The document describes various statistical methods for describing and analyzing data, including measures of central tendency (mean, median), variability (range, standard deviation, interquartile range), and distribution (histograms, boxplots). It provides examples of calculating these statistics and interpreting them for real data sets. Comparisons are made between the sample mean and median, and between theoretical descriptions of data distributions (Chebyshev's Rule and the Empirical Rule) and actual data analyses.

1. Chapter 4 Numerical Methods for Describing Data

2. Describing the Center of a Data Set with the arithmetic mean

3. Describing the Center of a Data Set with the arithmetic mean The population mean is denoted by µ , is the average of all x values in the entire population.

8. Describing the Center of a Data Set with the median The sample median is obtained by first ordering the n observations from smallest to largest (with any repeated values included, so that every sample observation appears in the ordered list). Then

9. Example of Median Calculation Consider the Fancytown data. First, we put the data in numerical increasing order to get 231,000 285,000 287,000 294,000 297,000 299,000 312,000 313,000 315,000 317,000 Since there are 10 (even) data values, the median is the mean of the two values in the middle.

10. Example of Median Calculation Consider the Lowtown data. We put the data in numerical increasing order to get 93,000 95,000 97,000 99,000 100,000 110,000 113,000 121,000 122,000 2,000,000 Since there are 10 (even) data values, the median is the mean of the two values in the middle.

11. Comparing the Sample Mean & Sample Median

12. Comparing the Sample Mean & Sample Median

15. Example of Trimmed Mean

16. Example of Trimmed Mean

18. Categorical Data - Sample Proportion

19. Categorical Data - Sample Proportion If we look at the student data sample, consider the variable gender and treat being female as a success, we have 25 of the sample of 79 students are female, so the sample proportion (of females) is

26. Calculations Revisited The values for s 2 and s are exactly the same as were obtained earlier.

29. Quartiles and IQR Example With 15 data values, the median is the 8 th value. Specifically, the median is 10. 2, 4, 7, 8, 9, 10, 10, 10, 11, 12, 12, 14, 15, 19, 25 Lower quartile = 8 Upper quartile = 14 Iqr = 14 - 8 = 6 Median Lower Half Upper Half Lower quartile Q 1 Upper quartile Q 3

36. Modified Boxplot Example Here is the modified boxplot for the student age data. Smallest data value that isn’t an outlier Largest data value that isn’t an outlier Mild Outliers Extreme Outliers 15 20 25 30 35 40 45 50

37. Modified Boxplot Example Here is the same boxplot reproduced with a vertical orientation. 50 45 40 35 30 25 20 15

38. Comparative Boxplot Example By putting boxplots of two separate groups or subgroups we can compare their distributional behaviors. Notice that the distributional pattern of female and male student weights have similar shapes, although the females are roughly 20 lbs lighter (as a group). 100 120 140 160 180 200 220 240 Females Males G e n d e r Student Weight

39. Comparative Boxplot Example

40. Interpreting Variability Chebyshev’s Rule

45. Z Scores The z score is how many standard deviations the observation is from the mean. A positive z score indicates the observation is above the mean and a negative z score indicates the observation is below the mean.

46. Z Scores Computing the z score is often referred to as standardization and the z score is called a standardized score .

47. Example A sample of GPAs of 38 statistics students appear below (sorted in increasing order) 2.00 2.25 2.36 2.37 2.50 2.50 2.60 2.67 2.70 2.70 2.75 2.78 2.80 2.80 2.82 2.90 2.90 3.00 3.02 3.07 3.15 3.20 3.20 3.20 3.23 3.29 3.30 3.30 3.42 3.46 3.48 3.50 3.50 3.58 3.75 3.80 3.83 3.97

48. Example The following stem and leaf indicates that the GPA data is reasonably symmetric and unimodal. 2 0 2 233 2 55 2 667777 2 88899 3 0001 3 2222233 3 444555 3 7 3 889 Stem: Units digit Leaf: Tenths digit

49. Example

50. Example Notice that the empirical rule gives reasonably good estimates for this example. 38/38 = 100%  99.7% within 3 standard deviations of the mean 37/38 = 97%  95% within 2 standard deviations of the mean 27/38 = 71%  68% within 1 standard deviation of the mean Actual Empirical Rule Interval

51. Comparison of Chebyshev’s Rule and the Empirical Rule The following refers to the weights in the sample of 79 students. Notice that the stem and leaf diagram suggest the data distribution is unimodal but is positively skewed because of the outliers on the high side. Nevertheless, the results for the Empirical Rule are good. 10 3 11 37 12 011444555 13 000000455589 14 000000000555 15 000000555567 16 000005558 17 0000005555 18 0358 19 5 20 00 21 0 22 55 23 79 Stem: Hundreds & tens digits Leaf: Units digit

52. Comparison of Chebyshev’ Rule and the Empirical Rule Notice that even with moderate positive skewing of the data, the Empirical Rule gave a much more usable and meaningful result.  99.7%  95%  68% Empirical Rule 79/79 = 100%  88.8% within 3 standard deviations of the mean 75/79 = 94.9%  75% within 2 standard deviations of the mean 56/79 = 70.9%  0% within 1 standard deviation of the mean Actual Chebyshev’s Rule Interval