Measures of Central Measures of Central TendencyTendency

IntroductionIntroduction

Three measures of central tendencyThree measures of central tendency All three summarize an entire distribution All three summarize an entire distribution

of scoresof scores By describing the most typical, central, or By describing the most typical, central, or

representative value of that distributionrepresentative value of that distribution So, they reduce large data sets by describing So, they reduce large data sets by describing

them using just a few numbersthem using just a few numbers All three define All three define typicaltypical in different ways in different ways

ApplicationsApplications

Nominal Nominal VariablesVariables

Ordinal Ordinal VariablesVariables

Interval Interval VariablesVariables

ModeMode ModeMode ModeMode

MedianMedian MedianMedian

MeanMean

ModeMode

This is the value or score that occurs most This is the value or score that occurs most frequentlyfrequently For example, scores on the first exam: For example, scores on the first exam: 35, 36,36,37,38,3935, 36,36,37,38,39 The modal scores on the exam is 36The modal scores on the exam is 36 It tells you that more people received that It tells you that more people received that

score than any otherscore than any other

Uses for the ModeUses for the Mode

The mode has two usesThe mode has two uses A “quick and easy” indicator of typical scoresA “quick and easy” indicator of typical scores When you are working with nominal-level When you are working with nominal-level

variablesvariables

Limitations of the ModeLimitations of the Mode

Some distributions have no mode at all, or so Some distributions have no mode at all, or so many that the statistic loses its meaningmany that the statistic loses its meaning For example, if there were an equal number of males For example, if there were an equal number of males

and females, there is no modeand females, there is no mode

The second limitation is when you report the The second limitation is when you report the mode for ordinal or interval-ratio data, the modal mode for ordinal or interval-ratio data, the modal score may be far from the center, so it gives score may be far from the center, so it gives very little informationvery little information

So, you need other clues to complete the pictureSo, you need other clues to complete the picture

MedianMedian

The median represents the exact center of The median represents the exact center of a distribution of scoresa distribution of scores

It is the It is the scorescore of the of the casecase having half the having half the cases above it and half below itcases above it and half below it After all the cases have been ordered from low After all the cases have been ordered from low

to highto high In 2009, the median per capita income was In 2009, the median per capita income was

$26,178; the median household income was $26,178; the median household income was $50,007, what does that tell you?$50,007, what does that tell you? So, the median is the score associated with the So, the median is the score associated with the

central or middle casecentral or middle case

MedianMedian

When the number of cases (N) is odd, the When the number of cases (N) is odd, the median is the middle casemedian is the middle case But when the number of cases (N) is even, the But when the number of cases (N) is even, the

median is the average of the two middle scoresmedian is the average of the two middle scores For a large sample, there is a formula for finding For a large sample, there is a formula for finding

the the positionposition of the median, but the median is the of the median, but the median is the score that occupies that positionscore that occupies that position The formula for the position is N + 1 divided by 2The formula for the position is N + 1 divided by 2 So for the above example, N = 7, so the position is So for the above example, N = 7, so the position is

8/2=48/2=4 The median is the number occupying the fourth The median is the number occupying the fourth

positionposition

MedianMedian If N is even, you still use the formula N+1/2 to find the If N is even, you still use the formula N+1/2 to find the

position position For the above example with a 21 added to the endFor the above example with a 21 added to the end

2,4,5,9,14,18,20,21 N = 82,4,5,9,14,18,20,21 N = 8 So, 8 + 1 divided by 2 = 4.5So, 8 + 1 divided by 2 = 4.5 The median will be the number between 9 and 14The median will be the number between 9 and 14 Add 9 + 14 and divide by 2 to find the average between the two Add 9 + 14 and divide by 2 to find the average between the two

scoresscores The median will be 23/2 = 11.5The median will be 23/2 = 11.5

If the two middle cases are the same score, the median If the two middle cases are the same score, the median is that numberis that number

The median cannot be calculated for variables measured The median cannot be calculated for variables measured at the nominal level, because they cannot be ordered or at the nominal level, because they cannot be ordered or ranked, so there is no middleranked, so there is no middle

Percentiles and QuartilesPercentiles and Quartiles

Introduced here because it is similar to the Introduced here because it is similar to the median, before going on to the last measure of median, before going on to the last measure of central tendencycentral tendency

PercentilesPercentiles A percentile identifies the point below which a specific A percentile identifies the point below which a specific

percentage of cases fallpercentage of cases fall If you have a percentile score on an exam of 82, it If you have a percentile score on an exam of 82, it

means that 82% of the people taking that exam means that 82% of the people taking that exam scored lower than you didscored lower than you did

The median is the 50The median is the 50thth percentile percentile

PercentilesPercentiles

To find a position associated with a percentile:To find a position associated with a percentile: Multiply the number of cases (N) by the proportional Multiply the number of cases (N) by the proportional

value of the percentilevalue of the percentile For example, 46For example, 46thth percentile = .46 percentile = .46 The resultant value identifies the number of the case The resultant value identifies the number of the case

that occupies that percentile scorethat occupies that percentile score Example to find raw score in 90Example to find raw score in 90 thth percentile percentile

If N = 150, we want the 90If N = 150, we want the 90thth percentile score percentile score Will multiply 150 by .90 = 135Will multiply 150 by .90 = 135 If we order the cases from lowest to highest, the 135If we order the cases from lowest to highest, the 135thth

person would have a percentile score of 90, and person would have a percentile score of 90, and whatever their raw score was, it would have 90% of whatever their raw score was, it would have 90% of people below that scorepeople below that score

QuartilesQuartiles

Quartiles divide the distribution into quartersQuartiles divide the distribution into quarters So, the first quartile is the 25So, the first quartile is the 25 thth percentile percentile Computers will report the score occupying the Computers will report the score occupying the

first quartile, the median, and the score first quartile, the median, and the score occupying the third quartileoccupying the third quartile

To interpret, 50% of all the people surveyed fell To interpret, 50% of all the people surveyed fell between the first quartile and the third quartilebetween the first quartile and the third quartile Since they occupy the 25% position, and the 75% Since they occupy the 25% position, and the 75%

position, there are 50% of the scores between themposition, there are 50% of the scores between them

Another reason why they are reported, is to Another reason why they are reported, is to eliminate the extreme eliminate the extreme

MeanMean

The mean is the arithmetic averageThe mean is the arithmetic average It is the most commonly used measure of It is the most commonly used measure of

central tendencycentral tendency To compute the mean, add up the scores and To compute the mean, add up the scores and

then divide by the number of scores (N)then divide by the number of scores (N) You should always look at the mean to see if it You should always look at the mean to see if it

is a reasonable statistic given the data with is a reasonable statistic given the data with which you beganwhich you began Also a good idea to do all of the math twiceAlso a good idea to do all of the math twice

Interpretation of the MeanInterpretation of the Mean

What, exactly, happens every time we What, exactly, happens every time we divide by Ndivide by N

If you substituted the mean for each of the If you substituted the mean for each of the six scores, and added them together, will six scores, and added them together, will get the same totalget the same total

Characteristics of the MeanCharacteristics of the Mean

The mean is generally more reliable than the median or The mean is generally more reliable than the median or the modethe mode It will vary less among samples drawn from the same population, It will vary less among samples drawn from the same population,

if you keep drawing more samplesif you keep drawing more samples

Second, the mean is the point around which all the Second, the mean is the point around which all the scores cancel outscores cancel out

Third characteristic of the mean is expressed in the Third characteristic of the mean is expressed in the statement: if the differences between the scores and the statement: if the differences between the scores and the mean are squared and then added, the resultant sum will mean are squared and then added, the resultant sum will be less than the sum of the squared differences between be less than the sum of the squared differences between the scores and any other point in the distributionthe scores and any other point in the distribution The mean is closer to all of the scores than the other measures The mean is closer to all of the scores than the other measures

of central tendencyof central tendency

Characteristics of the MeanCharacteristics of the Mean

The fourth characteristic of the mean is The fourth characteristic of the mean is that it is affected by every score in the that it is affected by every score in the distributiondistribution The mode and the median are not as much The mode and the median are not as much

affectedaffected Advantage of this:Advantage of this:

The mean uses all available information—every The mean uses all available information—every score in the distribution affects the meanscore in the distribution affects the mean

DisadvantageDisadvantage When a distribution has a few extreme cases, the When a distribution has a few extreme cases, the

mean becomes a very misleading measure of mean becomes a very misleading measure of central tendency, especially with a small samplecentral tendency, especially with a small sample

SkewnessSkewness

The mean is always pulled in the direction of The mean is always pulled in the direction of extreme scores, if they are only on one end (low extreme scores, if they are only on one end (low or high)or high)

The mean, median, and mode will only be the The mean, median, and mode will only be the same when a distribution is symmetricalsame when a distribution is symmetrical

When a distribution has some extremely high When a distribution has some extremely high scores (a positive scores (a positive skew) skew) the mean will have a the mean will have a greater value than the mediangreater value than the median

If the distribution has some very low scores (a If the distribution has some very low scores (a negative skew), the mean will be lower in value negative skew), the mean will be lower in value than the medianthan the median

Two Reasons for Comparing the Two Reasons for Comparing the Mean to the MedianMean to the Median

Gives you a quick way to determine if a distribution is Gives you a quick way to determine if a distribution is skewed, and tells you in which direction (since you don’t skewed, and tells you in which direction (since you don’t see the raw data)see the raw data)It gives people a simple way to “lie” with statisticsIt gives people a simple way to “lie” with statistics

for example, income is usually positively skewed (skewed to the for example, income is usually positively skewed (skewed to the right), so the mean will be higher than the median, since the right), so the mean will be higher than the median, since the extreme scores are the small percentage of people making over extreme scores are the small percentage of people making over $300,000 per year$300,000 per year

So, the Chamber of Commerce may report the mean income to So, the Chamber of Commerce may report the mean income to give the impression the community is wealthier than it really isgive the impression the community is wealthier than it really is

You would be interested in the median income if you were going to You would be interested in the median income if you were going to open a business in the community, since your shoppers would open a business in the community, since your shoppers would be average people, not the wealthybe average people, not the wealthy

As a researcher, you should report both numbers, and let the As a researcher, you should report both numbers, and let the reader decide which to usereader decide which to use