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1、Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-1Chapter 3Numerical Descriptive MeasuresBusiness Statistics:A First CourseFifth EditionChoice is yours, part 2Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-3In this chapter, you learn: nTo describe the pr
2、operties of central tendency, variation, and shape in numerical datanTo calculate descriptive summary measures for a populationnTo construct and interpret a boxplotnTo calculate the covariance and the coefficient of correlationLearning ObjectivesBusiness Statistics: A First Course, 5e 2009 Prentice-
3、Hall, Inc.Chap 3-4Summary DefinitionsThe central tendency is the extent to which all the data values group around a typical or central value.The variation is the amount of dispersion, or scattering, of values The shape is the pattern of the distribution of values from the lowest value to the highest
4、 value.Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-5Measures of Central Tendency:The MeannThe arithmetic mean (often just called “mean”) is the most common measure of central tendencynFor a sample of size n:Sample sizeObserved valuesThe ith valuePronounced x-barMeasures of
5、 Central Tendency:The MeannExample volume of Coke Listed below are the volumes (in ounces) of the Coke in five different cans. Find the mean for this sample.12.3 12.1 12.2 12.3 12.2Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-7Measures of Central Tendency:The MeannThe most
6、common measure of central tendencynMean = sum of values divided by the number of valuesnAffected by extreme values (outliers)(continued)0 1 2 3 4 5 6 7 8 9 10Mean = 3 0 1 2 3 4 5 6 7 8 9 10Mean = 4Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-8Measures of Central Tendency:Lo
7、cating the MediannThe location of the median when the values are in numerical order (smallest to largest):nIf the number of values is odd, the median is the middle numberBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-9Measures of Central Tendency:Locating the MediannIf the nu
8、mber of values is even, the median is the average of the two middle numbersNote that is not the value of the median, only the position of the median in the ranked dataBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-10Measures of Central Tendency:The MediannIn an ordered array,
9、 the median is the “middle” number (50% above, 50% below) nNot affected by extreme values0 1 2 3 4 5 6 7 8 9 10Median = 3 0 1 2 3 4 5 6 7 8 9 10Median = 3Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-11Measures of Central Tendency:The ModenValue that occurs most oftennNot af
10、fected by extreme valuesnUsed for either numerical or categorical datanThere may be no modenThere may be several modes0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 90 1 2 3 4 5 6No ModeMeasures of Central Tendency:The Moden Mean Mode ModeBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Ch
11、ap 3-13Measures of Central Tendency:Review ExampleHouse Prices: $2,000,000 $500,000 $300,000 $100,000 $100,000Sum $3,000,000Mean: ($3,000,000/5) = $600,000Median: middle value of ranked data = $300,000Mode: most frequent value = $100,000Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc
12、.Chap 3-14Measures of Central Tendency:Which Measure to Choose?The mean is generally used, unless extreme values (outliers) exist.The median is often used, since the median is not sensitive to extreme values. For example, median home prices may be reported for a region; it is less sensitive to outli
13、ers.In some situations it makes sense to report both the mean and the median.Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-15Measures of Central Tendency:SummaryCentral TendencyArithmetic MeanMedianModeMiddle value in the ordered arrayMost frequently observed valueBusiness S
14、tatistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-16Same center, different variationMeasures of VariationnMeasures of variation give information on the spread or variability or dispersion of the data values.VariationStandard DeviationCoefficient of VariationRangeVarianceBusiness Statistic
15、s: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-17Measures of Variation:The RangeSimplest measure of variationDifference between the largest and the smallest values:Range = Xlargest Xsmallest0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 13 - 1 = 12Example:Business Statistics: A First Course, 5e 20
16、09 Prentice-Hall, Inc.Chap 3-18Measures of Variation:Why The Range Can Be MisleadingIgnores the way in which data are distributedSensitive to outliers7 8 9 10 11 12Range = 12 - 7 = 57 8 9 10 11 12Range = 12 - 7 = 51,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,51,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2
17、,3,3,3,3,4,120Range = 5 - 1 = 4Range = 120 - 1 = 119Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-19nAverage (approximately) of squared deviations of values from the meannSample variance:Measures of Variation:The VarianceWhere = arithmetic meann = sample sizeXi = ith value o
18、f the variable XBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-20Measures of Variation:The Standard DeviationnMost commonly used measure of variationnShows variation about the meannIs the square root of the variancenHas the same units as the original datanSample standard devi
19、ation:Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-21Measures of Variation:The Standard DeviationSteps for Computing Standard Deviation1.Compute the difference between each value and the mean.2.Square each difference.3.Add the squared differences.4.Divide this total by n-1
20、to get the sample variance.5.Take the square root of the sample variance to get the sample standard deviation.Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-22Measures of Variation:Sample Standard DeviationSample Data (Xi) : 10 12 14 15 17 18 18 24 n = 8 Mean = X = 16A measur
21、e of the “average” scatter around the meanVariance of the Getting-Ready TimeBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-24Measures of Variation:Comparing Standard DeviationsMean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 2111 12 13 14 15 16 17 18 19 20 21Data BData AMe
22、an = 15.5 S = 0.92611 12 13 14 15 16 17 18 19 20 21Mean = 15.5 S = 4.570Data CBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-25Measures of Variation:Comparing Standard DeviationsSmaller standard deviationLarger standard deviationBusiness Statistics: A First Course, 5e 2009 Pr
23、entice-Hall, Inc.Chap 3-26Measures of Variation:Summary CharacteristicsThe more the data are spread out, the greater the range, variance, and standard deviation.The more the data are concentrated, the smaller the range, variance, and standard deviation.If the values are all the same (no variation),
24、all these measures will be zero.None of these measures are ever negative.Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-27Measures of Variation:The Coefficient of VariationnMeasures relative variationnAlways in percentage (%)nShows variation relative to meannCan be used to co
25、mpare the variability of two or more sets of data measured in different units Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-28Measures of Variation:Comparing Coefficients of VariationnStock A:nAverage price last year = $50nStandard deviation = $5nStock B:nAverage price last
26、year = $100nStandard deviation = $5Both stocks have the same standard deviation, but stock B is less variable relative to its priceBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-29Locating Extreme Outliers:Z-ScoreTo compute the Z-score of a data value, subtract the mean and d
27、ivide by the standard deviation.The Z-score is the number of standard deviations a data value is from the mean.A data value is considered an extreme outlier if its Z-score is less than -3.0 or greater than +3.0.The larger the absolute value of the Z-score, the farther the data value is from the mean
28、.Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-30Locating Extreme Outliers:Z-Scorewhere X represents the data value X is the sample mean S is the sample standard deviationBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-31Locating Extreme Outliers:Z-Sco
29、reSuppose the mean math SAT score is 490, with a standard deviation of 100.Compute the Z-score for a test score of 620.A score of 620 is 1.3 standard deviations above the mean and would not be considered an outlier.Z Score for the 10 Getting Ready TimeShape of a DistributionnDescribes how data are d
30、istributednMeasures of shapenSymmetric or skewedMean = Median Mean Median Median 1)n Examples:(1 - 1/22) x 100% = 75% . k=2 ( 2)(1 - 1/32) x 100% = 89% . k=3 ( 3)Chebyshev RulewithinAt leastHow Data Vary Around the MeanBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-48Quartile
31、 MeasuresnQuartiles split the ranked data into 4 segments with an equal number of values per segment25%nThe first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are largernQ2 is the same as the median (50% of the observations are smaller and 50% are larger)nOnly 25%
32、 of the observations are greater than the third quartile Q3Q1Q2Q325%25%25%Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-49Quartile Measures:Locating QuartilesFind a quartile by determining the value in the appropriate position in the ranked data, where First quartile positio
33、n: Q1 = (n+1)/4 ranked value Second quartile position: Q2 = (n+1)/2 ranked value Third quartile position: Q3 = 3(n+1)/4 ranked value where n is the number of observed valuesBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-50Quartile Measures:Calculation RulesnWhen calculating t
34、he ranked position use the following rulesnIf the result is a whole number then it is the ranked position to usenIf the result is a fractional half (e.g. 2.5, 7.5, 8.5, etc.) then average the two corresponding data values.nIf the result is not a whole number or a fractional half then round the resul
35、t to the nearest integer to find the ranked position.Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-51 (n = 9) Q1 is in the (9+1)/4 = 2.5 position of the ranked dataso use the value half way between the 2nd and 3rd values,so Q1 = 12.5Quartile Measures:Locating QuartilesSample
36、 Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Q1 and Q3 are measures of non-central location Q2 = median, is a measure of central tendencyBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-52 (n = 9)Q1 is in the (9+1)/4 = 2.5 position of the ranked data,so Q1 = (12+13)/2 = 1
37、2.5Q2 is in the (9+1)/2 = 5th position of the ranked data,so Q2 = median = 16Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data,so Q3 = (18+21)/2 = 19.5Quartile MeasuresCalculating The Quartiles: ExampleSample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 Q1 and Q3 are measures of non-centr
38、al location Q2 = median, is a measure of central tendencyBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-53Quartile Measures:The Interquartile Range (IQR)nThe IQR is Q3 Q1 and measures the spread in the middle 50% of the datanThe IQR is also called the midspread because it cov
39、ers the middle 50% of the datanThe IQR is a measure of variability that is not influenced by outliers or extreme valuesnMeasures like Q1, Q3, and IQR that are not influenced by outliers are called resistant measuresBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-54The Five Num
40、ber SummaryThe five numbers that help describe the center, spread and shape of data are:XsmallestFirst Quartile (Q1)Median (Q2)Third Quartile (Q3)XlargestBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-55Calculating The Interquartile RangeMedian(Q2)XmaximumXminimumQ1Q3Example:
41、25% 25% 25% 25%11 12.5 16 19.5 22Interquartile range = 19.5 12.5 = 7Five Number Summary andThe BoxplotnThe Boxplot: A Graphical display of the data based on the five-number summary:Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-56Example:Xsmallest - Q1 - Median - Q3 - Xlarges
42、t 25% of data 25% 25% 25% of data of data of dataXsmallest Q1 Median Q3 XlargestBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-57Five Number Summary:Shape of BoxplotsnIf data are symmetric around the median then the box and central line are centered between the endpointsnA Bo
43、xplot can be shown in either a vertical or horizontal orientationXsmallest Q1 Median Q3 XlargestBoxplots for Funds 2006 ReturnBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-59Distribution Shape and The BoxplotRight-SkewedLeft-SkewedSymmetricQ1Q2Q3Q1Q2Q3Q1Q2Q3Business Statisti
44、cs: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-60Boxplot ExamplenBelow is a Boxplot for the following data: 0 2 2 2 3 3 4 5 5 9 27nThe data are right skewed, as the plot depicts0 2 3 5 27Xsmallest Q1 Q2 Q3 XlargestBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-61Boxplo
45、t example showing an outlierThe boxplot below of the same data shows the outlier value of 27 plotted separatelyA value is considered an outlier if it is more than 1.5 times the interquartile range below Q1 or above Q3 Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-62The Covar
46、iancenThe covariance measures the strength of the linear relationship between two numerical variables (X & Y)nThe sample covariance:nOnly concerned with the strength of the relationship nNo causal effect is impliedBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-63nCovariance b
47、etween two variables:cov(X,Y) 0 X and Y tend to move in the same directioncov(X,Y) 0 X and Y tend to move in opposite directionscov(X,Y) = 0 X and Y are independentnThe covariance has a major flaw:nIt is not possible to determine the relative strength of the relationship from the size of the covaria
48、nceInterpreting CovarianceBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-64Coefficient of CorrelationnMeasures the relative strength of the linear relationship between two numerical variablesnSample coefficient of correlation: whereBusiness Statistics: A First Course, 5e 2009
49、 Prentice-Hall, Inc.Chap 3-65Features of theCoefficient of CorrelationnThe population coefficient of correlation is referred as .nThe sample coefficient of correlation is referred to as r.nEither or r have the following features:nUnit freenRanges between 1 and 1nThe closer to 1, the stronger the neg
50、ative linear relationshipnThe closer to 1, the stronger the positive linear relationshipnThe closer to 0, the weaker the linear relationshipBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-66Scatter Plots of Sample Data with Various Coefficients of CorrelationYXYXYXYXr = -1r =
51、-.6r = +.3r = +1YXr = 0Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-67The Coefficient of CorrelationUsing Microsoft Excel1.Select Tools/Data Analysis2.Choose Correlation from the selection menu3.Click OK . . .Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.C
52、hap 3-68The Coefficient of CorrelationUsing Microsoft Excel4.Input data range and select appropriate options5.Click OK to get outputBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-69Interpreting the Coefficient of CorrelationUsing Microsoft Excelr = .733There is a relatively s
53、trong positive linear relationship between test score #1 and test score #2.Students who scored high on the first test tended to score high on second test.Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-70Pitfalls in Numerical Descriptive MeasuresnData analysis is objectivenSho
54、uld report the summary measures that best describe and communicate the important aspects of the data setnData interpretation is subjectivenShould be done in fair, neutral and clear mannerBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-71Ethical ConsiderationsNumerical descript
55、ive measures:nShould document both good and bad resultsnShould be presented in a fair, objective and neutral mannernShould not use inappropriate summary measures to distort factsBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-72Chapter SummarynDescribed measures of central ten
56、dencynMean, median, modenDescribed measures of variationnRange, interquartile range, variance and standard deviation, coefficient of variation, Z-scoresnIllustrated shape of distributionnSymmetric, skewednDescribed data using the 5-number summarynBoxplotsBusiness Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.Chap 3-73Chapter SummarynDiscussed covariance and correlation coefficientnAddressed pitfalls in numerical descriptive measures and ethical considerations(continued)
