Describing Data: Displaying and Exploring Data

1. Describing Data: Displaying and Exploring Data Chapter 4 Modified by Boris Velikson, 2009

2. GOALS • Develop and interpret a dot plot. • Develop and interpret a stem-and-leaf display. • Compute and understand quartiles, deciles, and percentiles. • Construct and interpret box plots. • Compute and understand the coefficient of skewness. • Draw and interpret a scatter diagram. • Construct and interpret a contingency table.

3. Dot Plots • A dot plot groups the data as little as possible.The identity of an individual observation is not lost. • Each observation is simply displayed as a dot along a horizontal line indicating the possible values of the data. • If there are identical observations or the observations are too close to be shown individually, the dots are “piled” on top of each other. • Dot plots are useful for comparing two or more data sets. • Dot plots can be used only for discrete data. • The usefulness of a dot plot depends on the nature of the data set.

4. Dot Plots - Example Reported below are the number of vehicles sold in the last 24 months at Smith Ford Mercury Jeep, Inc., in Kane, Pennsylvania, and Brophy Honda Volkswagen in Greenville, Ohio. Construct dot plots and report summary statistics for the two small-town Auto USA lots. (Excel file on CDROM: 4 Dotplots.XLS)

5. Dot Plot – Minitab results (you can’t do this unless you install Minitab!)

6. WARNING about Dot Plots • You do not have MiniTab (it is not free). • On some computers, MegaStat does not work well when you try to obtain Dot Plots with it. (You can try it on 4 Dotplot.xls, figuring among Chapter 4 Excel files) • Therefore, you draw them by hand. • This is not so bad: anyway, Dot Plots are useful only for small sets of data.

7. Reminder: Visual Display of Data • We showed in Chapter 2 that one can see the data at a glance if one transforms it into a Frequency Distribution. The Data: 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. get transformed into afrequency distribution: And then into a histogram:

8. Stem-and-Leaf Display of Data • There exists another technique used to display quantitative information in a condensed form: the stem-and-leaf display. • Each numerical value is divided into two parts. The leading digit(s) becomes the stem and the trailing digit the leaf. The stems are located along the vertical axis, and the leaf values are stacked against each other along the horizontal axis. • Advantage of the stem-and-leaf display over a frequency distribution is that the identity of each observation is not lost.

9. Stem-and-Leaf – Example Suppose the seven observations in the 90 up to 100 class are: 96, 94, 93, 94, 95, 96, and 97. The stem value is the leading digit or digits, in this case 9. The leaves are the trailing digits. The stem is placed to the left of a vertical line and the leaf values to the right. The values in the 90 up to 100 class would appear as Then, we sort the values within each stem from smallest to largest. Thus, this row of the stem-and-leaf display would appear as follows:

10. Stem-and-Leaf – Another Example Stock prices on twelve consecutive days for a major publicly traded company: 86,79,92,84,69,88,91,83,96,78,82,85.

11. Stem-and-leaf: Example with additional information in the 1st column (Data: p.102, Table 4-1)

12. Stem-and-leaf: The look of MiniTab Output (You can’t obtain this output unless you install Minitab and unless it works on your computer, but you can do the same thing by hand.)

13. In-Class Exercises and Homework (dot plots and steam-and-leaf plots) • In class: P. 105 No. 3, 5, 7, 9 • Homework: No. 4, 6, 8, 10

14. Quartiles, Deciles and Percentiles • The standard deviation is the most widely used measure of dispersion. • One more way of describing the spread of data is determining the location (thesequential number in the set) of values that divide a set of observations into equal parts. • The most often used are quartiles, deciles, and percentiles.

15. Quartiles - Example Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney’s Oakland, California, office. Salomon Smith Barney is an investment company with offices located throughout the United States. $2,038 $1,758 $1,721 $1,637 $2,097 $2,047 $2,205 $1,787 $2,287 $1,940 $2,311 $2,054 $2,406 $1,471 $1,460 .

16. Quartiles– Example (cont.) Step 1: Organize the data from lowest to largest value median The middle value (the median) location is found by Lmed = (15+1)/2 = 8. There are 7 values below the median and 7 values above. Because the median divides the data into two halves, we also call it the 50th percentile (Q50) and we say that it is located at L50. In our case, L50=Lmed= 8, and Q50=med=$2038.

17. Quartiles– Example (cont.) Step 2: Compute the first and third quartiles. Locate L25 and L75 using: Therefore, the first and the fourth quartile are the 4th and the 12th observation in the array:

18. Quartiles – Example (cont.) Step 2: Compute the first and third quartiles. median 3d quartile 1st quartile In a similar way, we can split the values in 4 parts. The values separating the 4 parts are called quartiles. The 1st quartile is located at the position L25=(n+1)/4=(n+1)(25/100). Here, L25=(15+1)/4=4. The value of the 1st quartile is Q1 =$1721. The 3d quartile is located at the position L75=(n+1)(3/4) =(n+1)(75/100).Here, L75=(15+1)(3/4)=12. The value of the 3d quartile is Q3=$2205.

19. Percentiles Percentiles: median 3d quartile 1st quartile We also call the 1st quartile the 25th percentile, and the 3d quartile, the 75th percentile. (Then we write Q25 instead of Q1, and Q75 instead of Q3). In general,

20. Percentiles – Example (cont.) What if the number of data can’t be divided by 4? Suppose we have 12 observations in a sample. Then But there is no observation no. 3.25 or 9.75! So what do we do? We take X3 and X4, and form a value which is ¼-way between the two. This will be Q1. Then we take X9 and X10 and form a value which is ¾ way between the two. This will be Q3.

21. Percentiles – Example (cont.) 50th percentile: Median Price at 6.50th observation = 85 + .5(85-84) = 84.50 25th percentile: 1st quartile Price at 3.25th observation = 79 + .25(82-79) = 79.75 75th percentile: 3d quartile Price at 9.75th observation = 88 + .75(91-88) = 90.25

22. Deciles; Percentiles other than Quartiles We call deciles percentiles that are multiples of 10. L10 is the location of the first decile, L30 of the 3d decile, etc.

23. Percentiles – Example (Excel Data Analysis) Read p.109

24. Percentiles – Example (MegaStat)

25. Interquartile Range The Interquartile range is the distance between the third quartile Q3 and the first quartile Q1. This distance will include the middle 50 percent of the observations. Interquartile range = Q3 - Q1 Here Q1 and Q3 are the values of the quartiles.

26. In-Class Exercises and Homework (percentiles) • In class: P. 109 No. 11, 13 • Homework: No. 12,14

27. Boxplot - Example

28. Boxplot Example

29. Outliers Outliers are the data that lie more than 1.5 times the interquartile distance on the left of Q1 or on the right of Q3. In this example, Q1=$20000, Q3=$26000. So an outlier would be 1.5×6000=9000 away, i.e. less than 20000-9000=$11000 or more than 26000+9000=$35000. We see an asterisk marking an outlier. Attn.: different authors may define outliers in a slightly different way (not necessarily exactly 1.5 times the interquartile distance away)

30. In-Class Exercises and Homework (boxplots) • In class: P. 112 No. 15, 17 • Homework: No. 16,18

31. Skewness • In Chapter 3, measures of central location for a set of observations (the mean, median, and mode) and measures of data dispersion (e.g. range and the standard deviation) were introduced. • Another characteristic of a set of data is the shape. • There are four shapes commonly observed: • symmetric, • positively skewed, • negatively skewed, • bimodal.

32. Skewness - Formulas for Computing The coefficient of skewness can range from -3 up to 3. • A value near -3, such as -2.57, indicates considerable negative skewness. • A value such as 1.63 indicates moderate positive skewness. • A value of 0, which will occur when the mean and median are equal, indicates the distribution is symmetrical and that there is no skewness present. There are various formulas for the coefficient of skewness. The main two are: (in fact, the software coefficient works better, but if you have only a simple calculator, it is easier to calculate Pearson’s)

33. Commonly Observed Shapes

34. Skewness – An Example • Following are the earnings per share for a sample of 15 software companies for the year 2005. The earnings per share are arranged from smallest to largest. • Compute the mean, median, and standard deviation. Find the coefficient of skewness using Pearson’s estimate. What is your conclusion regarding the shape of the distribution?

35. Skewness –Using Pearson’s Coefficient

36. Skewness – Megastat Result You can see that the dot plot does not make any sense here, while the box plot conveys a lot of information.

37. In-Class Exercises and Homework (skewness) • In class: P. 117 No. 19 • Homework: No. 22

38. Describing Relationship between Two Variables • One graphical technique we use to show the relationship between variables is called a scatter diagram. • To draw a scatter diagram we need two variables. We scale one variable along the horizontal axis (X-axis) of a graph and the other variable along the vertical axis (Y-axis).

39. Describing Relationship between Two Variables – Scatter Diagram Examples

40. Describing Relationship between Two Variables – Scatter Diagram Excel Example In the Introduction to Chapter 2 we presented data from AutoUSA. In this case the information concerned the prices of 80 vehicles sold last month at the Whitner Autoplex lot in Raytown, Missouri. The data shown include the selling price of the vehicle as well as the age of the purchaser. Is there a relationship between the selling price of a vehicle and the age of the purchaser? Would it be reasonable to conclude that the more expensive vehicles are purchased by older buyers?

41. Describing Relationship between Two Variables – Scatter Diagram Excel Example

42. Contingency Tables • A scatter diagram requires that both of the variables be at least interval scale. • What if we wish to study the relationship between two variables when one or both are nominal or ordinal scale? In this case we tally the results in a contingency table.

43. Contingency Tables – An Example A manufacturer of preassembled windows produced 50 windows yesterday. This morning the quality assurance inspector reviewed each window for all quality aspects. Each was classified as acceptable or unacceptable and by the shift on which it was produced. Thus we reported two variables on a single item. The two variables are shift and quality. The results are reported in the following table.

44. In-Class Exercises and Homework (scatter diagrams and contingency tables) • In class: P. 121 No. 23 and 25 • Homework: No. 24 and 26

45. More Homework • Homework: No. 37,42,43

46. End of Chapter 4