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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 First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-3,In this chapter, you learn: To describe
2、the properties of central tendency, variation, and shape in numerical dataTo calculate descriptive summary measures for a populationTo construct and interpret a boxplotTo calculate the covariance and the coefficient of correlation,Learning Objectives,Business Statistics: A First Course, 5e 2009 Pren
3、tice-Hall, Inc.,Chap 3-4,Summary 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 the distribution of values from the lowest value to the
4、 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 called “mean”) is the most common measure of central tendencyFor a sample of size n:,Sample size,Observed values,The ith value,Pronounced x-
5、bar,Measures of Central Tendency:The Mean,Example 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 12.2,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-7,Measures of Central Tendency:
6、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 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
7、 of Central Tendency:Locating the Median,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 Prentice-Hall, Inc.,Chap 3-9,Measures of Central Tendency:Loca
8、ting 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, only 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
9、 Median,In an ordered array, the median is the “middle” 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 Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-11,Measures of Central Tendency:The Mode,Va
10、lue 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 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,Business Statistics: A First Cour
11、se, 5e 2009 Prentice-Hall, Inc.,Chap 3-13,Measures of Central Tendency:Review Example,House Prices: $2,000,000 $500,000 $300,000 $100,000 $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 Statistics: A Fi
12、rst 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, since the median is not sensitive to extreme values. For example, median home prices may be reported fo
13、r 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 3-15,Measures of Central Tendency:Summary,Central Tendency,Arithmetic Mean,Median,Mode,Middle value in the orde
14、red 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 the spread or variability or dispersion of the data values.,Business Statistics: A Firs
15、t Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-17,Measures of Variation:The Range,Simplest measure of variationDifference between 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 2009
16、 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 - 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,
17、2,2,2,2,3,3,3,3,4,120,Range = 5 - 1 = 4,Range = 120 - 1 = 119,Business 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 meann = sample sizeXi =
18、 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 variationShows variation about the meanIs the square root of the varianceHas the same units as the original dataSample st
19、andard deviation:,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-21,Measures 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 thi
20、s 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 3-22,Measures of Variation:Sample Standard Deviation,Sample Data (Xi) : 10 12 14 15 17 18 18 24,n = 8 Me
21、an = X = 16,A measure of the “average” scatter around the mean,Variance of the Getting-Ready 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 20 21,11 12 13 14 15 16 17
22、 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 = 4.570,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 S
23、tatistics: 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 are concentrated, the smaller the range, variance, and standard deviation.If the
24、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 3-27,Measures of Variation:The Coefficient of Variation,Measures relative variationAlways in percentage (%)Shows var
25、iation relative to meanCan be used to compare 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 A:Average price last year = $50Standard dev
26、iation = $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 First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-29,Locating Extreme Outliers:Z-Score,To compute the Z-scor
27、e 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 absolute value of the Z-score, t
28、he 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 sample mean S is the sample standard deviation,Business Statistics: A First Course, 5e 2009 Prentice-Hal
29、l, Inc.,Chap 3-31,Locating Extreme Outliers:Z-Score,Suppose 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
30、Time,Shape of a Distribution,Describes how data are distributedMeasures of shapeSymmetric or skewed,Mean = Median,Mean Median,Median Mean,Right-Skewed,Left-Skewed,Symmetric,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-34,General Descriptive Stats Using Microsoft Excel,Sele
31、ct Tools.Select Data Analysis.Select Descriptive Statistics and click OK.,Business Statistics: A First Course, 5e 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 de
32、scriptive statistics 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 2009 Prentice-Hall, Inc.,Minitab Output,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-37,Descriptive Statistic
33、s: House Price TotalVariable Count Mean SE Mean StDev Variance Sum MinimumHouse Price 5 600000 357771 800000 6.40000E+11 3000000 100000 N forVariable Median Maximum Range Mode Skewness KurtosisHouse Price 300000 2000000 1900000 100000 2.01 4.13,Business Statistics: A First Course, 5e 2009 Prentice-H
34、all, Inc.,Chap 3-38,Numerical Descriptive Measures for a Population,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 the population mean,
35、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 ,The population mean is the sum of the values in the population divided by the population size, N, = population meanN = population si
36、zeXi = ith value of the variable X,Where,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-40,Average of squared deviations of values from the meanPopulation variance:,Numerical Descriptive Measures For A Population: The Variance 2,Where, = population meanN = population sizeXi
37、= ith 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 ,Most commonly used measure of variationShows variation about the meanIs the square root of the population varianceHas the
38、same units as the original dataPopulation standard deviation:,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,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 varia
39、tion of data in a bell-shaped 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, Inc.,Chap 3-44,Approximately 95% of the data in a bell-shaped distrib
40、ution lies within two standard deviations of the mean, or 2Approximately 99.7% of the data in a bell-shaped distribution lies within three standard deviations of the mean, or 3,The Empirical Rule,99.7%,95%,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-45,Using the Empirical
41、 Rule,Suppose 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 between 320 and 680 (500 180).99.7% of all test takers scored between 230 and 770 (500 270).
42、,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-46,Regardless of how the 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 ( 2)(1 - 1/32) x 100% = 89% . k=3 ( 3),
43、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 data into 4 segments with an equal number of values per segment,The first quartile, Q1, is the value for which 25% of t
44、he observations are smaller and 75% are largerQ2 is the same as the median (50% of the observations 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
45、:Locating 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 value Third quartile position: Q3 = 3(n+1)/4 ranked value where n is the number of obser
46、ved values,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-50,Quartile Measures: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
47、, 8.5, etc.) then average the two corresponding data values.If the result is not a whole number or 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
48、 position of the ranked dataso use the value half way between the 2nd and 3rd values,so Q1 = 12.5,Quartile Measures:Locating Quartiles,Sample 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 tendency,Business Statis
49、tics: 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 = 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 = 19.5
50、,Quartile MeasuresCalculating The Quartiles: Example,Sample 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 tendency,Business Statistics: A First Course, 5e 2009 Prentice-Hall, Inc.,Chap 3-53,Quartile Measures:The
