Chap 004

A PowerPoint Presentation A PowerPoint Presentation Package to AccompanyPackage to Accompany

Applied Statistics in Applied Statistics in Business & Economics, Business & Economics,

33rd rd edition edition David P. Doane and Lori E. SewardDavid P. Doane and Lori E. Seward

Prepared by Lloyd R. Jaisingh Prepared by Lloyd R. Jaisingh

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

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Descriptive Statistics Descriptive Statistics

Chapter ContentsChapter Contents

4.1 Numerical Description4.1 Numerical Description

4.2 Central Tendency4.2 Central Tendency

4.3 Dispersion4.3 Dispersion

4.4 Standardized Data4.4 Standardized Data

4.5 Percentiles, Quartiles, and Box Plots4.5 Percentiles, Quartiles, and Box Plots

4.6 Correlation and Covariance4.6 Correlation and Covariance

4.7 Grouped Data4.7 Grouped Data

4.8 Skewness and Kurtosis4.8 Skewness and Kurtosis

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Chapter Learning Objectives (LO’s)Chapter Learning Objectives (LO’s)

LO1:LO1: Explain the concepts of central tendency, dispersion, and shape.Explain the concepts of central tendency, dispersion, and shape.

LO2:LO2: Use Excel to obtain descriptive statistics and visual displays.Use Excel to obtain descriptive statistics and visual displays.

LO3:LO3: Calculate and interpret common measures of central tendency.Calculate and interpret common measures of central tendency.

LO4:LO4: Calculate and interpret common measures of dispersion.Calculate and interpret common measures of dispersion.

LO5: LO5: Transform a data set into standardized values.Transform a data set into standardized values.

LO6:LO6: Apply the Empirical Rule and recognize outliers.Apply the Empirical Rule and recognize outliers.

LO7: LO7: Calculate quartiles and other percentiles.Calculate quartiles and other percentiles.

LO8:LO8: Make and interpret box plots.Make and interpret box plots.

LO9:LO9: Calculate and interpret a correlation coefficient and covariance.Calculate and interpret a correlation coefficient and covariance.

LO10:LO10: Calculate the mean and standard deviation from grouped data.Calculate the mean and standard deviation from grouped data.

LO11:LO11: Explain the concepts of skewness and kurtosis.Explain the concepts of skewness and kurtosis.

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Descriptive Statistics Descriptive Statistics

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4.1 Numerical Description4.1 Numerical Description

LO1:LO1: Explain the concepts of central tendency, dispersion, and shape.Explain the concepts of central tendency, dispersion, and shape.

Three key characteristics of numerical data:Three key characteristics of numerical data:

LO1LO1

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4.1 Numerical Description4.1 Numerical Description

LO2:LO2: Use Excel to obtain descriptive statistics and visual displays.Use Excel to obtain descriptive statistics and visual displays.

LO2LO2

EXCEL Displays for Tables 4.2 and 4.3EXCEL Displays for Tables 4.2 and 4.3

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4.2 Central Tendency4.2 Central TendencyLO3LO3

LO3:LO3: Calculate and interpret common measures of central tendency.Calculate and interpret common measures of central tendency.

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• Compare mean and median or look at histogram to determine degree of skewness.Compare mean and median or look at histogram to determine degree of skewness.

ShapeShape

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4.2 Central Tendency4.2 Central TendencyLO1LO1

LO1:LO1: Explain the concepts of central tendency, dispersion, and shape.Explain the concepts of central tendency, dispersion, and shape.

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• VariationVariation is the “spread” of data points about the center of the distribution in a sample. is the “spread” of data points about the center of the distribution in a sample. Consider the following measures of dispersion:Consider the following measures of dispersion:

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4.3 Dispersion4.3 DispersionLO4LO4

LO4: LO4: Calculate and interpret common measures of dispersion.Calculate and interpret common measures of dispersion.

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• For any population with mean and standard deviation , the percentage of observations that lie within k standard deviations of the mean must be at least 100[1 – 1/k2].

• For k = 2 standard deviations, 100[1 – 1/22] = 75%. So, at least 75.0% will lie within + 2• For k = 3 standard deviations,

100[1 – 1/32] = 88.9%• So, at least 88.9% will lie within + 3• Although applicable to any data set, these limits tend to be too wide to be useful.

Chebyshev’s TheoremChebyshev’s Theorem

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4.4 Standardized Data4.4 Standardized DataLO5LO5

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LO6:LO6: Apply the Empirical Rule and recognize outliersApply the Empirical Rule and recognize outliers

The Empirical RuleThe Empirical Rule

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4.4 Standardized Data4.4 Standardized DataLO6LO6

UnusualUnusual observations observations

are those that lie are those that lie

beyond beyond ++ 2 2..

OutliersOutliers are are

observations observations

that lie beyond that lie beyond ++ 3 3..

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• A standardized variablestandardized variable (Z) redefines each observation in terms the number of standard deviations from the mean.

iix

z

Standardization formula for a population:

Standardization formula for a sample:

iix x

zs

Defining a Standardized VariableDefining a Standardized Variable

A negative A negative zz

value means thevalue means the

observation isobservation is

below the mean.below the mean.

Positive Positive zz means means

the observation is the observation is

above the mean. above the mean.

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4.4 Standardized Data4.4 Standardized Data

LO5:LO5: Transform a data set into standardized values.Transform a data set into standardized values.

LO5LO5

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• PercentilesPercentiles are data that have been divided into 100 groups.

• For example, you score in the 83For example, you score in the 83rdrd percentile on a standardized test. That means percentile on a standardized test. That means

that 83% of the test-takers scored below youthat 83% of the test-takers scored below you..• DecilesDeciles are data that have been divided into 10 groups. are data that have been divided into 10 groups.• QuintilesQuintiles are data that have been divided into 5 groups. are data that have been divided into 5 groups.• QuartilesQuartiles are data that have been divided into 4 groups. are data that have been divided into 4 groups.

PercentilesPercentiles

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4.5 Percentiles, Quartiles, and Box-Plots4.5 Percentiles, Quartiles, and Box-PlotsLO7LO7

LO7:LO7: Calculate quartiles and other percentilesCalculate quartiles and other percentiles

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• A useful tool of exploratory data analysisexploratory data analysis (EDA).• Also called a box-and-whisker plotbox-and-whisker plot..

• Based on a five-number summaryfive-number summary: : Xmin, Q1, Q2, Q3, Xmax

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4.5 Percentiles, Quartiles, and Box-Plots4.5 Percentiles, Quartiles, and Box-PlotsLO8LO8

LO8:LO8: Make and interpret box plots.Make and interpret box plots.

• A box plot shows central tendencycentral tendency, dispersiondispersion, and shapeshape..

Fences and Unusual Data ValuesFences and Unusual Data Values

Values outside the inner fences are unusualunusual while those outside the outer fences are outliersoutliers

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• The average of the first and third quartiles.The average of the first and third quartiles.

Midhinge = Midhinge = 1 3

2

Q Q

• The name “midhingemidhinge” derives from the idea that, if the “box” were folded in half, it

would resemble a “hinge”.

Box Plots: Box Plots: MidhingeMidhinge

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4.5 Percentiles, Quartiles, and Box-Plots4.5 Percentiles, Quartiles, and Box-PlotsLO8LO8

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• The The sample correlation coefficient r sample correlation coefficient r is a statistic that describes the degree of linearity is a statistic that describes the degree of linearity between paired observations on two quantitative variables X and Y. Note: -1 ≤ r ≤ +1.between paired observations on two quantitative variables X and Y. Note: -1 ≤ r ≤ +1.

Correlation CoefficientCorrelation Coefficient

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4.6 Correlation and Covariance4.6 Correlation and CovarianceLO9LO9

LO9:LO9: Calculate and interpret a correlation coefficient and covariance.Calculate and interpret a correlation coefficient and covariance.

The covariance of two random variables X and Y (denoted σXY ) measures the degree to which the values of X and Y change together.

Population Sample

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A correlation coefficient is the covariance divided by the product of the standard deviations of X and Y.

CovarianceCovariance

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4.6 Correlation and Covariance4.6 Correlation and CovarianceLO9LO9

LO9:LO9: Calculate and interpret a correlation coefficient and covariance.Calculate and interpret a correlation coefficient and covariance.

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Group Mean and Standard DeviationGroup Mean and Standard Deviation

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4.7 Grouped Data4.7 Grouped DataLO10LO10

LO10:LO10: Calculate the mean and standard deviation from grouped data.Calculate the mean and standard deviation from grouped data.

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SkewnessSkewness

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4.8 Skewness and Kurtosis4.8 Skewness and KurtosisLO11LO11

LO11:LO11: Explain the concepts of skewness and kurtosis.Explain the concepts of skewness and kurtosis.

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KurtosisKurtosis

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4.8 Skewness and Kurtosis4.8 Skewness and KurtosisLO11LO11

LO11:LO11: Explain the concepts of skewness and kurtosis.Explain the concepts of skewness and kurtosis.