学习大二下商统夏南新-statistics for business11for ln2012ch.3descriptive numerical measures

1、中山大学岭南学院教授 夏南新博士 制作Chapter 3 DESCRIPTIVE STATISTICS: NUMERICAL METHODS 3.1 MEASURES OF LOCATIONMean Sample Mean Population MeanMedianThe median is the value in the middle when the data are arranged in ascending order, so it is a measure of location. For example: 32 42 46 46 542710 2755 2850 2880 288

2、0 2890 2920 2940 2950 3050 3130 ModeThe mode is the value that occurs with greatest frequency, so it also is a measure of location.E.g. 32 42 46 46 54TABLE 3.1 RELATRIVE AND PERCENT FREQUCENCY DISTRIBUTIONS FOR THE AUDIT-TIME DATA Audit Time Middle (days) Value Frequency 10-14 12 4 15-19 17 8 20-24

3、22 5 25-29 27 2 30-34 32 1 Total - 20PercentilesA percentile provides information about how the data are spread over the interval from the smallest value to the largest value.Calculating the th Percentile2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 Because index is not an integer, round up

4、. The position of the 85th percentile is the next integer greater than 10.2, the 11th position.QuartilesA data set divided into four parts, the division points are referred to as the quartiles and are defined as first quartile, or 25th percentile second quartile, or 50th percentile (also the median)

5、 third quartile, or 75th percentile. Figure 3.1 Location of the QuartilesForForFor3.2 MEASURES OF VARIABILITYRangeInterquartile RangeThe interquartile range es the influence of dependency on extreme values.Variance Population Mean Sample Mean TABLE 3. 2 COMPUTATION DEVIATIONS AND SQUARED DEVIATIONS

6、ABOUT THE MEAN FOR THE CLASS-SIZE DATA Number of Mean Deviation Squared DeviationStudents in Class About the Mean About the MeanClass ( ) Size ( ) 46 44 2 4 54 44 10 100 42 44 -2 4 46 44 2 4 32 44 -12 144Total 0 256Standard Deviation Sample Mean Population MeanE.g. Coefficient of VariationIt indicat

7、es how large the standard deviation is in relation to the mean (that is , the standard deviation relative to the mean) as follows. Coefficient of Variation 3.3 MEASURES OF RELATIVE LOCATION AND DETECTING OUTLIERS ScoresAssociated with each value, (i.e. with the values denoted by ), is another value

8、called its Scores. Chebyshevs TheoremAt least ( ) of the data values must be within standard deviations of the mean. Where is any value greater than 1. For an example using Chebyshevs theorem, assume that the midterm test scores for 100 students in a course had a mean of 70 and a standard deviation

9、of 5. How many students had test scores between 60 and 80? We note that 60 is two (i.e. ) standard deviations below the mean and 80 is two (i.e. ) standard deviations above the mean.We have At least 75% of the students must have test scores between 60 and 80.Notes: e.g.Empirical RuleFIGURE 3.2 A MOU

10、ND-SHAPED OR BELL-SHAPED DISTRIBUTION 95.45% For data having a bell-shaped distribution: Approximately 68.27%, 95.45%, and 99.73% of the data values will be within one, two and three standard deviation of the mean respectively. Detecting OutliersExtreme values such as unusually large or unusually sm

11、all values in a data set are called outliers.3.4 EXPLORATORY DATA ANALYSIS Five-Number SummaryBox Plot Lower Upper Limit Limit 3.5 MEASURES OF ASSOCIATION BETWEEN TWO VARIABLES Covariance Sample Mean Population Mean TABLE 3.3 CALCULATIONS FOR THE SAMPLE COVARIANCE 2 50 -1 -1 1 5 57 2 6 12 1 41 -2 -1

12、0 20 3 54 0 3 0 4 54 1 3 3 1 38 -2 -13 26 5 63 2 12 24 3 48 0 -3 0 4 59 1 8 8 2 46 -1 -5 5Totals 30 510 0 0 99 Interpretation of the CovarianceThe covariance is a measure of the linear association between two variables. Positive: (x and y are positively linearly related) FIGURE 3.3 INTERPRETATION OF

13、 SAMPLE CONVARIANCE Approximately 0: (x and y are not linearly related)FIGURE 3.4 INTERPRETATION OF SAMPLE CONVARIANCE Negative: (x and y are negatively linearly related) FIGURE 3.5 INTERPRETATION OF SAMPLE CONVARIANCECorrelation CoefficientPearson Product Moment Correlation Coefficient: Sample Data

14、Pearson Product Moment Correlation Coefficient: Population DataInterpretation of the Correlation Coefficient3.6 THE WEIGHTED MEAN AND WORKING WITH GROUPED DATA Weighted Mean Weighted Mean where value of observation weight for observationGrouped Data Sample Mean for Grouped Data where the midpoint for class the frequency for class the sample sizeTABLE 3.4 RELATRIVE AND PERCENT FREQUCENCY DISTRIBUTIONS FOR THE AUDIT-TIME DATA Audit Time lass Midpoin