McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

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Descriptive Statistics Descriptive Statistics

Chapter ContentsChapter Contents

4.1 Numerical Description

4.2 Measures of Center

4.3 Measures of Variability

4.4 Standardized Data

4.5 Percentiles, Quartiles, and Box Plots

4.6 Correlation and Covariance

4.7 Grouped Data

4.8 Skewness and Kurtosis

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Chapter Learning Objectives Chapter Learning Objectives

LO4-1:LO4-1: Explain the concepts of center, variability, and shape.Explain the concepts of center, variability, and shape.

LO4-2:LO4-2: Use Excel to obtain descriptive statistics and visual displays.Use Excel to obtain descriptive statistics and visual displays.

LO4-3:LO4-3: Calculate and interpret common measures of center.Calculate and interpret common measures of center.

LO4-4:LO4-4: Calculate and interpret common measures of variability.Calculate and interpret common measures of variability.

LO4-5: LO4-5: Transform a data set into standardized values.Transform a data set into standardized values.

LO4-6:LO4-6: Apply the Empirical Rule and recognize outliers.Apply the Empirical Rule and recognize outliers.

LO4-7:LO4-7: Calculate quartiles and other percentiles.Calculate quartiles and other percentiles.

LO4-8:LO4-8: Make and interpret box plots.Make and interpret box plots.

LO4-9:LO4-9: Calculate and interpret a correlation coefficient and covariance.Calculate and interpret a correlation coefficient and covariance.

LO4-10:LO4-10: Calculate the mean and standard deviation from grouped data.Calculate the mean and standard deviation from grouped data.

LO4-11:LO4-11: Assess skewness and kurtosis in a sampleAssess skewness and kurtosis in a sample

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Descriptive Statistics Descriptive Statistics

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4.1 Numerical Description4.1 Numerical Description

LO4-1: LO4-1: Explain the concepts of center, variability, and shape.Explain the concepts of center, variability, and shape.

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

LO4-1LO4-1

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LO4-2: LO4-2: Use Excel to obtain descriptive statistics and visual displays.Use Excel to obtain descriptive statistics and visual displays.

EXCEL Displays for Table 4.3EXCEL Displays for Table 4.3

LO4-2LO4-2 4.1 Numerical Description4.1 Numerical Description

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4.2 Measures of Center4.2 Measures of Center

LO4-3: LO4-3: Calculate and interpret common measures of center.Calculate and interpret common measures of center.

LO4-3LO4-3

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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 Measures of Center4.2 Measures of CenterLO4-1LO4-1

LO4-1: LO4-1: Explain the concepts of center, variability, and shape.Explain the concepts of center, variability, 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 variability:Consider the following measures of variability:

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4.3 Measures of Variability4.3 Measures of Variability

LO4-4: LO4-4: Calculate and interpret common measures of variability.Calculate and interpret common measures of variability.

LO4-4LO4-4

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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 Data

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LO4-6: LO4-6: Apply the Empirical Rule and recognize outliers.Apply the Empirical Rule and recognize outliers.

The Empirical RuleThe Empirical Rule

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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..

4.4 Standardized Data4.4 Standardized DataLO4-6LO4-6

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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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LO4-5LO4-5 4.4 Standardized Data4.4 Standardized Data

LO4-5: LO4-5: Transform a data set into standardized values.Transform a data set into standardized values.

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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-Plots

LO4-7: LO4-7: Calculate quartiles and other percentiles.Calculate quartiles and other percentiles.

LO4-7LO4-7

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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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• 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

4.5 Percentiles, Quartiles, and Box-Plots4.5 Percentiles, Quartiles, and Box-PlotsLO4-8LO4-8

LO4-8: LO4-8: Make and interpret box plots.Make and interpret box plots.

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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 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

LO4-9LO4-9

LO4-9: LO4-9: Calculate and interpret a correlation coefficient and covariance.Calculate and interpret a correlation coefficient and covariance.

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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 CovarianceLO4-9LO4-9

LO4-9: LO4-9: 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 DataLO4-10LO4-10

LO4-10: LO4-10: 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 KurtosisLO4-11LO4-11

LO4-11: LO4-11: Assess skewness and kurtosis in a sample.Assess skewness and kurtosis in a sample.

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KurtosisKurtosis

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LO4-11LO4-11 4.8 Skewness and Kurtosis4.8 Skewness and Kurtosis

LO4-11: LO4-11: Assess skewness and kurtosis in a sample.Assess skewness and kurtosis in a sample.