Chapter 4: Displaying Quantitative Data

This chapter revolves around four types of plots: histograms, stem-and-leaf displays, dotplots, and timeplots. In the notes that follow, I will concentrate on instructions for entering data into the TI-83 or TI-84 and then generating a histogram from that data, and generating a histogram using Data Desk. Stem-and-leaf displays are most easily drawn by hand; see the links below for more information on constructing these. ActivStats provides a lesson on dotplots. We'll look at timeplots a bit more when we reach Chapter 7.

Property taxes

For our first example with a quantitative variable, let's revisit the data about the single-family residences on the street where I live in Edmonds, first presented in the Chapter 2 Resources. For reference, here is the data one more time:

house	size	assess	lot	taxes	stories
20911	1561	304.0	0.20	2604	1
20912	1038	297.6	0.20	280	1
20918	1224	289.5	0.17	2353	1
20921	1232	292.8	0.17	756	1
20924	1995	314.6	0.17	2620	2
20927	1714	322.7	0.18	2632	1
20930	1832	336.1	0.18	2779	2
21003	1095	279.0	0.18	2321	1
21006	2011	319.5	0.18	2663	2
21015	1366	289.3	0.18	2415	1
21018	1292	301.4	0.18	2477	1
21023	1458	314.3	0.18	1386	1
21028	2031	320.9	0.18	2676	2
21105	1366	304.0	0.18	2473	1

Let's graph one of the quantitative variables. We'll start with the 2007 assessed value.

Histograms with the TI-83 and TI-84

Before we can graph data using the calculator, we need to enter the data into a list. The TI-83 and TI-84 have six built-in lists, called L1, L2, L3, L4, L5 and L6. To access the lists, press STAT:

(To complicate matters there is also a LIST menu above the STAT key, but we don't want to use this right now, so ignore it for the time being.)

Next press ENTER and you should be in the list editor:

Use the arrow keys to move up and down in a list, or the left and right arrows to move from one list to another.

If a list you want to use already contains data and you want to clear the list, move the cursor up so that the list name (e.g. L1) is highlighted:

then press CLEAR and ENTER; all of the data in the list should disappear. DO NOT press the DEL key when the list name is highlighted: this will delete the entire list, rather than its contents. To restore a deleted list move the cursor to the name of another list, press 2ND and then INS (for insert, above the DEL key), then press 2ND and then L1 (above the 1 key, or another key for the name of the list you want to insert). You can, however, use the DEL key to delete a single entry in a list.

To enter data once the entries in the list have been cleared, move the cursor to the first position in the list, type a number, then press ENTER. Repeat this until you have entered all of the data. If you enter the 2007 assessed value data into list L1 on your TI-84 (from here on out I'll just write "TI-84" so that I don't have to type "TI-83 or TI-84" repeatedly) your screen should look something like this:

Notice that when you move the cursor over the last entry in the list it says L1(14) = 304. This means that the 14th entry in list L1 is 304. Since there are 14 houses in the data set, this is what we want to see.

To exit the list editor, press 2ND then QUIT (above the MODE key).

We can now draw a histogram with the TI-84 using the assessed value data into list L1. Press 2ND then STAT PLOT (above the Y= key) to get to the STAT PLOTS menu:

then press ENTER to select Plot1:

Move the cursor over On and press ENTER to turn on Plot1:

Now move the cursor down and then left two spaces so that it highlights the histogram icon, then press ENTER:

If Xlist is not already set to L1 (or the name of the list with your data), move the cursor down, then type L1 (2ND and then the 1 key) and press ENTER to designate L1 as the Xlist.

If Freq is not automatically set to 1 (it usually is) then you'll need to reset it to 1. For some reason the TI-84 defaults to the alpha-lock when this entry is highlighted; if you need to change the frequency to 1 from something else, you'll need to press ALPHA to turn off the alpha-lock, then type the number 1. Your screen should then look something like what you see above.

Now press ZOOM, then move the cursor down to ZoomStat:

and press ENTER. This is usually the most expedient way to choose an appropriate window, but you may still need to adjust the window slightly. Here is the histogram you should get using ZoomStat:

To manually adjust the window settings, press the WINDOW button, then enter suitable values and press GRAPH. You can experiment with different WINDOW values, which may give you different histograms. If you change the WINDOW values as follows:

and press GRAPH you should get a histogram that looks like this:

Note that we sometimes get vastly different histograms just by adjusting the starting point and the bar width. We might say that the above histogram appears bimodal, while the original histogram appeared to have an outlier that is not visible in our second histogram.

Note that there is no scale indicated in this histogram, nor is the horizontal axis labeled with a variable name or units. Whenever we draw a histogram we should include all of these labels if technology permits.

Histograms from frequency tables

On occasion, quantitative data is displayed in a frequency tables, much like those we created for categorical variables in Chapter 3. This ussually occurs when the quantitative variable only takes on discrete values, or is conveniently rounded to an integer.

For example, in Fall 2006 students in my online class were given an unlimited number of attempts to take a 5-point quiz. We could display the number of attempts like this:

attempts	count
0	3
1	8
2	8
3	4
4	2
5	2
6	1

If we wanted to enter this data into the TI-84 to create a histogram, we could enter 0 three times, then 1 eight times, then 2 eight times, and so on into a list until the list had 28 entries, one for each student. But there's another way. We can enter the number of attempts (the left column) into one list (say L1) and the counts into the next list (say L2):

We then follow the same steps as before, except we type L2 for Freq in the STAT PLOT menu:

To get a histogram we can use ZoomStat:

Of course, it would have been much simpler in this case to create a stem-and-leaf display by hand:

0	000
0	11111111
0	22222222
0	3333
0	44
0	55
0	6	Key: 0|6 means 6 attempts

Notice that if you rotate your head to the right this stem-and-leaf display appears to have the same shape as the histogram. The advantage is that all of the original data values are still apparent.

Histograms with Data Desk

If you want to use a computer to make a histogram, use Data Desk. (Excel has something that it calls a histogram, but it's not a histogram; while it is possible to make histograms with Excel, it's complicated and takes about 100 times longer than it does with Data Desk.)

First, save the houses.txt data file to your desktop, either by right-clicking the link in this sentence or grabbing it from the Data Sets folder.

Now start Data Desk, click File and Import..., navigate to houses.txt and click once on the file name, then click Open. Click Use these variable names; see the instructions in the Chapter 3 Resources for importing data into Data Desk. At this point you should see all of the variables from the houses.txt data set displayed like this:

Now click on the assess variable so that the variable's icon is designated as Y:

then click on Plot and select Histograms:

The following histogram should appear:

You can make the window larger by clicking on the lower right corner of the window and dragging it across the screen:

You can adjust the plot options by clicking on the hyperview menu (the little triangle icon in the upper-left corner of the histogram window) and selecting Plot Scale:

If we use 270 for Align bars at and 12 for Bar width:

we get this histogram:

which should remind you of one we created with the TI-84. You can also hold down the CTRL key and click and drag the icon in the lower-right corner of the histogram window to adjust the bar widths.

Note that Data Desk includes a scale on each axis and names the variable, which is at least better than the TI-84. The units and a label for the vertical axis are still missing. Something like this:

would be better, although I had to hack the histogram using Photoshop to include the new information along the axes.

As before, to copy and paste a Data Desk histogram to another application, make sure the histogram window is selected, then click Edit and Copy Window. Then go to another application (like Microsoft Word) and paste.

Try using the TI-84 and Data Desk to create displays of the other quantitative variables in the house data set.

Positively Skewed vs. Negatively Skewed

Once we have made a histogram (or stemplot or dotplot), we want to be able to describe the shape, center and spread of the graph. In Chapter 4 we concentrate on the shape (uniform, unimodal, bimodal; symmetric, skewed to the right, skewed to the left); we'll discuss more precise ways of measuring the center and spread in the next chapter.

Sometimes we use the term "positively skewed" in place of "skewed to the right", and "negatively skewed" in place of "skewed to the left"; the terms "positive" and "negative" are especially appropriate when a histogram or stem-and-leaf display is constructed with the quantitative variable along the vertical axis rather than the horizontal axis, as is the case with this stem-and-leaf display of the number of attempts made on quiz by my students during Fall 2006:

0	6
0	55
0	44
0	3333
0	22222222
0	11111111
0	000	Key: 0|6 means 6 attempts

Notice that this is the same data as in the previous stem-and-leaf example, but here I've put the higher numbers at the top and the lower numbers at the bottom. We would call this data set unimodal and positively skewed, since the longer tail is skewed in the positive direction. It would not be wrong to say that the data is skewed to the right, but since there is no right or left here (just up and down), the phrase "positively skewed" is more precise.

Homework

Work the following problems in Chapter 4: 5, 7, 9, 13, 15, 17, 23, 25, 27 and 29. (As usual, you are encouraged to work additional problems.) You can safely ignore the discussion about "Re-expressing Skewed Data to Improve Symmetry" on pp. 56-57 for now, although be aware that the some of the unassigned exercises (e.g. 39-42) relate to this topic. If you have time, though, you might want to at least read about re-expression; these ideas will return in some later chapters (including all of Chapter 10) but all such material will remain optional and will not appear on the exams (except possibly as extra credit).

Errata

There is a typo on page 48: the top row of the first stem-and-leaf display should read:

8|000448

rather than:

8|0004444

(Somewhat less importantly, the first line in the last sidebar on the previous page should read "stem-and-leaf display" rather than "Stem-and-Leaf display.")

ActivStats

Work the activities on pages 4-1 through 4-3 in the ActivStats lesson book, as time permits. These contain further information about constructing and using stem-and-leaf displays, as well as dotplots and histograms.

Quiz

The Chapter 4 Quiz offers additional practice with describing the shapes of distributions displayed in histograms and stem-and-leaf displays.

Additional Resources

Picturing Distributions

Episode 2 from Against All Odds features a discussion of histograms and stemplots.

Decisions Through Data: Stemplots

Unit 2 of Decisions Through Data talks about stemplots

Carnegie Mellon: Introduction to Statistics

This open source course has a lesson called "One Categorical Variable" that may be of interest (see Unit 2, Module 1).

Sofia: Elementary Statistics

Lesson 2.1 of the Sofia Open Content Initiative's Elementary Statistics course includes a discussion of stemplots and histograms (ignore the discussion of boxplots until you reach Chapter 5).

Histogram tool

A Java applet for creating histograms. View histograms for built-in or user-specified data, and experiment with how the size of the class intervals influences the appearance of a histogram.

Histogram applet

A Java applet that allows you to play with histogram bin widths.

Stem-and-leaf tool

A Java applet for creating stem-and-leaf displays.

Stemplot tutorial

A detailed tutorial on constructing stem-and-leaf displays, from Statistics Canada.

IDB Population Pyramids

This interactive Web site from census.gov allows you to create population pyramids for any country and any year (and even animate the changes).

TI-83 Resource: Histograms

Instructions on creating a histogram with the TI-83; check out the link about entering data into lists if you having difficulty with that part of the process.

Simple Histograms on the TI-83/84

Another set of instructions for creating a histogram on the TI-83. Contrary to what the author of this page states, go ahead and use ZoomStat.

TI-83/84 Troubleshooting

Guide to some common errors encountered when using the TI-84.