Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CHAPTER 4 EXPLORING QUANTITATIVE DATA Slide 4- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Think Before You Draw, Again Remember the “Make a picture” rule? Now that we are adding to our repertoire of 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: Be sure that the data are values of a quantitative variable whose units are known.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Histogram Histograms are used to show the frequency of data. Very similar to bar graphs, but use intervals on the X axis. Bars do touch. Histograms have a title. Histograms have two axes which are labeled.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The histogram is a tool for presenting the distribution of a quantitative variable in graphical form. For example, suppose the following data is the number of hours worked in a week by a group of nurses:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The vertical axis is frequency. So, for example, there are two nurses who worked from 25 to less than 30 hours that week. The data values are grouped in intervals of width five hours. The first interval includes the values from 25 to less than 30 hours. The second interval includes values from 30 to less than 35 and so on. The intervals are shown on the horizontal axis These data are displayed in the following histogram:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The choice of interval width will affect the appearance of the histogram. Here it is again, to the right, presented in a histogram of interval width of 10 and then 2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 – Here is a histogram of heart rates. Describe what this histogram tells you. Figure 2 Histogram of heart rate data cont’d

+ 8 1)Divide the range of data into classes of equal width. Class width is equal to the Range divided by # of desired bars and then bounce that quotient up one integer. 2)Find the count (frequency) or percent (relative frequency) of individuals in each class. 3)Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals. Displaying Quantitative Data How to Make a Histogram

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Making a Histogram This table presents data on the percent of residentsfrom each state who were born outside of the U.S. Displaying Quantitative Data Frequency Table ClassCount 0 to <520 5 to < to < to < to < to <301 Total50 Percent of foreign-born residents Number of States

+ 10 Using the TI-83 to make histograms The TI-83 can be used to make histograms, and will allow you to change the location and widths of the ranges. See TI-Tips Page 47 in your textbook.

+ 11 Using the TI-83 to make histograms Start by entering data into a list Example: Enter the data from P. 47 into any list; in this case, we will use L 1

+ 12 Using the TI-83 to make histograms Choose 2 nd :Stat Plot to choose a histogram plot Caution: Watch out for other plots that might be “turned on” or equations that might be graphed

+ 13 Using the TI-83 to make histograms Turn the plot “on”, Choose the histogram plot. Xlist should point to the location of the data.

+ 14 Using the TI-83 to make histograms Under the “Zoom” menu, choose option 9: ZoomStat

+ 15 Using the TI-83 to make histograms The result is a histogram where the calculator has decided the width and location of the ranges You can use the Trace key to get information about the ranges and the frequencies

+ 16 Using the TI-83 to make histograms You can change the size and location of the ranges by using the Window button Use the scale key to change the number of classes. Enter the CLASS WIDTH. Press the Graph button to see the results

+ 17 Using the TI-83 to make histograms Voila! Of course, you can still change the ranges if you don’t like the results.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide In Summary: Histograms are useful when dealing with a large set of data and you wish to organize that data to show frequencies. This histogram is displaying the number of employees and what salary they earn in the thousands.

Larson/Farber 4th ed. 19 Graphing Quantitative Data Sets Stem-and-leaf plot Each number is separated into a stem and a leaf. Similar to a histogram, however it still contains original data values. Larson/Farber 4th ed. 19 Data: 21, 25, 25, 26, 27, 28, 30, 36, 36,

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Stem-and-Leaf Example This is a stem-and-leaf display for the pulse rates of 24 women at a health clinic. How do you read this display? 8/8= 88 b/m

Larson/Farber 4th ed. 21 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?

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Larson/Farber 4th ed. 23 Example: Constructing a Stem-and-Leaf Plot The following are the numbers of text messages sent last month by the cellular phone users on one floor of a college dormitory. Display the data in a stem-and-leaf plot. Larson/Farber 4th ed

Larson/Farber 4th ed. 24 Solution: Constructing a Stem-and-Leaf Plot Larson/Farber 4th ed. 24 The data entries go from a low of 78 to a high of 159. Use the rightmost digit as the leaf.  For instance, 78 = 7 | 8 and 159 = 15 | 9 List the stems, 7 to 15, to the left of a vertical line. For each data entry, list a leaf to the right of its stem

Larson/Farber 4th ed. 25 Solution: Constructing a Stem-and-Leaf Plot Larson/Farber 4th ed. 25 Include a key to identify the values of the data. From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Shape, Center, and Spread When describing a distribution, make sure to always tell about three things: shape, center, and spread…

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What is the Shape of the Distribution? 1.Does the histogram have a single, central mountain-like bump or several separated bumps? 2.Is the histogram symmetric? 3.Do any unusual features stick out?

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Modes of a Histogram 1.Does the histogram have a single, central Bump or several separated bumps? Bumps 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Humps and Bumps (cont.) A bimodal histogram has two apparent peaks:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Anything Unusual? (cont.) The following histogram has outliers—there are three cities in the leftmost bar:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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? Shortly we will talk in more detail about spread…

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Comparing Distributions (cont.) Compare the following distributions of ages for female and male heart attack patients:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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).