business statistics

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© 2003 Thomson/South-Western© 2003 Thomson/South-Western

Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

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

MethodsMethodsPart APart A

Measures of LocationMeasures of Location Measures of VariabilityMeasures of Variability

xx %%

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Measures of LocationMeasures of Location

MeanMean MedianMedian ModeMode PercentilesPercentiles QuartilesQuartiles

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Example: Apartment RentsExample: Apartment Rents

Given below is a sample of monthly rent Given below is a sample of monthly rent values ($)values ($)

for one-bedroom apartments. The data is a for one-bedroom apartments. The data is a sample of 70sample of 70

apartments in a particular city. The data are apartments in a particular city. The data are presentedpresented

in ascending order. 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

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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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.

If the data are from a sample, the mean is If the data are from a sample, the mean is denoted by denoted by

..

If the data are from a population, the mean is If the data are from a population, the mean is denoted by denoted by (mu). (mu).

xxnixxni

xNi x

Ni

xx

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MeanMean

xxni

34 35670

490 80,

.xxni

34 35670

490 80,

.

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

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MedianMedian

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

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

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MedianMedian

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

For an odd number of observations, the For an odd number of observations, the median is the middle value.median is the middle value.

For an even number of observations, the For an even number of observations, the median is the average of the two middle median is the average of the two middle values.values.

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MedianMedian

Median = 50th percentileMedian = 50th percentile

i i = (= (pp/100)/100)nn = (50/100)70 = 35.5 = (50/100)70 = 35.5 Averaging the 35th and Averaging the 35th and

36th data values:36th data values:

Median = (475 + 475)/2 = 475Median = (475 + 475)/2 = 475425 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

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ModeMode

The The modemode of a data set is the value that occurs of a data set is the value that occurs with greatest frequency.with greatest frequency.

The greatest frequency can occur at two or The greatest frequency can occur at two or more different values.more different values.

If the data have exactly two modes, the data If the data have exactly two modes, the data are are bimodalbimodal..

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

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ModeMode

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

Mode = 450Mode = 450425 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

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PercentilesPercentiles

A percentile provides information about how A percentile provides information about how the data are spread over the interval from the the data are spread over the interval from the smallest value to the largest value.smallest value to the largest value.

Admission test scores for colleges and Admission test scores for colleges and universities are frequently reported in terms of universities are frequently reported in terms of percentiles.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.

• 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 th percentile.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 th percentile is the value in the percentile 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 th percentile is the average of the values in positionsaverage of the values in positions i i and and ii +1.+1.

PercentilesPercentiles

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90th Percentile90th Percentile

ii = ( = (pp/100)/100)nn = (90/100)70 = 63 = (90/100)70 = 63

Averaging the 63rd and 64th data Averaging the 63rd and 64th data values:values:

90th Percentile = (580 + 590)/2 = 90th Percentile = (580 + 590)/2 = 585585425 430 430 435 435 435 435 435 440 440

440 440 440 445 445 445 445 445 450 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

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QuartilesQuartiles

Quartiles are specific percentilesQuartiles 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 = = (75/100)70 = 52.5 = 5353

Third quartile = 525Third quartile = 525425 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

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Measures of VariabilityMeasures of Variability

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

For example, in choosing supplier A or supplier For example, in choosing supplier A or supplier B we might consider not only the average B we might consider not only the average delivery time for each, but also the variability delivery time for each, but also the variability in delivery time for each. 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 of a data set is the difference between the largest and smallest data values.between the 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 to the smallest and largest

data values.data values.

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RangeRange

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

Range = 615 - 425 = 190Range = 615 - 425 = 190425 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

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Interquartile RangeInterquartile Range

The The interquartile rangeinterquartile range of a data set is the of a data set is the difference between the third quartile and the first difference between the third quartile and the first quartile.quartile.

It is the range for the It is the range for the middle 50%middle 50% of the data. of the data. It It overcomes the sensitivityovercomes the sensitivity to extreme data to extreme data

values.values.

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Interquartile RangeInterquartile Range

3rd Quartile (3rd Quartile (QQ3) = 5253) = 525

1st Quartile (1st Quartile (QQ1) = 4451) = 445

Interquartile Range = Interquartile Range = QQ3 - 3 - QQ1 = 525 - 445 = 1 = 525 - 445 = 8080

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

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VarianceVariance

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

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

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VarianceVariance

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

If the data set is a sample, the variance is If the data set is a sample, the variance is denoted by denoted by ss22. .

If the data set is a population, the variance is If the data set is a population, the variance is denoted by denoted by 22..

sxi x

n2

2

1

( )s

xi x

n2

2

1

( )

22

( )xNi 2

2

( )xNi

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Standard DeviationStandard Deviation

The The standard deviationstandard deviation of a data set is the of a data set is the positive square root of the variance.positive 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 it more easily comparable, than the making it more easily comparable, than the variance, to the mean.variance, to the mean.

If the data set is a sample, the standard If the data set is a sample, the standard deviation is denoted deviation is denoted ss..

If the data set is a population, the standard If the data set is a population, the standard deviation is denoted deviation is denoted (sigma). (sigma).

s s 2s s 2

2 2

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Coefficient of VariationCoefficient of Variation

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

If the data set is a sample, the coefficient of If the data set is a sample, the coefficient of variation is computed as follows:variation is computed as follows:

If the data set is a population, the coefficient If the data set is a population, the coefficient of variation is computed as follows:of variation is computed as follows:

sx

( )100sx

( )100

( )100

( )100

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VarianceVariance

Standard DeviationStandard Deviation

Coefficient of VariationCoefficient of Variation

sxi x

n2

2

12 996 16

( ), .s

xi x

n2

2

12 996 16

( ), .

s s 2 2996 47 54 74. .s s 2 2996 47 54 74. .

sx

10054 74490 80

100 11 15..

.sx

10054 74490 80

100 11 15..

.

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

business statistics

Technology

data set

data items

data arebimodal

data aremultimodal

property value data

largest data values

smallest data values

apartment rents range