every score in the distribution Every score in the distribution contributes to the value of the mean.contributes to the value of the mean. Every score must be added into the total Every score must be added into the total

in order to compute the mean.in order to compute the mean.

►Changing the value of the score will Changing the value of the score will change the meanchange the mean

► Introducing a new score or removing a Introducing a new score or removing a score will change the value of the score will change the value of the meanmean

MedianMedian

►The score that divides a distribution The score that divides a distribution exactly in half.exactly in half.

►No symbols or notationsNo symbols or notations►Definition and computations are Definition and computations are

identical for a sample and for a identical for a sample and for a populationpopulation

►Goal of a median is to determine the Goal of a median is to determine the precise midpoint of a distribution.precise midpoint of a distribution.

ExampleExample

►When N is an odd number When N is an odd number

3, 5, 8, 10, 113, 5, 8, 10, 11►When N is an even numberWhen N is an even number

3, 3, 4, 5, 7, 83, 3, 4, 5, 7, 8

Median = 8

Median = 4.5

Median (cont.)Median (cont.)

►Used when a researcher wants to Used when a researcher wants to divide the sample or population into divide the sample or population into two groups that are exactly the same two groups that are exactly the same size.size.

►Median splitMedian split Where one group is above the median line Where one group is above the median line

and the other is belowand the other is below For example: one of high-scoring subjects For example: one of high-scoring subjects

and one of low-scoring subjectsand one of low-scoring subjects

Figure 3.5Figure 3.5

The median divides the area in the graph in halfThe median divides the area in the graph in half

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ModeMode

►The score or category that has the The score or category that has the greatest frequencygreatest frequency

►No symbols or notation to identify the No symbols or notation to identify the modemode

►The definition is the same for either a The definition is the same for either a population or a sample distribution.population or a sample distribution.

Mode (cont.)Mode (cont.)

►Can be used to determine the typical Can be used to determine the typical or average value for any scale of or average value for any scale of measurement, including a nominal measurement, including a nominal scale (chapter 1)scale (chapter 1)

► It is possible to have more than one It is possible to have more than one modemode

Table 3.4Table 3.4

Favorite restaurantsFavorite restaurants

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Figure 3.6Figure 3.6

A bimodal distributionA bimodal distribution

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vs. multimodal

Selecting a Measure of Selecting a Measure of Central TendencyCentral Tendency

►Could be possible to compute two or Could be possible to compute two or three measures of central tendency three measures of central tendency with a set of data.with a set of data.

►Often get similar results.Often get similar results.

MeanMean

► Mean is the most preferred measure.Mean is the most preferred measure. Usually a good representative valueUsually a good representative value Goal is to find the single value that best Goal is to find the single value that best

represents the entire distribution.represents the entire distribution.

► Mean has the added advantage of being Mean has the added advantage of being closely related to variance and standard closely related to variance and standard deviation (the most common measures of deviation (the most common measures of variability)variability)

► This relationship makes the mean a valuable This relationship makes the mean a valuable measure for purposes of inferential statisticsmeasure for purposes of inferential statistics

When to Use the MedianWhen to Use the Median

►Three situations in which the median Three situations in which the median serves as a valuable alternative to the serves as a valuable alternative to the mean.mean. Extreme scores or skewed distributionsExtreme scores or skewed distributions Undetermined valuesUndetermined values Open-ended distributionsOpen-ended distributions

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When to Use the ModeWhen to Use the Mode

►Three situations in which the mode is Three situations in which the mode is commonly used as an alternative to commonly used as an alternative to the mean, or is used in conjunction the mean, or is used in conjunction with the mean to describe central with the mean to describe central tendencytendency Nominal scalesNominal scales Discrete variablesDiscrete variables Describing shapeDescribing shape

Figure 3.10Figure 3.10

Central tendency and symmetrical distributionsCentral tendency and symmetrical distributions

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Normal BimodalRectangular

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Figure 3.11Figure 3.11

Central tendency and skewed distributionsCentral tendency and skewed distributions

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