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Statistical MethodsStatistical MethodsStatistical MethodsStatistical Methods

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

StatisticalMethods

DescriptiveStatistics

InferentialStatistics

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Descriptive StatisticsDescriptive StatisticsDescriptive StatisticsDescriptive Statistics

InvolvesInvolves Collecting dataCollecting data Presenting dataPresenting data Characterizing Characterizing

datadata

PurposePurpose Describe dataDescribe data

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Inferential StatisticsInferential StatisticsInferential StatisticsInferential Statistics

InvolvesInvolves EstimationEstimation Hypothesis Hypothesis

testingtesting

PurposePurpose Make decisions Make decisions

about population about population characteristicscharacteristics

Population?Population?

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Population (universe)Population (universe) All items of interestAll items of interest

SampleSample Portion of populationPortion of population

ParameterParameter Summary measure about populationSummary measure about population

StatisticStatistic Summary measure about sampleSummary measure about sample

Key TermsKey TermsKey TermsKey Terms

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Data TypesData TypesData TypesData Types

Data

Numerical(Quantitative)

Categorical(Qualitative)

Discrete Continuous

Data

Numerical(Quantitative)

Categorical(Qualitative)

Discrete Continuous

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Data Type ExamplesData Type ExamplesData Type ExamplesData Type Examples

NumericalNumerical DiscreteDiscrete

To how many magazines do you subscribe To how many magazines do you subscribe currently? ___ (Number)currently? ___ (Number)

Continuous Continuous How tall are you? ___ (Inches)How tall are you? ___ (Inches)

CategoricalCategorical Do you own savings bonds? __ Yes __ NoDo you own savings bonds? __ Yes __ No

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How Are Data How Are Data Measured?Measured?

How Are Data How Are Data Measured?Measured?

Nominal scaleNominal scale

Ordinal scaleOrdinal scale

Interval scaleInterval scale

Ratio scaleRatio scale

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Standard NotationStandard NotationStandard NotationStandard Notation

Measure Sample Population

Mean X

Stand. Dev. S

Variance S2 2

Size n N

Measure Sample Population

Mean X

Stand. Dev. S

Variance S2 2

Size n N

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Numerical Data Numerical Data PropertiesProperties

Numerical Data Numerical Data PropertiesProperties

Central Tendency Central Tendency (Location)(Location)

Variation Variation (Dispersion)(Dispersion)

ShapeShape

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Numerical DataNumerical DataProperties & MeasuresProperties & Measures

Numerical DataNumerical DataProperties & MeasuresProperties & Measures

Numerical DataProperties

Mean

Median

Mode

Midrange

Midhinge

CentralTendency

Range

Interquartile RangeVariance

Standard Deviation

Coeff. of Variation

Variation

Skew

Kurtosis

Shape

Numerical DataProperties

Mean

Median

Mode

Midrange

Midhinge

CentralTendency

Range

Interquartile RangeVariance

Standard Deviation

Coeff. of Variation

Variation

Skew

Kurtosis

Shape

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Central TendencyCentral TendencyCentral TendencyCentral Tendency

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MeanMeanMeanMean

Measure of central tendencyMeasure of central tendency Most common measureMost common measure Acts as ‘balance point’Acts as ‘balance point’ Affected by extreme values (‘outliers’)Affected by extreme values (‘outliers’) Formula (sample mean):Formula (sample mean):

X

X

nX X X

n

ii

n

n

1 1 2

X

X

nX X X

n

ii

n

n

1 1 2

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MedianMedianMedianMedian

Measure of central tendencyMeasure of central tendency Middle value in ordered sequenceMiddle value in ordered sequence

If odd n, middle value of sequenceIf odd n, middle value of sequence If even n, average of 2 middle valuesIf even n, average of 2 middle values

Not affected by extreme values Not affected by extreme values Position of median in sequencePosition of median in sequence

Positioning Point n 12

Positioning Point n 12

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ModeModeModeMode

Measure of central tendencyMeasure of central tendency Value that occurs most oftenValue that occurs most often Not affected by extreme valuesNot affected by extreme values May be no mode or several modesMay be no mode or several modes May be used for numerical & May be used for numerical &

categorical datacategorical data

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QuartilesQuartilesQuartilesQuartiles

Measure of Measure of noncentralnoncentral tendency tendency Split ordered data into 4 quartersSplit ordered data into 4 quarters

25%25% 25%25% 25%25% 25%25%

QQ11 QQ22 QQ33

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Summary of Summary of Central TendencyCentral Tendency

Summary of Summary of Central TendencyCentral Tendency

Measure Equation DescriptionMean Xi / n Balance pointMedian (n+1) Position

2Middle value whenordered

Mode none Most frequentMidrange X Xsmallest l est arg

2

Middle of smallest& largest

Midhinge Q Q1 3

2 Middle of 1st &

3rd quartile whereQi = i (n+1)/4

Measure Equation DescriptionMean Xi / n Balance pointMedian (n+1) Position

2Middle value whenordered

Mode none Most frequentMidrange X Xsmallest l est arg

2

Middle of smallest& largest

Midhinge Q Q1 3

2 Middle of 1st &

3rd quartile whereQi = i (n+1)/4

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VariationVariationVariationVariation

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7 8 9 10 7 8 9 107 8 9 10 7 8 9 10

RangeRangeRangeRange

Measure of dispersionMeasure of dispersion Difference between largest & Difference between largest &

smallest observationssmallest observations

Ignores how data are distributedIgnores how data are distributed

Range X Xl est smallestargRange X Xl est smallestarg

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Interquartile RangeInterquartile RangeInterquartile RangeInterquartile Range

Measure of dispersionMeasure of dispersion Also called midspreadAlso called midspread Spread in middle 50%Spread in middle 50% Not affected by extreme valuesNot affected by extreme values Formula:Formula:

Interquartile Range Q Q3 1Interquartile Range Q Q3 1

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4 6 8 10 12

X = 8.3

4 6 8 10 12

X = 8.3

Variance & Variance & Standard DeviationStandard Deviation

Variance & Variance & Standard DeviationStandard Deviation

Measures of dispersionMeasures of dispersion Most common measuresMost common measures Consider how data are distributedConsider how data are distributed Show variation about mean (Show variation about mean (XX or or ))

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DISPERSION ABOUT DISPERSION ABOUT THE MEANTHE MEAN

•VarianceVariance

ss22== X-X

n -1

2

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DISPERSION ABOUT DISPERSION ABOUT MEAN (cont.)MEAN (cont.)

•Standard deviation (sd)Standard deviation (sd)

X-X

n -1

2

s =s =

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PROPERTIES OF THE PROPERTIES OF THE STANDARD DEVIATION IN STANDARD DEVIATION IN

A NORMAL CURVEA NORMAL CURVE

X

X

X

1s1s 68%68%

2s2s 95%95%

3s3s 99%99%

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Summary of Summary of Variation Measures Variation Measures

Summary of Summary of Variation Measures Variation Measures

Measure Equation Description

Range X largest - Xsmallest Total spread

Interquartile range Q3 - Q1 Spread of middle 50%

Standard deviation(Sample)

X X

ni

2

1

Dispersion aboutsample mean

Standard deviation(Population)

X

N

i

2 Dispersion aboutpopulation mean

Variance(Sample)

(X i -X)2

n - 1Squared dispersionabout sample mean

Coeff. of variation (S /X)100% Relative variation

Measure Equation Description

Range X largest - Xsmallest Total spread

Interquartile range Q3 - Q1 Spread of middle 50%

Standard deviation(Sample)

X X

ni

2

1

Dispersion aboutsample mean

Standard deviation(Population)

X

N

i

2 Dispersion aboutpopulation mean

Variance(Sample)

(X i -X)2

n - 1Squared dispersionabout sample mean

Coeff. of variation (S /X)100% Relative variation

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ShapeShapeShapeShape

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ShapeShapeShapeShape

Describes how data are distributedDescribes how data are distributed Measures of shapeMeasures of shape

Skew: SymmetrySkew: Symmetry

Right-SkewedLeft-Skewed Symmetric

Mean = Median = ModeMean Median Mode Mode Median Mean

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Median

4 6 8 10 12

Q 3Q 1 X largestX smallestMedian

4 6 8 10 12

Q 3Q 1 X largestX smallest

Box-and-Whisker PlotBox-and-Whisker PlotBox-and-Whisker PlotBox-and-Whisker Plot

Graphical display of data usingGraphical display of data using5-number summary5-number summary

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Right-SkewedLeft-Skewed Symmetric

Q 1 Median Q 3Q 1 Median Q 3 Q 1 Median Q 3

Shape & Shape & Box-and-Whisker PlotBox-and-Whisker Plot

Shape & Shape & Box-and-Whisker PlotBox-and-Whisker Plot

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ConclusionConclusionConclusionConclusion

Explained numerical data propertiesExplained numerical data properties Described summary measuresDescribed summary measures

Central tendencyCentral tendency VariationVariation ShapeShape

Analyzed numerical data using Analyzed numerical data using summary measures summary measures

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This Class...This Class...This Class...This Class...

What was the most important thing What was the most important thing you learned in class today?you learned in class today?

What do you still have questions What do you still have questions about?about?

How can today’s class be improved?How can today’s class be improved?

Please take a moment to answer the Please take a moment to answer the following questions in writing:following questions in writing: