1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1

2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 4 Displaying Quantitative Data

3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Dealing With a Lot of Numbers… Summarizing the data will help us when we look at large sets of quantitative data. Without summaries of the data, it’s hard to grasp what the data tell us. The best thing to do is to make a picture… We can’t use bar charts or pie charts for quantitative data, since those displays are for categorical variables.

4. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Histograms: Displaying the Distribution of Price Changes The chapter example discusses the changes in Enron’s stock price from 1997 – 2001. First, slice up the entire span of values covered by the quantitative variable into equal-width piles called bins. The bins and the counts in each bin give the distribution of the quantitative variable.

5. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Histograms: Displaying the Distribution of Price Changes (cont.) A histogram plots the bin counts as the heights of bars (like a bar chart). Here is a histogram of the monthly price changes in Enron stock:

6. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Histograms: Displaying the Distribution of Price Changes (cont.) A relative frequency histogram displays the percentage of cases in each bin instead of the count. In this way, relative frequency histograms are faithful to the area principle. Here is a relative frequency histogram of the monthly price changes in Enron stock:

7. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Be Careful A histogram gives the number of data points that fall into equal intervals. Care must be taken in choosing the intervals because it can affect the shape of the graph and misrepresent the true data.

8. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 1 st graph The first graph is uses intervals of size 10 yielding the intervals 40-49, 50-59, etc.

9. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 2 nd Graph In the second graph, the intervals are 40-45, 45- 50, 50-55, etc. This affects the shape of the graph.

10. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10 Histogram A histogram is more convenient than a dot-plot or a stem and leaf plot because you don't have to represent each data point. However, you don't get to see the value of each data point. So a table of data and summary statistics would help people interpret the data.

11. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11 Stem-and-Leaf Displays Stem-and-leaf displays show the distribution of a quantitative variable, like histograms do, while preserving the individual values. Stem-and-leaf displays contain all the information found in a histogram and, when carefully drawn, satisfy the area principle and show the distribution.

12. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12 Stem-and-Leaf Example Compare the histogram and stem-and-leaf display for the pulse rates of 24 women at a health clinic. Which graphical display do you prefer?

13. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13 Stem and Leaf Displays Shows quantitative data values in a way that sketches the distribution of the data. The stem-and-leaf plot below shows the number of students enrolled in a dance class in the past 12 years. The number of students are 81, 84, 85, 86, 93, 94, 97, 100, 102, 103, 110, and 111.

14. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14 Constructing a Stem-and-Leaf Display First, cut each data value into leading digits (“stems”) and trailing digits (“leaves”). Use the stems to label the bins. Use only one digit for each leaf—either round or truncate the data values to one decimal place after the stem.

15. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15 Dotplots A dotplot is a simple display. It just places a dot along an axis for each case in the data. The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot. You might see a dotplot displayed horizontally or vertically.

16. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16 Dotplot Graphs a dot for each case against a single axis Graph the following number 5, 5,5,5,5,5,5,10,10,10,10,10 etc

17. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17 Dotplot with two sets of data Example

18. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18 Think Before You Draw, Again Remember the “Make a picture” rule? Now that we have options for data displays, you need to Think carefully about which type of display to make. Before making a stem-and-leaf display, a histogram, or a dotplot, check the Quantitative Data Condition: The data are values of a quantitative variable whose units are known.

19. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19 Shape, Center, and Spread When describing a distribution, make sure to always tell about three things: shape, center, and spread…

20. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20 What is the Shape of the Distribution? 1.Does the histogram have a single, central hump or several separated bumps? 2.Is the histogram symmetric? 3.Do any unusual features stick out?

21. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 21 Humps and Bumps 1.Does the histogram have a single, central hump or several separated bumps? Humps in a histogram are called modes. A histogram with one main peak is dubbed unimodal; histograms with two peaks are bimodal; histograms with three or more peaks are called multimodal.

22. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 Unimodal Having one mode.

23. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 23 Bimodal Distribution with two modes

24. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 24 Humps and Bumps (cont.) A bimodal histogram has two apparent peaks:

25. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 25 Humps and Bumps (cont.) A histogram that doesn’t appear to have any mode and in which all the bars are approximately the same height is called uniform:

26. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 26 Symmetry 2.Is the histogram symmetric? If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, the histogram is symmetric.

27. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 27 Symmetrical Symmetrical Histogram

28. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 28 Dotplot Dots are mirrored images

29. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 29 Stem and leaf Example

30. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 30 Symmetry (cont.) The (usually) thinner ends of a distribution are called the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail. In the figure below, the histogram on the left is said to be skewed left, while the histogram on the right is said to be skewed right.

31. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 31 Examples Skewed right

32. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 32 Skewed left Left

33. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 33 All three Examples

34. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 34 Funny example http://www.herkimershideaway.org/apstatistics/y mmsum99/ymm111.htm

35. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 35 Anything Unusual? 3.Do any unusual features stick out? Sometimes it’s the unusual features that tell us something interesting or exciting about the data. You should always mention any stragglers, or outliers, that stand off away from the body of the distribution. Are there any gaps in the distribution? If so, we might have data from more than one group.

36. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 36 Anything Unusual? (cont.) The following histogram has outliers—there are three cities in the leftmost bar:

37. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 37 Outliers The following histogram has outliers— there are three cities in the leftmost bar:

38. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 38 Where is the Center of the Distribution? If you had to pick a single number to describe all the data what would you pick? It’s easy to find the center when a histogram is unimodal and symmetric—it’s right in the middle. On the other hand, it’s not so easy to find the center of a skewed histogram or a histogram with more than one mode. For now, we will “eyeball” the center of the distribution. In the next chapter we will find the center numerically.

39. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 39 How Spread Out is the Distribution? Variation matters, and Statistics is about variation. Are the values of the distribution tightly clustered around the center or more spread out? In the next two chapters, we will talk about spread…

40. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 40 Comparing Distributions Often we would like to compare two or more distributions instead of looking at one distribution by itself. When looking at two or more distributions, it is very important that the histograms have been put on the same scale. Otherwise, we cannot really compare the two distributions. When we compare distributions, we talk about the shape, center, and spread of each distribution.

41. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 41 Comparing Distributions (cont.) Compare the following distributions of ages for female and male heart attack patients:

42. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 42 Timeplots: Order, Please! For some data sets, we are interested in how the data behave over time. In these cases, we construct timeplots of the data.

43. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 43 *Re-expressing Skewed Data to Improve Symmetry Figure 4.11

44. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 44 *Re-expressing Skewed Data to Improve Symmetry (cont.) One way to make a skewed distribution more symmetric is to re-express or transform the data by applying a simple function (e.g., logarithmic function). Note the change in skewness from the raw data (Figure 4.11) to the transformed data (Figure 4.12): Figure 4.12

45. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 45 What Can Go Wrong? Don’t make a histogram of a categorical variable— bar charts or pie charts should be used for categorical data. Don’t look for shape, center, and spread of a bar chart.

46. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 46 What Can Go Wrong? (cont.) Don’t use bars in every display—save them for histograms and bar charts. Below is a badly drawn timeplot and the proper histogram for the number of eagles sighted in a collection of weeks:

47. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 47 What Can Go Wrong? (cont.) Choose a bin width appropriate to the data. Changing the bin width changes the appearance of the histogram:

48. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 48 What Can Go Wrong? (cont.) Avoid inconsistent scales, either within the display or when comparing two displays. Label clearly so a reader knows what the plot displays. Good intentions, bad plot:

49. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 49 What have we learned? We’ve learned how to make a picture for quantitative data to help us see the story the data have to Tell. We can display the distribution of quantitative data with a histogram, stem-and-leaf display, or dotplot. Tell about a distribution by talking about shape, center, spread, and any unusual features. We can compare two quantitative distributions by looking at side-by-side displays (plotted on the same scale). Trends in a quantitative variable can be displayed in a timeplot.