Slides by JOHN LOUCKS St. Edward’s University

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Slides by

JOHNLOUCKSSt. Edward’sUniversity

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Chapter 3, Part AChapter 3, Part A Descriptive Statistics: Numerical Descriptive Statistics: Numerical

MeasuresMeasures

Measures of LocationMeasures of Location Measures of VariabilityMeasures of Variability

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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.

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

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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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Seventy efficiency apartments were Seventy efficiency apartments were randomlyrandomly

sampled in a small college town. The sampled in a small college town. The monthly rentmonthly rent

prices for these apartments are listed below.prices for these apartments are listed below.

Sample MeanSample Mean

Example: Apartment RentsExample: Apartment Rents

445 615 430 590 435 600 460 600 440 615440 440 440 525 425 445 575 445 450 450465 450 525 450 450 460 435 460 465 480450 470 490 472 475 475 500 480 570 465600 485 580 470 490 500 549 500 500 480570 515 450 445 525 535 475 550 480 510510 575 490 435 600 435 445 435 430 440

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Sample MeanSample Mean

34,356 490.80

70ix

xn

34,356 490.80

70ix

xn

445 615 430 590 435 600 460 600 440 615440 440 440 525 425 445 575 445 450 450465 450 525 450 450 460 435 460 465 480450 470 490 472 475 475 500 480 570 465600 485 580 470 490 500 549 500 500 480570 515 450 445 525 535 475 550 480 510510 575 490 435 600 435 445 435 430 440


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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 1414 1919 2626 27271818 2727

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

in ascending orderin ascending order

2626 1818 2727 1212 1414 2727 1919 7 observations7 observations

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

Median = 19Median = 19

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1212 1414 1919 2626 27271818 2727

MedianMedian

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

in ascending orderin ascending order

2626 1818 2727 1212 1414 2727 3030 8 observations8 observations

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

Median = (19 + 26)/2 = 22.5Median = (19 + 26)/2 = 22.5

1919

3030

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MedianMedian

Averaging the 35th and 36th data values:Averaging the 35th and 36th data values:Median = (475 + 475)/2 = 475Median = (475 + 475)/2 = 475

Note: Data is in ascending order.Note: Data is in ascending order.

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615


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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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ModeMode

450 occurred most frequently (7 times)450 occurred most frequently (7 times)

Mode = 450Mode = 450


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615


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Excel Formula WorksheetExcel Formula Worksheet

Note: Rows 7-71 are not shown.Note: Rows 7-71 are not shown.

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean =AVERAGE(B2:B71)3 2 440 Median =MEDIAN(B2:B71)4 3 450 Mode =MODE(B2:B71)5 4 6156 5 480

Using Excel to ComputeUsing Excel to Computethe Mean, Median, and Modethe Mean, Median, and Mode

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Value WorksheetValue Worksheet

Using Excel to ComputeUsing Excel to Computethe Mean, Median, and Modethe Mean, Median, and Mode

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean 490.803 2 440 Median 475.004 3 450 Mode 450.005 4 6156 5 480


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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. 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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PercentilesPercentiles

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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8080thth Percentile Percentile

ii = ( = (pp/100)/100)nn = (80/100)70 = 56 = (80/100)70 = 56Averaging the 56Averaging the 56thth and 57 and 57thth data values: data values:

80th Percentile = (535 + 549)/2 = 54280th Percentile = (535 + 549)/2 = 542


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615


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8080thth Percentile Percentile

““At least 80% of theAt least 80% of the items take on aitems take on a

value of 542 or less.”value of 542 or less.”

““At least 20% of theAt least 20% of theitems take on aitems take on a

value of 542 or more.”value of 542 or more.”

56/70 = .8 or 80%56/70 = .8 or 80% 14/70 = .2 or 20%14/70 = .2 or 20%

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615


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Using Excel’s Using Excel’s PercentilePercentile Function Function

The formula Excel uses to compute the location (The formula Excel uses to compute the location (LLpp))of the of the ppth percentile is th percentile is

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

Excel would compute the location of the 80Excel would compute the location of the 80thth percentile for the apartment rent data as follows:percentile for the apartment rent data as follows:

LL8080 = (80/100)70 + (1 – 80/100) = 56 + .2 = 56.2 = (80/100)70 + (1 – 80/100) = 56 + .2 = 56.2

The 80The 80thth percentile would be percentile would be

535 + .2(549 - 535) = 535 + 2.8 = 537.8535 + .2(549 - 535) = 535 + 2.8 = 537.8

Using Excel’s Rank and Percentile ToolUsing Excel’s Rank and Percentile Toolto Compute Percentiles and Quartilesto Compute Percentiles and Quartiles

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A B C D E

1Apart-ment

Monthly Rent ($) 80th Percentile

2 1 525 =PERCENTILE(B2:B71,.8) 3 2 440 4 3 450 5 4 6156 5 480



It is not necessaryIt is not necessaryto put the datato put the data

in ascending order.in ascending order.

8080thth percentilepercentile


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Excel Value WorksheetExcel Value Worksheet

A B C D E

1Apart-ment

Monthly Rent ($) 80th Percentile

2 1 525 537.8 3 2 440 4 3 450 5 4 6156 5 480



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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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Third QuartileThird Quartile

Third quartile = 75th percentileThird quartile = 75th percentile

i i = (= (pp/100)/100)nn = (75/100)70 = 52.5 = 53 = (75/100)70 = 52.5 = 53Third quartile = 525Third quartile = 525


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615


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Using Excel’s Using Excel’s QuartileQuartile Function Function

Excel computes the locations of the 1Excel computes the locations of the 1stst, 2, 2ndnd, and 3, and 3rdrd

quartiles by first converting the quartiles toquartiles by first converting the quartiles topercentiles and then using the following formula topercentiles and then using the following formula tocompute the location (compute the location (LLpp) of the ) of the ppth percentile: th percentile:

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

Excel would compute the location of the 3Excel would compute the location of the 3rdrd quartile quartile(75(75thth percentile) for the rent data as follows: percentile) for the rent data as follows:

LL7575 = (75/100)70 + (1 – 75/100) = 52.5 + .25 = 52.75 = (75/100)70 + (1 – 75/100) = 52.5 + .25 = 52.75

The 3The 3rdrd quartile would be quartile would be

515 + .75(525 - 515) = 515 + 7.5 = 522.5515 + .75(525 - 515) = 515 + 7.5 = 522.5


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A B C D E

1Apart-ment

Monthly Rent ($) Third Quartile

2 1 525 =QUARTILE(B2:B71,3) 3 2 440 4 3 450 5 4 6156 5 480



It is not necessaryIt is not necessaryto put the datato put the data

in ascending order.in ascending order.


33rdrd quartilequartile

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A B C D E

1Apart-ment

Monthly Rent ($) Third Quartile

2 1 525 522.5 3 2 440 4 3 450 5 4 6156 5 480


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Excel’s Excel’s Rank and PercentileRank and Percentile Tool Tool

Step 1Step 1 Click the Click the DataData tab on the Ribbon tab on the Ribbon

Step 2Step 2 In the In the AnalysisAnalysis group, click group, click Data AnalysisData AnalysisStep 3Step 3 Choose Choose Rank and PercentileRank and Percentile from the list of from the list of Analysis ToolsAnalysis ToolsStep 4Step 4 When the Rank and Percentile dialog box appears When the Rank and Percentile dialog box appears (see details on next slide)(see details on next slide)

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Step 4Step 4 Complete the Rank and Percentile dialog Complete the Rank and Percentile dialog box as follows:box as follows:

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B C D E F G1 Rent Point Rent Rank Percent2 525 4 615 1 98.50%3 440 63 615 1 98.50%4 450 35 600 3 92.70%5 615 42 600 3 92.70%6 480 49 600 3 92.70%7 510 56 600 3 92.70%8 575 28 590 7 91.30%9 430 21 580 8 89.80%10 440 7 575 9 86.90%

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Geometric Mean (GM)

• The Geometric Mean is useful in finding the averages of increases in:– Percents– Ratios– Indexes– Growth Rates

• The Geometric Mean will always be less than or equal to (never more than) the arithmetic mean

• The GM gives a more conservative figure that is not drawn up by large values in the set

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Geometric Mean

• The GM of a set of n positive numbers is defined as the nth root of the product of n values. The formula is:

n nXXXXGM ))...()()((1% 321

1))...()()((% 321 n nXXXXGM

GM = Geometric MeanX 1 = A particular number (1 + %)X 2 = A particular number (1 + %)n = Number of postive numbers in set

Define Variables & Symbols

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Geometric Mean Example 1:Percentage Increase

09886.1)15.1)(05.1(2 GM

Starting Salary $41,000.00Increase in salary Year 1 5%Increase in salary Year 2 15%

1.05 * 1.15 = 1.2075GM = 1.2075 ^ (1/2) - 1 = 9.886%

In Excel:

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Raise 1 = $41,000.00 * 10% = $4,100.00Raise 2 = 45,100.00 * 10% = 4,510.00Total $8,610.00

If We used Arithmetic Mean (5%+15%)/2 = 10%

Verify Geometric Mean Example

Raise 1 = $41,000.00 * 5% = $2,050.00Raise 2 = 43,050.00 * 15% = 6,457.50Total $8,507.50

Verify 1:

Raise 1 = $41,000.00 * 0.09886 = $4,053.39Raise 2 = 45,053.39 * 0.09886 = 4,454.11Total $8,507.50

Verify 2:

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Another Use Of GM:Ave. % Increase Over Time

• Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another

• Where n = number of periods

1periods) theall of beginningat (Value

periods) theall of endat Value(nGM

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Example for GM: Ave. % Increase Over Time

• The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%.

0127.1000,755

000,8358 GM

•The annual rate of increase is 1.27%The annual rate of increase is 1.27%

•For the years 1992 through 2000, the rate of For the years 1992 through 2000, the rate of female enrollment growth at American colleges female enrollment growth at American colleges was 1.27% per yearwas 1.27% per year

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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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RangeRange

Range = largest value - smallest valueRange = largest value - smallest value

Range = 615 - 425 = 190Range = 615 - 425 = 190


425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615


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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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425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615


3rd Quartile (3rd Quartile (QQ3) = 5253) = 5251st Quartile (1st Quartile (QQ1) = 4451) = 445

Interquartile Range = Interquartile Range = QQ3 - 3 - QQ1 = 525 - 445 = 801 = 525 - 445 = 80



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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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54.74100 % 100 % 11.15%

490.80sx

54.74100 % 100 % 11.15%

490.80sx

22 ( )

2,996.161

ix xs

n

2

2 ( ) 2,996.16

1ix x

sn

2 2996.16 54.74s s 2 2996.16 54.74s s

the the standardstandard

deviation isdeviation isabout 11% about 11%

of the of the mean mean

• VarianceVariance

• Standard DeviationStandard Deviation

• Coefficient of VariationCoefficient of Variation

Sample Variance, Standard Deviation,Sample Variance, Standard Deviation,And Coefficient of VariationAnd Coefficient of Variation


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Using Excel to Compute the Sample Using Excel to Compute the Sample Variance, Standard Deviation, and Variance, Standard Deviation, and

Coefficient of VariationCoefficient of Variation Formula WorksheetFormula Worksheet


A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean =AVERAGE(B2:B71)3 2 440 Median =MEDIAN(B2:B71)4 3 450 Mode =MODE(B2:B71)5 4 615 Variance =VAR(B2:B71)6 5 480 Std. Dev. =STDEV(B2:B71)7 6 510 C.V. =E6/E2*100

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Value WorksheetValue Worksheet

Using Excel to Compute the Sample Using Excel to Compute the Sample Variance, Standard Deviation, and Variance, Standard Deviation, and

Coefficient of VariationCoefficient of Variation

A B C D E

1Apart-ment

Monthly Rent ($)

2 1 525 Mean 490.803 2 440 Median 475.004 3 450 Mode 450.005 4 615 Variance 2996.166 5 480 Std. Dev. 54.747 6 510 C.V. 11.15


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Using Excel’sUsing Excel’sDescriptive Statistics ToolDescriptive Statistics Tool

Step 1Step 1 Click the Click the DataData tab on the Ribbon tab on the Ribbon

Step 2Step 2 In the In the AnalysisAnalysis group, click group, click Data AnalysisData AnalysisStep 3Step 3 Choose Choose Descriptive StatisticsDescriptive Statistics from the list of from the list of

Analysis ToolsAnalysis Tools

Step 4Step 4 When the Descriptive Statistics dialog box appears: When the Descriptive Statistics dialog box appears: (see details on next slide)(see details on next slide)

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Excel’s Excel’s Descriptive StatisticsDescriptive Statistics Dialog Box Dialog Box

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Excel Value Worksheet (Partial)Excel Value Worksheet (Partial)



A B C D E

1Apart-ment

Monthly Rent ($) Monthly Rent ($)

2 1 5253 2 440 Mean 490.84 3 450 Standard Error 6.5423481145 4 615 Median 4756 5 480 Mode 4507 6 510 Standard Deviation 54.737211468 7 575 Sample Variance 2996.162319

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Excel Value Worksheet (Partial)Excel Value Worksheet (Partial)


Note: Rows 1-8 and 17-71 are not shown.Note: Rows 1-8 and 17-71 are not shown.

A B C D E9 8 430 Kurtosis -0.33409329810 9 440 Skewness 0.92433047311 10 450 Range 19012 11 470 Minimum 42513 12 485 Maximum 61514 13 515 Sum 3435615 14 575 Count 7016 15 430

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End of Chapter 3, Part AEnd of Chapter 3, Part A

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Chapter 3, Part BChapter 3, Part B Descriptive Statistics: Numerical Descriptive Statistics: Numerical

MeasuresMeasures Measures of Distribution Shape, Relative Measures of Distribution Shape, Relative

Location, and Detecting OutliersLocation, and Detecting Outliers Exploratory Data AnalysisExploratory Data Analysis

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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 Chebyshev’s

TheoremTheorem 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 Excel’s SKEW function can be used to compute thethe

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

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Symmetric (not skewed)Symmetric (not skewed)

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Skewness = Skewness = 0 0

• Skewness is zero.Skewness is zero.

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

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Moderately Skewed LeftModerately Skewed Left

Skewness = Skewness = .31 .31

• Skewness is negative.Skewness is negative.

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

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Moderately Skewed RightModerately Skewed Right

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Skewness = .31 Skewness = .31

• Skewness is positive.Skewness is positive.

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

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Highly Skewed RightHighly Skewed RightR

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Skewness = 1.25 Skewness = 1.25

• Skewness is positive (often above 1.0).Skewness is positive (often above 1.0).

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

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Seventy efficiency apartments were Seventy efficiency apartments were randomlyrandomly

sampled in a college town. The monthly rent sampled in a college town. The monthly rent pricesprices

for the apartments are listed below in for the apartments are listed below in ascending order. ascending order.



425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Skewness = .92 Skewness = .92



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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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• z-Score of Smallest Value (425)z-Score of Smallest Value (425)

425 490.80 1.20

54.74ix x

zs

425 490.80

1.2054.74

ix xz

s

z-Scoresz-Scores

Standardized Values for Apartment RentsStandardized Values for Apartment Rents-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27


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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






89%89%89%89%






94%94%94%94%


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Let Let zz = 1.5 with = 490.80 and = 1.5 with = 490.80 and ss = 54.74 = 54.74xx

At least (1 At least (1 1/(1.5) 1/(1.5)22) = 1 ) = 1 0.44 = 0.56 or 56% 0.44 = 0.56 or 56%

of the rent values must be betweenof the rent values must be between

xx - - zz((ss) = 490.80 ) = 490.80 1.5(54.74) = 409 1.5(54.74) = 409

andandxx + + zz((ss) = 490.80 + 1.5(54.74) = 573) = 490.80 + 1.5(54.74) = 573

(Actually, 86% of the rent values(Actually, 86% of the rent values are between 409 and 573.)are between 409 and 573.)


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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

0000

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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Detecting OutliersDetecting Outliers

• The most extreme z-scores are -1.20 and 2.27The most extreme z-scores are -1.20 and 2.27

• Using |Using |zz| | >> 3 as the criterion for an outlier, there 3 as the criterion for an outlier, there are no outliers in this data set.are no outliers in this data set.

-1.20 -1.11 -1.11 -1.02 -1.02 -1.02 -1.02 -1.02 -0.93 -0.93-0.93 -0.93 -0.93 -0.84 -0.84 -0.84 -0.84 -0.84 -0.75 -0.75-0.75 -0.75 -0.75 -0.75 -0.75 -0.56 -0.56 -0.56 -0.47 -0.47-0.47 -0.38 -0.38 -0.34 -0.29 -0.29 -0.29 -0.20 -0.20 -0.20-0.20 -0.11 -0.01 -0.01 -0.01 0.17 0.17 0.17 0.17 0.350.35 0.44 0.62 0.62 0.62 0.81 1.06 1.08 1.45 1.451.54 1.54 1.63 1.81 1.99 1.99 1.99 1.99 2.27 2.27

Standardized Values for Apartment RentsStandardized Values for Apartment Rents


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Exploratory Data AnalysisExploratory Data Analysis

Five-Number SummaryFive-Number Summary Box PlotBox Plot

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Five-Number SummaryFive-Number Summary

1111 Smallest ValueSmallest Value Smallest ValueSmallest Value

First QuartileFirst Quartile First QuartileFirst Quartile

MedianMedian MedianMedian

Third QuartileThird Quartile Third QuartileThird Quartile

Largest ValueLargest Value Largest ValueLargest Value

2222

3333

4444

5555

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Five-Number SummaryFive-Number Summary

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

425 430 430 435 435 435 435 435 440 440440 440 440 445 445 445 445 445 450 450450 450 450 450 450 460 460 460 465 465465 470 470 472 475 475 475 480 480 480480 485 490 490 490 500 500 500 500 510510 515 525 525 525 535 549 550 570 570575 575 580 590 600 600 600 600 615 615

Lowest Value = 425Lowest Value = 425 First Quartile = 445First Quartile = 445

Median = 475Median = 475

Third Quartile = 525Third Quartile = 525Largest Value = 615Largest Value = 615


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400400

425425

450450

475475

500500

525525

550550

575575

600600

625625

• A box is drawn with its ends located at the first andA box is drawn with its ends located at the first and third quartiles.third quartiles.

Box PlotBox Plot

• A vertical line is drawn in the box at the location ofA vertical line is drawn in the box at the location of the median (second quartile).the median (second quartile).

Q1 = 445Q1 = 445 Q3 = 525Q3 = 525

Q2 = 475Q2 = 475


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Box PlotBox Plot

Limits are located (not drawn) using the Limits are located (not drawn) using the interquartile range (IQR).interquartile range (IQR).

Data outside these limits are considered Data outside these limits are considered outliersoutliers.. The locations of each outlier is shown with the The locations of each outlier is shown with the

symbolsymbol * * ..continuedcontinued

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Box PlotBox Plot

Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(80) = 325Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(80) = 325

Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(80) = 645Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(80) = 645

• The lower limit is located 1.5(IQR) below The lower limit is located 1.5(IQR) below QQ1.1.

• The upper limit is located 1.5(IQR) above The upper limit is located 1.5(IQR) above QQ3.3.

• There are no outliers (values less than 325 orThere are no outliers (values less than 325 or greater than 645) in the apartment rent data.greater than 645) in the apartment rent data.


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Box PlotBox Plot

• Whiskers (dashed lines) are drawn from the endsWhiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data valuesof the box to the smallest and largest data values inside the limits.inside the limits.

400400

425425

450450

475475

500500

525525

550550

575575

600600

625625

Smallest valueSmallest valueinside limits = 425inside limits = 425

Largest valueLargest valueinside limits = 615inside limits = 615


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End of Chapter 3, Part BEnd of Chapter 3, Part B

Slides by JOHN LOUCKS St. Edward’s University

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