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 properties
2、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-Hall,Inc.Chap
3、 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 value.Business
4、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 sizenXXXnXXn21n1iiObserved valuesThe ith valuePronounced x-barMeasures of Central
5、 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 12.2oz.12.22 is emean volum The22.1252.123.122.121.123.12nxxBusiness Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 3-7Measu
6、res of Central Tendency:The MeannThe most 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=4351555432145205104321Business Statistics:A First Course,5e 2009 Pren
7、tice-Hall,Inc.Chap 3-8Measures of Central Tendency:Locating 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 numberdataorderedtheinposition21npositionMedianBusiness Statistics:A First Course,5e
8、2009 Prentice-Hall,Inc.Chap 3-9Measures of Central Tendency:Locating the MediannIf the number 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 data21nBusiness Statistics:A First Course,5e 2009
9、Prentice-Hall,Inc.Chap 3-10Measures of Central Tendency:The MediannIn an ordered array,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 3-11M
10、easures of Central Tendency:The ModenValue that occurs most oftennNot affected 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
11、ModeBusiness Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 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
12、Statistics:A First Course,5e 2009 Prentice-Hall,Inc.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 reporte
13、d for a region;it is less sensitive to outliers.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 MeanMedianModenXXnii1Middle value in the order
14、ed arrayMost frequently observed valueBusiness Statistics: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
15、 of VariationRangeVarianceBusiness Statistics: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
16、Statistics:A First Course,5e 2009 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
17、,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120Range=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 Variance1-n)X(XSn1i2i2Where =arithmetic meann=sample sizeXi=i
18、th value of the variable XXBusiness 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 stand
19、ard deviation:1-n)X(XSn1i2iBusiness 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
20、this total by n-1 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=164
21、.309571301816)(2416)(1416)(1216)(101n)X(24)X(14)X(12)X(10S22222222A measure 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 1
22、5 16 17 18 19 20 2111 12 13 14 15 16 17 18 19 20 21Data BData AMean=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 de
23、viationBusiness Statistics:A First Course,5e 2009 Prentice-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
24、 the values are all the same(no variation),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 varia
25、tion relative to meannCan be used to compare the variability of two or more sets of data measured in different units 100%XSCVBusiness Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 3-28Measures of Variation:Comparing Coefficients of VariationnStock A:nAverage price last year=$50nStandard d
26、eviation=$5nStock B:nAverage price last year=$100nStandard deviation=$5Both stocks have the same standard deviation,but stock B is less variable relative to its price10%100%$50$5100%XSCVA5%100%$100$5100%XSCVBBusiness Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 3-29Locating Extreme Outli
27、ers:Z-ScoreTo compute the Z-score of a data value,subtract the mean and divide 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 abs
28、olute value of the Z-score,the farther the data value is from the mean.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 deviationSXXZBusiness Statistics:A First Cours
29、e,5e 2009 Prentice-Hall,Inc.Chap 3-31Locating Extreme Outliers:Z-ScoreSuppose the mean math SAT score is 490,with a standard deviation of 100.Compute the Z-score for a test score of 620.3.1100130100490620SXXZA score of 620 is 1.3 standard deviations above the mean and would not be considered an outl
30、ier.Z Score for the 10 Getting Ready TimeShape of a DistributionnDescribes how data are distributednMeasures 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 S
31、tatistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 3-48Quartile 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(5
32、0%of the observations are smaller and 50%are larger)nOnly 25%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
33、position in the ranked data,where First quartile position: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 M
34、easures:Calculation RulesnWhen calculating the 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
35、a fractional half then round the result 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 Measu
36、res:Locating QuartilesSample 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 Q
37、1=(12+13)/2=12.5Q2 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-centra
38、l 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 covers the m
39、iddle 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 Number SummaryThe
40、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:25%25%25%25%11 12.5 16
41、 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 -Xlargest 25%of data 25%25%25%of data of dat
42、a 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 Boxplot can be shown in either a vertical or
43、horizontal orientationXsmallest Q1 Median Q3 XlargestBoxplots for Funds 2019 ReturnBusiness Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 3-59Distribution Shape and The BoxplotRight-SkewedLeft-SkewedSymmetricQ1Q2Q3Q1Q2Q3Q1Q2Q3Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Ch
44、ap 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 depicts 0 2 3 5 270 2 3 5 27Xsmallest Q1 Q2 Q3 XlargestBusiness Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 3-61Boxplot example showing an outlierThe boxplot bel
45、ow 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 CovariancenThe covariance measures the strength of
46、the linear relationship between two numerical variables(X&Y)nThe sample covariance:nOnly concerned with the strength of the relationship nNo causal effect is implied1n)YY)(XX()Y,X(covn1iiiBusiness Statistics:A First Course,5e 2009 Prentice-Hall,Inc.Chap 3-63nCovariance between two variables:cov(X,Y)
47、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 covarianceInterpreting CovarianceBusines
48、s 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:whereYXSSY),(Xcovr 1n)X(XSn1i2iX1n)Y)(YX(XY),(Xcovn1iii1n)Y(YSn1i2iYBusiness Statistic
49、s:A First Course,5e 2009 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
50、,the stronger the negative 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 CorrelationYX
