[教育]應(yīng)用統(tǒng)計學(xué)英文課件businessstatisticsch03numericaldescriptivemeasu

1、Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-1,Chapter 3Numerical Descriptive Measures,Business Statistics:A First CourseFifth Edition,Choice is yours, part 2,Business Statistics: A

2、First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-3,In this chapter, you learn: To describe the properties of central tendency, variation, and shape in numerical dataTo calculate descriptive summary measures for

3、a populationTo construct and interpret a boxplotTo calculate the covariance and the coefficient of correlation,Learning Objectives,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-4,Summar

4、y Definitions,The 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 th

5、e distribution of values from the lowest value to the highest value.,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-5,Measures of Central Tendency:The Mean,The arithmetic mean (often just

6、 called “mean”) is the most common measure of central tendencyFor a sample of size n:,Sample size,,Observed values,,,The ith value,,Pronounced x-bar,,Measures of Central Tendency:The Mean,Example volume of Coke

7、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 12.2,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.

8、,Chap 3-7,Measures of Central Tendency:The Mean,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

9、 6 7 8 9 10,,,,,,,,Mean = 3,,0 1 2 3 4 5 6 7 8 9 10,,,,,,,Mean = 4,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-8,Measures of Central Tendency:Locating the Me

10、dian,The location of the median when the values are in numerical order (smallest to largest):If the number of values is odd, the median is the middle number,,Business Statistics: A First Course, 5e © 2009 Pren

11、tice-Hall, Inc.,Chap 3-9,Measures of Central Tendency:Locating the Median,If the number of values is even, the median is the average of the two middle numbersNote that is not the value of the median, on

12、ly the position of the median in the ranked data,,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-10,Measures of Central Tendency:The Median,In an ordered array, the median is the “middle”

13、 number (50% above, 50% below) 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: A First

14、Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-11,Measures of Central Tendency:The Mode,Value that occurs most oftenNot affected by extreme valuesUsed for either numerical or categorical dataThere may be no modeT

15、here may be several modes,,0 1 2 3 4 5 6 7 8 9 10 11 12 13 14,,,,,,,,,Mode = 9,,,,,,,,,0 1 2 3 4 5 6,,,,,,,No Mode,Measures of Central Tendency:The Mode,Mean Mode Mode,,,,,,,,

16、Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-13,Measures of Central Tendency:Review Example,House Prices: $2,000,000 $500,000 $300,000 $100,00

17、0 $100,000Sum $3,000,000,Mean: ($3,000,000/5) = $600,000Median: middle value of ranked data = $300,000Mode: most frequent value = $100,000,Business Stati

18、stics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-14,Measures of Central Tendency:Which Measure to Choose?,The mean is generally used, unless extreme values (outliers) exist.The median is often used, sin

19、ce the median is not sensitive to extreme values. For example, median home prices may be reported for a region; it is less sensitive to outliers.In some situations it makes sense to report both the mean and the median.

20、,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-15,Measures of Central Tendency:Summary,,Central Tendency,,Arithmetic Mean,Median,Mode,,,,,,,,,,,,,,,,,,,,,,,,,,Middle value in the ordered

21、 array,Most frequently observed value,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-16,Same center, different variation,Measures of Variation,Measures of variation give information on th

22、e spread or variability or dispersion of the data values.,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-17,Measures of Variation:The Range,Simplest measure of variationDifference betwee

23、n the largest and the smallest values:,Range = Xlargest – Xsmallest,,,,,,,,,,,,,,,,0 1 2 3 4 5 6 7 8 9 10 11 12 13 14,,,,Range = 13 - 1 = 12,,Example:,Business Statistics: A First Course, 5e

24、 © 2009 Prentice-Hall, Inc.,Chap 3-18,Measures of Variation:Why The Range Can Be Misleading,Ignores the way in which data are distributedSensitive to outliers,,,,,,,,7 8 9 10 11 12,Range = 12

25、- 7 = 5,,7 8 9 10 11 12,,,,,,,,Range = 12 - 7 = 5,,,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120,Range = 5 - 1 = 4,Range = 120 - 1 = 119,Business

26、 Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-19,Average (approximately) of squared deviations of values from the meanSample variance:,Measures of Variation:The Variance,Where,= arithmetic me

27、ann = sample sizeXi = ith value of the variable X,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-20,Measures of Variation:The Standard Deviation,Most commonly used measure of variation

28、Shows variation about the meanIs the square root of the varianceHas the same units as the original dataSample standard deviation:,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-21,Me

29、asures of Variation:The Standard Deviation,Steps 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 tot

30、al 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 3-22,,Measures of Va

31、riation:Sample Standard Deviation,Sample Data (Xi) : 10 12 14 15 17 18 18 24,n = 8 Mean = X = 16,,,A measure of the “average” scatter around the mean,,Variance of the Getting-Re

32、ady Time,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-24,Measures of Variation:Comparing Standard Deviations,Mean = 15.5 S = 3.338,,11 12 13 14 15 16 17 18 19

33、 20 21,,,,,,,,,11 12 13 14 15 16 17 18 19 20 21,Data B,Data A,,,,,,,,,,Mean = 15.5 S = 0.926,11 12 13 14 15 16 17 18 19 20 21,,,,,,,,,,Mean = 15.5 S = 4.5

34、70,Data C,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-25,Measures of Variation:Comparing Standard Deviations,Smaller standard deviationLarger standard deviation,,,Business Statistics

35、: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-26,Measures of Variation:Summary Characteristics,The more the data are spread out, the greater the range, variance, and standard deviation.The more the data

36、are concentrated, the smaller the range, variance, and standard deviation.If the values are all the same (no variation), all these measures will be zero.None of these measures are ever negative.,Business Statistics:

37、A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-27,Measures of Variation:The Coefficient of Variation,Measures relative variationAlways in percentage (%)Shows variation relative to meanCan be used to compar

38、e the variability of two or more sets of data measured in different units,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-28,Measures of Variation:Comparing Coefficients of Variation,Stock

39、 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

40、,,,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-29,Locating Extreme Outliers:Z-Score,To compute the Z-score of a data value, subtract the mean and divide by the standard deviation.The

41、 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-s

42、core, the farther the data value is from the mean.,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-30,Locating Extreme Outliers:Z-Score,where X represents the data value X is the sampl

43、e mean S is the sample standard deviation,,,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-31,Locating Extreme Outliers:Z-Score,Suppose the mean math SAT score is 490, with a standard

44、 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 Time,,,,Shape of a Dist

45、ribution,Describes how data are distributedMeasures of shapeSymmetric or skewed,,,,,,,Mean = Median,,,Mean < Median,,Median < Mean,,,,,,,,,,,,,,Right-Skewed,Left-Skewed,Symmetric,,,Business Statistics: A First Co

46、urse, 5e © 2009 Prentice-Hall, Inc.,Chap 3-34,General Descriptive Stats Using Microsoft Excel,Select Tools.Select Data Analysis.Select Descriptive Statistics and click OK.,,,,Business Statistics: A First Course, 5

47、e © 2009 Prentice-Hall, Inc.,Chap 3-35,General Descriptive Stats Using Microsoft Excel,4. Enter the cell range.5. Check the Summary Statistics box.6. Click OK,,,Excel output,Microsoft Excel descriptive statistics

48、 output, using the house price data:,House Prices: $2,000,000 500,000 300,000 100,000 100,000,,,,,,,,,,Chap 3-36,Business Statistics: A First Course, 5e © 20

49、09 Prentice-Hall, Inc.,Minitab Output,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-37,Descriptive Statistics: House Price TotalVariable Count Mean SE Mean

50、 StDev Variance Sum MinimumHouse Price 5 600000 357771 800000 6.40000E+11 3000000 100000 N forVariable Median Maximum Range Mod

51、e Skewness KurtosisHouse Price 300000 2000000 1900000 100000 2.01 4.13,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-38,Numerical Descriptive Measures for a Popula

52、tion,Descriptive statistics discussed previously described a sample, not the population.Summary measures describing a population, called parameters, are denoted with Greek letters.Important population parameters are

53、the population mean, variance, and standard deviation.,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-39,Numerical Descriptive Measures for a Population: The mean µ,The population m

54、ean is the sum of the values in the population divided by the population size, N,μ = population meanN = population sizeXi = ith value of the variable X,Where,Business Statistics: A First Course, 5e © 2009 Prentice

55、-Hall, Inc.,Chap 3-40,Average of squared deviations of values from the meanPopulation variance:,Numerical Descriptive Measures For A Population: The Variance σ2,Where,μ = population meanN = population sizeXi = ith

56、value of the variable X,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-41,Numerical Descriptive Measures For A Population: The Standard Deviation σ,Most commonly used measure of variation

57、Shows variation about the meanIs the square root of the population varianceHas the same units as the original dataPopulation standard deviation:,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, I

58、nc.,Chap 3-42,Sample statistics versus population parameters,,,,,,,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-43,,,The empirical rule approximates the variation of data in a bell-shape

59、d distributionApproximately 68% of the data in a bell shaped distribution is within 1 standard deviation of the mean or,The Empirical Rule,,,,,,,,,68%,,,,Business Statistics: A First Course, 5e © 2009 Prentice-Hall

60、, Inc.,Chap 3-44,,,Approximately 95% of the data in a bell-shaped distribution lies within two standard deviations of the mean, or µ ± 2σApproximately 99.7% of the data in a bell-shaped distribution lies with

61、in three standard deviations of the mean, or µ ± 3σ,The Empirical Rule,,,,,,,,,,,,,,,99.7%,95%,,,,,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-45,Using the Empirical Rule,Supp

62、ose that the variable Math SAT scores is bell-shaped with a mean of 500 and a standard deviation of 90. Then,68% of all test takers scored between 410 and 590 (500 ± 90).95% of all test takers scored bet

63、ween 320 and 680 (500 ± 180).99.7% of all test takers scored between 230 and 770 (500 ± 270).,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-46,,Regardless of how t

64、he data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1) Examples:(1 - 1/22) x 100% = 75% …........ k=2 (μ ± 2σ)(1 - 1/32) x 100

65、% = 89% ………. k=3 (μ ± 3σ),Chebyshev Rule,within,At least,,How Data Vary Around the Mean,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-48,Quartile Measures,Quartiles split the ranked

66、 data into 4 segments with an equal number of values per segment,,The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are largerQ2 is the same as the median (50% of the observation

67、s are smaller and 50% are larger)Only 25% of the observations are greater than the third quartile Q3,,,Q1,Q2,Q3,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-49,Quartile Measures:Locati

68、ng Quartiles,Find a quartile by determining the value in the appropriate position in the ranked data, where First quartile position: Q1 = (n+1)/4 ranked value Second quartile position: Q2 = (n+1)/2 ranked

69、value Third quartile position: Q3 = 3(n+1)/4 ranked value where n is the number of observed values,Business Statistics: A First Course, 5e © 2009 Prentice-Hall, Inc.,Chap 3-50,Quartile Measures:

70、Calculation Rules,When calculating the ranked position use the following rulesIf the result is a whole number then it is the ranked position to useIf the result is a fractional half (e.g. 2.5, 7.5, 8.5, etc.) then av