Basic Business Statistics,9th Edition - James Madison 基本业务统计第九版-杰姆斯麦迪逊

1、Chapter 3Numerical Descriptive MeasuresSummary MeasuresCentral TendencyMeanMedianModeQuartileGeometric MeanVariationVarianceStandard DeviationCoefficient of VariationRangeMeasures of Central TendencyVarious ways to describe the central, most common or middle value in a distribution or set of dataThe

2、 Mean (Arithmetic Mean)The MedianThe ModeThe Geometric MeanThe MeanEquals the sum of all observations or values divided by the number of valuesThe Most Common Measure of Central TendencyGenerally called the “Average in common usagePopulation mean vs. Sample meanThe MeanPopulation mean = (mu)Recall:

3、N = population sizeSample mean: X (x-bar)Recall n = sample size = Sumx = Sum of all values of xThe MeanAffected by Extreme Values (Outliers)Note how a difference of one value affects the mean in the example below0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5Mean = 6The MedianThe “Middle

4、 numberThe most “Robust measure of Central TendencyNot affected by extreme valuesIn an ordered array, the Median = (n+1)/2 ordered observation. If n or N is odd, the median is the middle numberIf n or N is even, the median is the average of the 2 middle numbers0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8

5、 9 10 12 14 Median = 5Median = 5ModeThe Value that Occurs Most OftenNot Affected by Extreme ValuesThere may be 1 ModesThere may be no ModeCan be used for Numerical or Categorical Data0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 90 1 2 3 4 5 6No ModeExampleProblem #Time Spent (Minutes)112243348576574899

6、11Jim has 20 problems to do for homework. Some are harder than others and take more time to solve. We take a random sample of 9 problems.Find the mean, median and mode for the number of minutes Jim spends on his homework. Solution: MeanProblem #Time Spent (Minutes)11224334857657489911Sample size (n)

7、 = 9Problems 1 through 9 = x1, x2, x3 x9, respectively.x = (12 + 4 + 3 + 8 + 7 + 5 + 4 + 9 + 11) = 63 minutesx/n = 63/9 = 7 minutesSolution: Median34457891112Place the data in ascending order as at right.(n+1)/2 = (9+1)/2 = 5The 5th ordered observation is 7 and so is the Median.Solution: ModeSince t

8、he data is already arranged in order from smallest to largest we will keep it that way.Only the value 4 occurs 1 time.The Mode is 4.34457891112Solution: Excel/PHStatEquations:Mean (=AVERAGE)Median (=MEDIAN)Mode (=MODE)Excel worksheetApproximating the Mean from a Frequency DistributionUsed when the o

9、nly source of data is a frequency distributionExampleClassMPFreq.10 but 2015320 but 3025630 but 4035540 but 5045450 but 60552Total20X= (15*3) + (25*6) + (35*5) + (45*4) + (55*2)/20= (45 + 150 + 175 + 180 + 110)/20= 660/20= 33Measures of VariationVariationVarianceStandard DeviationCoefficient of Vari

10、ationPopulationVarianceSample VariancePopulationStandardDeviationSample Standard DeviationRangeInterquartile RangeRangeDifference between the Largest and the Smallest Observations:Ignores the distribution of data7 8 9 10 11 12Range = 12 - 7 = 57 8 9 10 11 12Range = 12 - 7 = 5QuartilesQuartiles Split

11、 Ordered Data into 4 equal portionsQ1 and Q3 are Measures of Non-central LocationQ2 = the Median25%25%25%25%QuartilesEach Quartile has position and valueWith the data in an ordered array, the position of Qi is:The value of Qi is the value associated with that position in the ordered arrayExample:Dat

12、a in Ordered Array: 11 12 13 16 16 17 18 21 22 Quartiles ExampleFind the 1st and 3rd Quartiles in the ordered observations at right.Position of Q1 = 1(9+1)/4 = 2.5The 2.5th observation = (4+4)/2 = 4 Position of Q3 = 3(9+1)/4 = 3(Q1) = 7.5The 7.5th observation = (9+11)/2 = 1034457891112Interquartile

13、Range (IQR)The difference between Q1 and Q3 The middle 50% of the valuesAlso Known as Midspread: Resistant to extreme valuesExample:Q1 = 12.5, Q3 = 17.5 17.5 12.5 = 5IQR = 511 12 13 16 16 17 17 18 21Range and IQR ExampleFind the Range and the Interquartile Range in this distribution.Range = largest

14、smallest = 12 3 = 9.Position of Q1 = 1(9+1)/4 = 2.5The 2.5th observation = (4+4)/2 = 4 Position of Q3 = 3(9+1)/4 = 3(Q1) = 7.5The 7.5th observation = (9+11)/2 = 10IQR = 10 4 = 634457891112VarianceShows Variation about the MeanPopulation Variance (2): Sample Variance (S2):Standard Deviation (SD)Most

15、Important Measure of VariationThe square root of the varianceShows Variation about the MeanPopulation Standard Deviation ():Sample Standard Deviation (S):Variance and Standard DeviationBoth measure the average “scatter about the meanVariance computations produce “squared units which makes interpreta

16、tion more difficultFor example, $2 is meaningless.Since it is the square root of the Variance, the Standard Deviation is expressed in the same units as the original dataTherefore, the Standard Deviation is the most commonly used measure of variationA trivial Standard Deviation ExampleS= (3-4)2 + (4-

17、4)2 + (5-4)23-1S= -12 + 02 + 12 2S= 2/2 = 1Sample values: = 3, 4, 5Sample mean = 4n = 3Comparing 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 AMean = 15.5 s = .925811 12 13 14 15 16 17 18 19 20 21Mean = 15.5 s = 4.57Data CGreater

18、 S (or ) = more dispersion of dataApproximating SD from a Frequency DistributionUsed when the raw data are not available and the only source of data is a frequency distributionApproximating SD from a Frequency DistributionClassMPFreq.10 but 2015320 but 3025630 but 4035540 but 5045450 but 60552Total2

19、0X = 33S2 = (3(15-33)2 + 6(25-33)2 + 5(35-33)2 + 4(45-33)2 + 2(55-33)2)/ 20-1S2 =(972 + 384 + 20 + 576 + 968)/19 = 2920/19S2 = 153.7S = 12.4PhStatDispersionQi = QUARTILES2 = VARS = STDEV2 = VARP = STDEVPExcel worksheetCoefficient of VariationMeasure of Relative VariationShows Variation Relative to t

20、he MeanUsed to Compare Two or More Sets of Data Measured in Different Units S = Sample Standard DeviationX = Sample MeanComparing Coefficientof VariationStock A:Average price last year = $50Standard deviation = $2Stock B:Average price last year = $100Standard deviation = $5Coefficient of Variation:S

21、tock A:Stock B:The Z - ScoreZ Score = difference between the value and the mean, divided by the standard deviationUseful in identifying outliers (extreme values)Outliers are values in a data set that are located far from the meanThe larger the Z Score, the larger the distance from the meanA Z Score

22、is considered an outlier if it is +3.0Formulae:Z = X - X SORZ = X - Z Score ExampleIf our sample mean = 25, and our sample standard deviation is 10, is a value of 65 an outlier?4.0 3.0, so 65 is an outlier.Z = X - X S= 65 - 25 = 40 = 4.0 1010Shape of a DistributionThe Empirical RuleFor most data set

23、s: 68% of the Observations Fall Within () 1 Standard Deviation Around the Mean 95% of the Observations Fall Within () 2 SD Around the Mean 99.7% of the Observations Fall Within () 3 SD Around the MeanMore accurate for fairly symmetric data setsThe Empirical Rule -1sd +1sd68% 68% of the Observations

24、Fall Within () 1 Standard Deviation Around the MeanThe Empirical Rule -2sd -1sd +1sd +2sd95% 95% of the Observations Fall Within () 2 SD Around the MeanThe Empirical Rule -3sd -2sd -1sd +1sd +2sd +3sd99.7% 99.7% of the Observations Fall Within () 3 SD Around the MeanThe Bienayme-Chebyshev RuleThe Pe

25、rcentage of Observations Contained Within Distances of k Standard Deviations Around the Mean Must Be at Least:Applies regardless of the shape of the data setThe Bienayme-Chebyshev RuleAt least () 75% of the observations must be contained within distances of 2 SD around the meanAt least () 88.89% of

26、the observations must be contained within distances of 3 SD around the meanAt least () 93.75% of the observations must be contained within distances of 4 SD around the meanThe Bienayme-Chebyshev Rule- 2sd +2sd 75%The Bienayme-Chebyshev Rule - 3sd - 2sd +2sd +3sd 88.89% 75%The Bienayme-Chebyshev Rule

27、 - 4sd - 3sd - 2sd +2sd +3sd +4sd 88.89% 93.75% 75%Shape of a DistributionDescribe How Data are DistributedMeasures of ShapeSymmetric or Skewed (asymmetric)Mean = Median =Mode Mean Median Mode Mode Median MeanRight-SkewedLeft-SkewedSymmetricThe Box Plot (“Box-and-Whisker) 5 number summaryMedian, Q1, Q3, Xsmallest, XlargestBox PlotGraphical display of data using 5-number summaryMedian4 6 8 1012XlargestXsmallestDistribution Shape & Box PlotR