1 1 Slide © 2007 Thomson South-Western. All Rights Reserved

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© 2007 Thomson South-Western. All Rights Reserved© 2007 Thomson South-Western. All Rights Reserved

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Chapter 3Chapter 3 Descriptive Statistics: Numerical Descriptive Statistics: Numerical

MeasuresMeasures

Measures of LocationMeasures of Location Measures of VariabilityMeasures of Variability Measures of Distribution Shape, Relative Measures of Distribution Shape, Relative

Location, and Detecting OutliersLocation, and Detecting Outliers Measures of Association Between Two Measures of Association Between Two

VariablesVariables Weighted MeanWeighted Mean

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Measures of LocationMeasures of Location

If the measures are computedIf the measures are computed for data from a sample,for data from a sample,

they are called they are called sample statisticssample statistics..

If the measures are computedIf the measures are computed for data from a population,for data from a population,

they are called they are called population parameterspopulation parameters..

A sample statistic is referred toA sample statistic is referred toas the as the point estimatorpoint estimator of the of the

corresponding population parameter.corresponding population parameter.

MeanMean

MedianMedian ModeMode PercentilesPercentiles QuartilesQuartiles

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MeanMean

The The meanmean of a data set is the average of all of a data set is the average of all the data values.the data values.

The sample mean is the point estimator of The sample mean is the point estimator of the population mean the population mean . .

xx

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Sample Mean Sample Mean xx

Number ofNumber ofobservationsobservationsin the samplein the sample

Number ofNumber ofobservationsobservationsin the samplein the sample

Sum of the valuesSum of the valuesof the of the nn observations observations

Sum of the valuesSum of the valuesof the of the nn observations observations

ixx

n ix

xn

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Population Mean Population Mean

Number ofNumber ofobservations inobservations inthe populationthe population

Number ofNumber ofobservations inobservations inthe populationthe population

Sum of the valuesSum of the valuesof the of the NN observations observations

Sum of the valuesSum of the valuesof the of the NN observations observations

ix

N

ix

N

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MedianMedian

Whenever a data set has extreme values, the medianWhenever a data set has extreme values, the median is the preferred measure of central location.is the preferred measure of central location.

A few extremely large incomes or property valuesA few extremely large incomes or property values can inflate the mean.can inflate the mean.

The median is the measure of location most oftenThe median is the measure of location most often reported for annual income and property value data.reported for annual income and property value data.

The The medianmedian of a data set is the value in the middle of a data set is the value in the middle when the data items are arranged in ascending order.when the data items are arranged in ascending order.

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MedianMedian

1212 2222 2626 2727 27272424 2828

For an For an odd numberodd number of observations: of observations:

in ascending orderin ascending order

2626 2828 2727 2222 2424 2727 1212 7 observations7 observations

the median is the middle value.the median is the middle value.

Median = 26Median = 26

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2828

MedianMedian

For an For an even numbereven number of observations: of observations:

in ascending orderin ascending order

2727 8 observations8 observations

the median is the average of the middle two values.the median is the average of the middle two values.

Median = (26 + 27)/2 = 26.5Median = (26 + 27)/2 = 26.5

30301212 2222 2626 2727 27272424

2626 2828 2727 2222 2424 3030 1212

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Mean VS MedianMean VS Median

The mean IS affected by outliers (extreme The mean IS affected by outliers (extreme observations)observations)

The median IS NOT affected by outliersThe median IS NOT affected by outliers

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ModeMode

The The modemode of a data set is the value that occurs with of a data set is the value that occurs with greatest frequency.greatest frequency. The greatest frequency can occur at two or moreThe greatest frequency can occur at two or more different values.different values. If the data have exactly two modes, the data areIf the data have exactly two modes, the data are bimodalbimodal..

If the data have more than two modes, the data areIf the data have more than two modes, the data are multimodalmultimodal..

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PercentilesPercentiles

A percentile provides information about how theA percentile provides information about how the data are spread over the interval from the smallestdata are spread over the interval from the smallest value to the largest value.value to the largest value. Admission test scores for colleges and universitiesAdmission test scores for colleges and universities are frequently reported in terms of percentiles.are frequently reported in terms of percentiles.

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The The ppth percentileth percentile of a data set is a value such of a data set is a value such that at least that at least pp percent of the items take on this percent of the items take on this value or less and at least (100 - value or less and at least (100 - pp) percent of ) percent of the items take on this value or more.the items take on this value or more.


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Arrange the data in ascending order.Arrange the data in ascending order. Arrange the data in ascending order.Arrange the data in ascending order.

Compute index Compute index ii, the position of the , the position of the ppth percentile.th percentile. Compute index Compute index ii, the position of the , the position of the ppth percentile.th percentile.

ii = ( = (pp/100)/100)nn

If If ii is not an integer, round up. The is not an integer, round up. The pp th percentileth percentile is the value in the is the value in the ii th position.th position. If If ii is not an integer, round up. The is not an integer, round up. The pp th percentileth percentile is the value in the is the value in the ii th position.th position.

If If ii is an integer, the is an integer, the pp th percentile is the averageth percentile is the average of the values in positionsof the values in positions i i and and ii +1.+1. If If ii is an integer, the is an integer, the pp th percentile is the averageth percentile is the average of the values in positionsof the values in positions i i and and ii +1.+1.

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Note on Excel’s Note on Excel’s PercentilePercentile Function Function

The formula that Excel uses is The formula that Excel uses is differentdifferent from the one used in the textbook! from the one used in the textbook!

In order to find the observation where the median occurs, In order to find the observation where the median occurs, Excel uses the following formula:Excel uses the following formula:

LLpp = ( = (pp/100)/100)nn + (1 – + (1 – pp/100)/100)

Once the observation is identified Excel will: Once the observation is identified Excel will: 1.1. If If LLpp is a whole number (e.g. 12), is a whole number (e.g. 12),

Excel’s result will be the same as the textbook’s.Excel’s result will be the same as the textbook’s.2.2. If If LpLp is not a whole number (e.g. 12.3) Excel’s is not a whole number (e.g. 12.3) Excel’s

result will be different from the textbook’s.result will be different from the textbook’s.

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QuartilesQuartiles

Quartiles are specific percentiles.Quartiles are specific percentiles. First Quartile = 25th PercentileFirst Quartile = 25th Percentile

Second Quartile = 50th Percentile = MedianSecond Quartile = 50th Percentile = Median Third Quartile = 75th PercentileThird Quartile = 75th Percentile

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Measures of VariabilityMeasures of Variability

It is often desirable to consider measures of variabilityIt is often desirable to consider measures of variability (dispersion), as well as measures of location.(dispersion), as well as measures of location.

For example, in choosing supplier A or supplier B weFor example, in choosing supplier A or supplier B we might consider not only the average delivery time formight consider not only the average delivery time for each, but also the variability in delivery time for each.each, but also the variability in delivery time for each.

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Measures of VariabilityMeasures of Variability

RangeRange Interquartile RangeInterquartile Range VarianceVariance Standard DeviationStandard Deviation Coefficient of VariationCoefficient of Variation

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RangeRange

The The rangerange of a data set is the difference between the of a data set is the difference between the largest and smallest data values.largest and smallest data values.

It is the It is the simplest measuresimplest measure of variability. of variability. It is It is very sensitivevery sensitive to the smallest and largest data to the smallest and largest data values.values.

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Interquartile RangeInterquartile Range

The The interquartile rangeinterquartile range of a data set is the difference of a data set is the difference between the third quartile and the first quartile.between the third quartile and the first quartile. It is the range for the It is the range for the middle 50%middle 50% of the data. of the data.

It overcomes the sensitivity to extreme data values.It overcomes the sensitivity to extreme data values.

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The The variancevariance is a measure of variability that utilizes is a measure of variability that utilizes all the data.all the data.

VarianceVariance

It is based on the difference between the value ofIt is based on the difference between the value of each observation (each observation (xxii) and the mean ( for a sample,) and the mean ( for a sample, for a population).for a population).

xx

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VarianceVariance

The variance is computed as follows:The variance is computed as follows:

The variance is computed as follows:The variance is computed as follows:

The variance is the The variance is the average of the squaredaverage of the squared differencesdifferences between each data value and the mean. between each data value and the mean. The variance is the The variance is the average of the squaredaverage of the squared differencesdifferences between each data value and the mean. between each data value and the mean.

for afor asamplesample

for afor apopulationpopulation

22

( )xNi 2

2

( )xNis

xi x

n2

2

1

( )s

xi x

n2

2

1

( )

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Standard DeviationStandard Deviation

The The standard deviationstandard deviation of a data set is the positive of a data set is the positive square root of the variance.square root of the variance.

It is measured in the It is measured in the same units as the datasame units as the data, making, making it more easily interpreted than the variance.it more easily interpreted than the variance.

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The standard deviation is computed as follows:The standard deviation is computed as follows:

The standard deviation is computed as follows:The standard deviation is computed as follows:



Standard DeviationStandard Deviation

s s 2s s 2 2 2

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The coefficient of variation is computed as follows:The coefficient of variation is computed as follows:

The coefficient of variation is computed as follows:The coefficient of variation is computed as follows:

Coefficient of VariationCoefficient of Variation

100 %s

x

100 %s

x

The The coefficient of variationcoefficient of variation indicates how large the indicates how large the standard deviation is in relation to the mean.standard deviation is in relation to the mean. The The coefficient of variationcoefficient of variation indicates how large the indicates how large the standard deviation is in relation to the mean.standard deviation is in relation to the mean.



100 %

100 %

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Measures of Distribution Shape,Measures of Distribution Shape,Relative Location, and Detecting OutliersRelative Location, and Detecting Outliers

Distribution ShapeDistribution Shape z-Scoresz-Scores Chebyshev’s TheoremChebyshev’s Theorem Empirical RuleEmpirical Rule Detecting OutliersDetecting Outliers

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Distribution Shape: SkewnessDistribution Shape: Skewness

An important measure of the shape of a An important measure of the shape of a distribution is called distribution is called skewnessskewness..

The formula for computing skewness for a data The formula for computing skewness for a data set is somewhat complex.set is somewhat complex.

• Skewness can be easily computed using Skewness can be easily computed using statistical software.statistical software.

Excel’s SKEW function can be used to compute theExcel’s SKEW function can be used to compute the

skewness of a data set.skewness of a data set.

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Symmetric (not skewed)Symmetric (not skewed)

• Skewness is zero.Skewness is zero.

• Mean and median are equal.Mean and median are equal.R

ela

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Skewness = Skewness = 0 0

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Rela

tive F

req

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00


Moderately Skewed LeftModerately Skewed Left

• Skewness is negative.Skewness is negative.

• Mean will usually be less than the median.Mean will usually be less than the median.

Skewness = Skewness = .31 .31

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Moderately Skewed RightModerately Skewed Right

• Skewness is positive.Skewness is positive.

• Mean will usually be more than the median.Mean will usually be more than the median.R

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tive F

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

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Skewness = .31 Skewness = .31

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The The z-scorez-score is often called the standardized value. is often called the standardized value. The The z-scorez-score is often called the standardized value. is often called the standardized value.

It denotes the number of standard deviations a dataIt denotes the number of standard deviations a data value value xxii is from the mean. is from the mean. It denotes the number of standard deviations a dataIt denotes the number of standard deviations a data value value xxii is from the mean. is from the mean.

z-Scoresz-Scores

zx xsii

zx xsii

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z-Scoresz-Scores

A data value less than the sample mean will have aA data value less than the sample mean will have a z-score less than zero.z-score less than zero. A data value greater than the sample mean will haveA data value greater than the sample mean will have a z-score greater than zero.a z-score greater than zero. A data value equal to the sample mean will have aA data value equal to the sample mean will have a z-score of zero.z-score of zero.

An observation’s z-score is a measure of the relativeAn observation’s z-score is a measure of the relative location of the observation in a data set.location of the observation in a data set.

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Chebyshev’s TheoremChebyshev’s Theorem

At least (1 - 1/At least (1 - 1/zz22) of the items in ) of the items in anyany data set will be data set will be within within zz standard deviations of the mean, where standard deviations of the mean, where z z isis any value greater than 1.any value greater than 1.

At least (1 - 1/At least (1 - 1/zz22) of the items in ) of the items in anyany data set will be data set will be within within zz standard deviations of the mean, where standard deviations of the mean, where z z isis any value greater than 1.any value greater than 1.

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At least of the data values must beAt least of the data values must be

within of the mean.within of the mean.



75%75%75%75%

zz = 2 standard deviations = 2 standard deviations zz = 2 standard deviations = 2 standard deviations

Chebyshev’s TheoremChebyshev’s Theorem





89%89%89%89%






94%94%94%94%


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Empirical RuleEmpirical Rule

For data having a bell-shaped distribution:For data having a bell-shaped distribution:

of the values of a normal random variableof the values of a normal random variable are within of its mean.are within of its mean. of the values of a normal random variableof the values of a normal random variable are within of its mean.are within of its mean.68.26%68.26%68.26%68.26%

+/- 1 standard deviation+/- 1 standard deviation+/- 1 standard deviation+/- 1 standard deviation


+/- 2 standard deviations+/- 2 standard deviations+/- 2 standard deviations+/- 2 standard deviations


+/- 3 standard deviations+/- 3 standard deviations+/- 3 standard deviations+/- 3 standard deviations

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Empirical RuleEmpirical Rule

xx – – 33 – – 11

– – 22 + 1+ 1

+ 2+ 2 + 3+ 3

68.26%68.26%95.44%95.44%99.72%99.72%

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Detecting OutliersDetecting Outliers

An An outlieroutlier is an unusually small or unusually large is an unusually small or unusually large value in a data set.value in a data set. A data value with a z-score less than -3 or greaterA data value with a z-score less than -3 or greater than +3 might be considered an outlier.than +3 might be considered an outlier. It might be:It might be:

• an incorrectly recorded data valuean incorrectly recorded data value• a data value that was incorrectly included in thea data value that was incorrectly included in the data setdata set• a correctly recorded data value that belongs ina correctly recorded data value that belongs in the data setthe data set

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Measures of Association Measures of Association Between Two VariablesBetween Two Variables

CovarianceCovariance Correlation CoefficientCorrelation Coefficient

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CovarianceCovariance

Positive values indicate a positive relationship.Positive values indicate a positive relationship. Positive values indicate a positive relationship.Positive values indicate a positive relationship.

Negative values indicate a negative relationship.Negative values indicate a negative relationship. Negative values indicate a negative relationship.Negative values indicate a negative relationship.

The The covariancecovariance is a measure of the is a measure of the linearlinear association association between two variables.between two variables. The The covariancecovariance is a measure of the is a measure of the linearlinear association association between two variables.between two variables.

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CovarianceCovariance

The correlation coefficient is computed as follows:The correlation coefficient is computed as follows:


forforsamplessamples

forforpopulationspopulations

sx x y ynxy

i i

( )( )

1s

x x y ynxy

i i

( )( )

1

xyi x i yx y

N

( )( )

xy

i x i yx y

N

( )( )

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Correlation CoefficientCorrelation Coefficient

Values near +1 indicate a Values near +1 indicate a strong positive linearstrong positive linear relationshiprelationship.. Values near +1 indicate a Values near +1 indicate a strong positive linearstrong positive linear relationshiprelationship..

Values near -1 indicate a Values near -1 indicate a strong negative linearstrong negative linear relationshiprelationship. . Values near -1 indicate a Values near -1 indicate a strong negative linearstrong negative linear relationshiprelationship. .

The coefficient can take on values between -1 and +1.The coefficient can take on values between -1 and +1. The coefficient can take on values between -1 and +1.The coefficient can take on values between -1 and +1.

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forforsamplessamples

forforpopulationspopulations

rs

s sxyxy

x yrs

s sxyxy

x y

xyxy

x y

xyxy

x y


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Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of thedoes not mean that one variable is the cause of the other.other.

Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of thedoes not mean that one variable is the cause of the other.other.

Correlation is a measure of linear association and notCorrelation is a measure of linear association and not necessarily causation. necessarily causation. Correlation is a measure of linear association and notCorrelation is a measure of linear association and not necessarily causation. necessarily causation.

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A golfer is interested in investigatingA golfer is interested in investigating

the relationship, if any, between drivingthe relationship, if any, between driving

distance and 18-hole score.distance and 18-hole score.

277.6277.6259.5259.5269.1269.1267.0267.0255.6255.6272.9272.9

696971717070707071716969

Average DrivingAverage DrivingDistance (yds.)Distance (yds.)

AverageAverage18-Hole Score18-Hole Score

Covariance and Correlation CoefficientCovariance and Correlation Coefficient

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Weighted MeanWeighted Mean

When the mean is computed by giving each dataWhen the mean is computed by giving each data value a weight that reflects its importance, it isvalue a weight that reflects its importance, it is referred to as a referred to as a weighted meanweighted mean.. In the computation of a grade point average (GPA),In the computation of a grade point average (GPA), the weights are the number of credit hours earned forthe weights are the number of credit hours earned for each grade.each grade. When data values vary in importance, the analystWhen data values vary in importance, the analyst must choose the weight that best reflects themust choose the weight that best reflects the importance of each value.importance of each value.

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Weighted MeanWeighted Mean

i i

i

wxx

w

i i

i

wxx

w

where:where:

xxii = value of observation = value of observation ii

wwi i = weight for observation = weight for observation ii

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In class empirical exercisesIn class empirical exercises

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1 1 Slide © 2007 Thomson South-Western. All Rights Reserved