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1、Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-1,Business Statistics:A Decision-Making Approach7th Edition,Chapter 3Describing Data Using Numerical Measures,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-2,After completing this chap
2、ter,you should be able to:Compute and interpret the mean,median,and mode for a set of dataCompute the range,variance,and standard deviation and know what these values meanConstruct and interpret a box and whisker graph Compute and explain the coefficient of variation and z scoresUse numerical measur
3、es along with graphs,charts,and tables to describe data,Chapter Goals,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-3,Chapter Topics,Measures of Center and LocationMean,median,modeOther measures of LocationWeighted mean,percentiles,quartilesMeasures of VariationRan
4、ge,interquartile range,variance and standard deviation,coefficient of variationUsing the mean and standard deviation togetherCoefficient of variation,z-scores,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-4,Summary Measures,Center and Location,Mean,Median,Mode,Othe
5、r Measures of Location,Weighted Mean,Describing Data Numerically,Variation,Variance,Standard Deviation,Coefficient of Variation,Range,Percentiles,Interquartile Range,Quartiles,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-5,Measures of Center and Location,Center an
6、d Location,Mean,Median,Mode,Weighted Mean,Overview,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-6,Mean(Arithmetic Average),The Mean is the arithmetic average of data valuesPopulation meanSample mean,n=Sample Size,N=Population Size,Business Statistics:A Decision-Ma
7、king Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-7,Mean(Arithmetic Average),The most common measure of central tendencyMean=sum of values divided by the number of valuesAffected by extreme values(outliers),(continued),0 1 2 3 4 5 6 7 8 9 10,Mean=3,0 1 2 3 4 5 6 7 8 9 10,Mean=4,Business Statistics:A D
8、ecision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-8,Median,In an ordered array,the median is the“middle”number,i.e.,the number that splits the distribution in halfThe median is not affected by extreme values,0 1 2 3 4 5 6 7 8 9 10,Median=3,0 1 2 3 4 5 6 7 8 9 10,Median=3,Business Statistics:
9、A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-9,Median,To find the median,sort the n data values from low to high(sorted data is called a data array)Find the value in the i=(1/2)n positionThe ith position is called the Median Index PointIf i is not an integer,round up to next highest
10、integer,(continued),Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-10,Median Example,Note that n=13Find the i=(1/2)n position:i=(1/2)(13)=6.5Since 6.5 is not an integer,round up to 7The median is the value in the 7th position:Md=12,(continued),Data array:4,4,5,5,9,1
11、1,12,14,16,19,22,23,24,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-11,Shape of a Distribution,Describes how data is distributedSymmetric or skewed,Mean=Median,Mean Median,Median Mean,Right-Skewed,Left-Skewed,Symmetric,(Longer tail extends to left),(Longer tail ex
12、tends to right),Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-12,Mode,A measure of locationThe value that occurs most oftenNot affected by extreme valuesUsed for either numerical or categorical dataThere may be no modeThere may be several modes,0 1 2 3 4 5 6 7 8 9
13、10 11 12 13 14,Mode=5,0 1 2 3 4 5 6,No Mode,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-13,Weighted Mean,Used when values are grouped by frequency or relative importance,Example:Sample of 26 Repair Projects,Weighted Mean Days to Complete:,Business Statistics:A De
14、cision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-14,Five houses on a hill by the beach,Review Example,House Prices:$2,000,000 500,000 300,000 100,000 100,000,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-15,Summary Statistics,Mean:($3,000,000/5)=$600,000Med
15、ian:middle value of ranked data=$300,000Mode:most frequent value=$100,000,House Prices:$2,000,000 500,000 300,000 100,000 100,000Sum 3,000,000,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-16,Mean is generally used,unless extreme values(outliers)existThen Median is
16、 often used,since the median is not sensitive to extreme values.Example:Median home prices may be reported for a region less sensitive to outliers,Which measure of location is the“best”?,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-17,Other Location Measures,Other
17、 Measures of Location,Percentiles,Quartiles,1st quartile=25th percentile2nd quartile=50th percentile=median3rd quartile=75th percentile,The pth percentile in a data array:p%are less than or equal to this value(100 p)%are greater than or equal to this value(where 0 p 100),Business Statistics:A Decisi
18、on-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-18,Percentiles,The pth percentile in an ordered array of n values is the value in ith position,where,Example:Find the 60th percentile in an ordered array of19 values.,If i is not an integer,round up to the next higher integer value,So use value in
19、 the i=12th position,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-19,Quartiles,Quartiles split the ranked data into 4 equal groups:Note that the second quartile(the 50th percentile)is the median,25%,25%,25%,25%,Q1,Q2,Q3,Business Statistics:A Decision-Making Approa
20、ch,7e 2008 Prentice-Hall,Inc.,Chap 3-20,Quartiles,Sample Data in Ordered Array:11 12 13 16 16 17 18 21 22,Example:Find the first quartile,(n=9)Q1=25th percentile,so find i:i=(9)=2.25so round up and use the value in the 3rd position:Q1=13,25 100,Business Statistics:A Decision-Making Approach,7e 2008
21、Prentice-Hall,Inc.,Chap 3-21,Box and Whisker Plot,A graphical display of data using a central“box”and extended“whiskers”:,Example:,25%25%25%25%,Outliers Lower 1st Median 3rd Upper Limit Quartile Quartile Limit,*,*,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-22,Co
22、nstructing the Box and Whisker Plot,Outliers Lower 1st Median 3rd Upper Limit Quartile Quartile Limit,*,*,The lower limit is Q1 1.5(Q3 Q1),The upper limit is Q3+1.5(Q3 Q1),The center box extends from Q1 to Q3The line within the box is the medianThe whiskers extend to the smallest and largest values
23、within the calculated limitsOutliers are plotted outside the calculated limits,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-23,Shape of Box and Whisker Plots,The Box and central line are centered between the endpoints if data is symmetric around the median(A Box a
24、nd Whisker plot can be shown in either vertical or horizontal format),Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-24,Distribution Shape and Box and Whisker Plot,Right-Skewed,Left-Skewed,Symmetric,Q1,Q2,Q3,Q1,Q2,Q3,Q1,Q2,Q3,Business Statistics:A Decision-Making Ap
25、proach,7e 2008 Prentice-Hall,Inc.,Chap 3-25,Box-and-Whisker Plot Example,Below is a Box-and-Whisker plot for the following data:0 2 2 2 3 3 4 5 6 11 27This data is right skewed,as the plot depicts,0 2 3 6 12 27,Min Q1 Q2 Q3 Max,*,Upper limit=Q3+1.5(Q3 Q1)=6+1.5(6 2)=12,27 is above the upper limit so
26、 is shown as an outlier,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-26,Measures of Variation,Variation,Variance,Standard Deviation,Coefficient of Variation,PopulationVariance,Sample Variance,PopulationStandardDeviation,Sample Standard Deviation,Range,Interquartil
27、e Range,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-27,Measures of variation give information on the spread or variability of the data values.,Variation,Same center,different variation,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap
28、 3-28,Range,Simplest measure of variationDifference between the largest and the smallest observations:,Range=xmaximum xminimum,0 1 2 3 4 5 6 7 8 9 10 11 12 13 14,Range=14-1=13,Example:,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-29,Ignores the way in which data a
29、re distributedSensitive to outliers,7 8 9 10 11 12,Range=12-7=5,7 8 9 10 11 12,Range=12-7=5,Disadvantages of the Range,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120,Range=5-1=4,Range=120-1=119,Business Statistics:A Decision-Making Approach,7e 2
30、008 Prentice-Hall,Inc.,Chap 3-30,Interquartile Range,Can eliminate some outlier problems by using the interquartile range Eliminate some high-and low-valued observations and calculate the range from the remaining values.Interquartile range=3rd quartile 1st quartile,Business Statistics:A Decision-Mak
31、ing Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-31,Interquartile Range Example,Median(Q2),X,maximum,X,minimum,Q1,Q3,Example:,25%25%25%25%,12 30 45 57 70,Interquartile range=57 30=27,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-32,Average of squared deviations of va
32、lues from the meanPopulation variance:Sample variance:,Variance,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-33,Standard Deviation,Most commonly used measure of variationShows variation about the meanHas the same units as the original dataPopulation standard devia
33、tion:Sample standard deviation:,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-34,Calculation Example:Sample Standard Deviation,Sample Data(Xi):10 12 14 15 17 18 18 24,n=8 Mean=x=16,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-35,
34、Comparing Standard Deviations,Mean=15.5 s=3.338,11 12 13 14 15 16 17 18 19 20 21,11 12 13 14 15 16 17 18 19 20 21,Data B,Data A,Mean=15.5 s=.9258,11 12 13 14 15 16 17 18 19 20 21,Mean=15.5 s=4.57,Data C,Same mean,but different standard deviations:,Business Statistics:A Decision-Making Approach,7e 20
35、08 Prentice-Hall,Inc.,Chap 3-36,Coefficient of Variation,Measures relative variationAlways in percentage(%)Shows variation relative to meanIs used to compare two or more sets of data measured in different units,Population Sample,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,In
36、c.,Chap 3-37,Comparing Coefficients of Variation,Stock A:Average price last year=$50Standard deviation=$5Stock B:Average price last year=$100Standard deviation=$5,Both stocks have the same standard deviation,but stock B is less variable relative to its price,Business Statistics:A Decision-Making App
37、roach,7e 2008 Prentice-Hall,Inc.,Chap 3-38,If the data distribution is bell-shaped,then the interval:contains about 68%of the values in the population or the sample,The Empirical Rule,68%,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-39,contains about 95%of the val
38、ues in the population or the samplecontains about 99.7%of the values in the population or the sample,The Empirical Rule,99.7%,95%,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-40,Regardless of how the data are distributed,at least(1-1/k2)of the values will fall wit
39、hin k standard deviations of the mean Examples:(1-1/12)=0%.k=1(1)(1-1/22)=75%.k=2(2)(1-1/32)=89%.k=3(3),Tchebysheffs Theorem,within,At least,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-41,A standardized data value refers to the number of standard deviations a val
40、ue is from the meanStandardized data values are sometimes referred to as z-scores,Standardized Data Values,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-42,where:x=original data value=population mean=population standard deviationz=standard score(number of standard
41、deviations x is from),Standardized Population Values,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-43,where:x=original data valuex=sample means=sample standard deviationz=standard score(number of standard deviations x is from),Standardized Sample Values,Business St
42、atistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-44,IQ scores in a large population have a bell-shaped distribution with mean=100 and standard deviation=15Find the standardized score(z-score)for a person with an IQ of 121.Someone with an IQ of 121 is 1.4 standard deviations abov
43、e the mean,Standardized Value Example,Answer:,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-45,Using Microsoft Excel,Descriptive Statistics are easy to obtain from Microsoft ExcelUse menu choice:Data/data analysis/descriptive statisticsEnter details in dialog box,B
44、usiness Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-46,Using Excel,Select:Data/data analysis/descriptive statistics,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-47,Enter dialog box detailsCheck box for summary statisticsClick OK,Using E
45、xcel,(continued),Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-48,Excel output,Microsoft Excel descriptive statistics output,using the house price data:,House Prices:$2,000,000 500,000 300,000 100,000 100,000,Business Statistics:A Decision-Making Approach,7e 2008 P
46、rentice-Hall,Inc.,Chap 3-49,Chapter Summary,Described measures of center and locationMean,median,mode,weighted meanDiscussed percentiles and quartilesCreated Box and Whisker PlotsIllustrated distribution shapesSymmetric,skewed,Business Statistics:A Decision-Making Approach,7e 2008 Prentice-Hall,Inc.,Chap 3-50,Chapter Summary,Described measure of variationRange,interquartile range,variance,standard deviation,coefficient of variationDiscussed Tchebysheffs TheoremCalculated standardized data values,(continued),
