統計學 Fall 2003

1. 統計學 Fall 2003 授課教師：統計系余清祥 日期：2003年9月30日 第三週：初步資料分析

2. Chapter 3 Part A Descriptive Statistics: Numerical Methods • Measures of Location • Measures of Variability   x %

3. Measures of Location • Mean • Median • Mode • Percentiles • Quartiles

4. Example: Apartment Rents Given below is a sample of monthly rent values ($) for one-bedroom apartments. The data is a sample of 70 apartments in a particular city. The data are presented in ascending order.

5. Mean • The mean of a data set is the average of all the data values. • If the data are from a sample, the mean is denoted by . • If the data are from a population, the mean is denoted by m (mu).

6. Example: Apartment Rents • Mean

7. Median • The median is the measure of location most often reported for annual income and property value data. • A few extremely large incomes or property values can inflate the mean.

8. Median • The median of a data set is the value in the middle when the data items are arranged in ascending order. • For an odd number of observations, the median is the middle value. • For an even number of observations, the median is the average of the two middle values.

9. Example: Apartment Rents • Median Median = 50th percentile i = (p/100)n = (50/100)70 = 35.5 Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475

10. Mode • The mode of a data set is the value that occurs with greatest frequency. • The greatest frequency can occur at two or more different values. • If the data have exactly two modes, the data are bimodal. • If the data have more than two modes, the data are multimodal.

11. Example: Apartment Rents • Mode 450 occurred most frequently (7 times) Mode = 450

12. Percentiles • A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. • Admission test scores for colleges and universities are frequently reported in terms of percentiles.

13. Percentiles • The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more. • Arrange the data in ascending order. • Compute index i, the position of the pth percentile. i = (p/100)n • If i is not an integer, round up. The pth percentile is the value in the ith position. • If i is an integer, the pth percentile is the average of the values in positions i and i+1.

14. Example: Apartment Rents • 90th Percentile i = (p/100)n = (90/100)70 = 63 Averaging the 63rd and 64th data values: 90th Percentile = (580 + 590)/2 = 585

15. Quartiles • Quartiles are specific percentiles • First Quartile = 25th Percentile • Second Quartile = 50th Percentile = Median • Third Quartile = 75th Percentile

16. Example: Apartment Rents • Third Quartile Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525

17. Measures of Variability • It is often desirable to consider measures of variability (dispersion), as well as measures of location. • For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.

18. Measures of Variability • Range • Interquartile Range • Variance • Standard Deviation • Coefficient of Variation

19. Range • The range of a data set is the difference between the largest and smallest data values. • It is the simplest measure of variability. • It is very sensitive to the smallest and largest data values.

20. Example: Apartment Rents • Range Range = largest value - smallest value Range = 615 - 425 = 190

21. Interquartile Range • The interquartile range of a data set is the difference between the third quartile and the first quartile. • It is the range for the middle 50% of the data. • It overcomes the sensitivity to extreme data values.

22. Example: Apartment Rents • Interquartile Range 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80

23. Variance • The variance is a measure of variability that utilizes all the data. • It is based on the difference between the value of each observation (xi) and the mean (x for a sample, m for a population).

24. Variance • The variance is the average of the squared differences between each data value and the mean. • If the data set is a sample, the variance is denoted by s2. • If the data set is a population, the variance is denoted by  2.

25. Standard Deviation • The standard deviation of a data set is the positive square root of the variance. • It is measured in the same units as the data, making it more easily comparable, than the variance, to the mean. • If the data set is a sample, the standard deviation is denoted s. • If the data set is a population, the standard deviation is denoted  (sigma).

26. Coefficient of Variation • The coefficient of variation indicates how large the standard deviation is in relation to the mean. • If the data set is a sample, the coefficient of variation is computed as follows: • If the data set is a population, the coefficient of variation is computed as follows:

27. Example: Apartment Rents • Variance • Standard Deviation • Coefficient of Variation

28. End of Chapter 3, Part A

29. Chapter 3 Part B Descriptive Statistics: Numerical Methods • Measures of Relative Location and Detecting Outliers • Exploratory Data Analysis • Measures of Association Between Two Variables • The Weighted Mean and Working with Grouped Data  %  x

30. Measures of Relative Locationand Detecting Outliers • z-Scores • Chebyshev’s Theorem • Empirical Rule • Detecting Outliers

31. z-Scores • The z-score is often called the standardized value. • It denotes the number of standard deviations a data value xi is from the mean. • A data value less than the sample mean will have a z-score less than zero. • A data value greater than the sample mean will have a z-score greater than zero. • A data value equal to the sample mean will have a z-score of zero.

32. Example: Apartment Rents • z-Score of Smallest Value (425) Standardized Values for Apartment Rents

33. Chebyshev’s Theorem At least (1 - 1/k2) of the items in any data set will be within k standard deviations of the mean, where k is any value greater than 1. • At least 75% of the items must be within • k = 2 standard deviations of the mean. • At least 89% of the items must be within • k = 3 standard deviations of the mean. • At least 94% of the items must be within • k = 4 standard deviations of the mean.

34. Example: Apartment Rents • Chebyshev’s Theorem Let k = 1.5 with = 490.80 and s = 54.74 At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56% of the rent values must be between - k(s) = 490.80 - 1.5(54.74) = 409 and + k(s) = 490.80 + 1.5(54.74) = 573

35. Example: Apartment Rents • Chebyshev’s Theorem (continued) Actually, 86% of the rent values are between 409 and 573.

36. Empirical Rule For data having a bell-shaped distribution: • Approximately 68% of the data values will be within onestandard deviation of the mean.

37. Empirical Rule For data having a bell-shaped distribution: • Approximately 95% of the data values will be within twostandard deviations of the mean.

38. Empirical Rule For data having a bell-shaped distribution: • Almost all (99.7%) of the items will be within threestandard deviations of the mean.

39. Example: Apartment Rents • Empirical Rule Interval% in Interval Within +/- 1s 436.06 to 545.54 48/70 = 69% Within +/- 2s 381.32 to 600.28 68/70 = 97% Within +/- 3s 326.58 to 655.02 70/70 = 100%

40. Detecting Outliers • An outlier is an unusually small or unusually large value in a data set. • A data value with a z-score less than -3 or greater than +3 might be considered an outlier. • It might be an incorrectly recorded data value. • It might be a data value that was incorrectly included in the data set. • It might be a correctly recorded data value that belongs in the data set !

41. Example: Apartment Rents • Detecting Outliers The most extreme z-scores are -1.20 and 2.27. Using |z| > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents

42. Exploratory Data Analysis • Five-Number Summary • Box Plot

43. Five-Number Summary • Smallest Value • First Quartile • Median • Third Quartile • Largest Value

44. Example: Apartment Rents • Five-Number Summary Lowest Value = 425 First Quartile = 450 • Median = 475 • Third Quartile = 525 Largest Value = 615

45. Box Plot • A box is drawn with its ends located at the first and third quartiles. • A vertical line is drawn in the box at the location of the median. • Limits are located (not drawn) using the interquartile range (IQR). • The lower limit is located 1.5(IQR) below Q1. • The upper limit is located 1.5(IQR) above Q3. • Data outside these limits are considered outliers. • … continued

46. Box Plot (Continued) • Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. • The locations of each outlier is shown with the symbol* .

47. Example: Apartment Rents • Box Plot Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5 Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5 There are no outliers. 575 600 625 450 375 400 500 525 550 425 475

48. Measures of Association Between Two Variables • Covariance • Correlation Coefficient

49. Covariance • The covariance is a measure of the linear association between two variables. • Positive values indicate a positive relationship. • Negative values indicate a negative relationship.

50. Covariance • If the data sets are samples, the covariance is denoted by sxy. • If the data sets are populations, the covariance is denoted by .