1. Slide 4-1 Copyright © 2004 Pearson Education, Inc. 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 is difficult to grasp what the data tell us. In this chapter, we concentrate on graphical displays of quantitative data.

2. Slide 4-2 Copyright © 2004 Pearson Education, Inc. Distributions and Histograms 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.

3. Slide 4-3 Copyright © 2004 Pearson Education, Inc. Distributions and Histograms (cont.) A histogram plots the bin counts as the heights of bars (like a bar chart). 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.

4. Slide 4-4 Copyright © 2004 Pearson Education, Inc. Histogram Example The figure shows the first 36 months of Enron monthly stock price changes. (Later, we will examine these same data in something called a stem-and-leaf display.)

5. Slide 4-5 Copyright © 2004 Pearson Education, Inc. 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.

6. Slide 4-6 Copyright © 2004 Pearson Education, Inc. 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.

7. Slide 4-7 Copyright © 2004 Pearson Education, Inc. Stem-and-Leaf Example In the figure, 2|124 stands for the numbers $2.1, $2.2, and $2.4. –The stem tells us we are in the $2 range. –Each leaf gives us the Enron stock price change to the nearest dime.

8. Slide 4-8 Copyright © 2004 Pearson Education, Inc. Dotplots A dotplot is a simple display. It just places a dot for each case in the data. The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot.

9. Slide 4-9 Copyright © 2004 Pearson Education, Inc. Shape, Center, and Spread When describing a distribution, make sure to always tell about three things: shape, center, and spread…

10. Slide 4-10 Copyright © 2004 Pearson Education, Inc. The Shape of the Distribution When talking about the shape of the distribution, make sure to address the following three questions: 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?

11. Slide 4-11 Copyright © 2004 Pearson Education, Inc. 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.

12. Slide 4-12 Copyright © 2004 Pearson Education, Inc. Humps and Bumps (cont.) A bimodal histogram has two apparent peaks:

13. Slide 4-13 Copyright © 2004 Pearson Education, Inc. 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:

14. Slide 4-14 Copyright © 2004 Pearson Education, Inc. 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.

15. Slide 4-15 Copyright © 2004 Pearson Education, Inc. 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.

16. Slide 4-16 Copyright © 2004 Pearson Education, Inc. Anything Odd? 3.Do any unusual features stick out? –Believe it or not, 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.

17. Slide 4-17 Copyright © 2004 Pearson Education, Inc. Center and Spread Center: If you had to pick a single number to describe all the data what would you pick? Spread: Since statistics is about variation, how spread out is the distribution?

18. Slide 4-18 Copyright © 2004 Pearson Education, Inc. 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.

19. Slide 4-19 Copyright © 2004 Pearson Education, Inc. 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.

20. Slide 4-20 Copyright © 2004 Pearson Education, Inc. *Re-expressing Skewed Data Figure 4.12

21. Slide 4-21 Copyright © 2004 Pearson Education, Inc. *Re-expressing Skewed Data (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 4.12) to the transformed data (Figure 4.13): Figure 4.13

22. Slide 4-22 Copyright © 2004 Pearson Education, Inc. What Can Go Wrong? Don’t make a histogram of a categorical variable—bar charts or pie charts should be used for categorical data. Choose a scale appropriate to the data. Avoid inconsistent scales, either within the display or when comparing two displays. Label clearly so a reader knows what the plot displays.

23. Slide 4-23 Copyright © 2004 Pearson Education, Inc. Key Concepts Quantitative variables can be displayed using histograms, dotplots, and/or stem-and-leaf displays. These displays help us to see the distributions of the variables. Timeplots help us to see patterns in the data over time. Consider three things when looking at these displays: shape, center, and spread. Distributions can be classified as symmetric or skewed (look at how the tails behave with respect to the rest of the distribution).

24. Slide 4-24 Copyright © 2004 Pearson Education, Inc. Key Concepts (cont.) A mode is a hump or local high point in the shape of the distribution: –unimodal (one mode) –bimodal (two modes) –multimodal (more than two modes) –uniform (relatively flat, no mode) Be on the lookout for outliers (extreme values that stand off away from the bulk of the data).