1. Analyzing One-Variable Data
Lesson 1.5
Displaying Quantitative Data: Histograms
Statistics and Probability with Applications, 3rd Edition
Starnes, Tabor
Bedford Freeman Worth Publishers

2. Displaying Quantitative Data: Histograms
Learning Targets
After this lesson, you should be able to:
Make histograms of quantitative data.
Interpret histograms.
Compare distributions of quantitative data with histograms.

3. Displaying Quantitative Data: Histograms
You can use a dotplot or stemplot to display quantitative data. Both graphs show every individual data value. For large data sets, this can make it difficult to see the overall pattern in the graph. We often get a cleaner picture of the distribution by grouping nearby values together. Doing so allows us to make a new type of graph: a histogram.
Histogram
A histogram shows each interval as a bar. The heights of the bars show the frequencies or relative frequencies of values in each interval.

4. Displaying Quantitative Data: Histograms
(a) Dotplot and (b) histogram of the duration (in minutes) of 220 eruptions of the Old Faithful geyser.

5. Displaying Quantitative Data: Histograms
How to Make a Histogram
Choose equal-width intervals that span the data. Five intervals is a good minimum.
Make a table that shows the frequency (count) or relative frequency (percent or proportion) of individuals in each interval.
Draw and label the axes. Put the name of the quantitative variable under the horizontal axis. To the left of the vertical axis, indicate if the graph shows the frequency or relative frequency of individuals in each interval.
Scale the axes. Place equally spaced tick marks at the smallest value in each interval along the horizontal axis. On the vertical axis, start at 0 and place equally spaced tick marks until you exceed the largest frequency or relative frequency in any interval.
Draw bars above the intervals. Make the bars equal in width and leave no gaps between them. The height of each bar corresponds to the frequency or relative frequency of individuals in that interval. An interval with no data values will appear as a bar of height 0 on the graph.

6. For example, suppose the following data is the number of hours worked in a week by a group of nurses:

7. These data are displayed in the following histogram:
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. 26 28 30 35 36 37 38 39 40 41 42 43 45 47 48 50 53 72

8. 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 43 in your textbook. 8

9. 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 L1 9

10. Using the TI-83 to make histograms
Choose 2nd:Stat Plot to choose a histogram plot
Caution: Watch out for other plots that might be “turned on” or equations that might be graphed 10

11. Using the TI-83 to make histograms
Turn the plot “on”,
Choose the histogram plot.
Xlist should point to the location of the data. 11

12. Using the TI-83 to make histograms
Under the “Zoom” menu, choose
option 9: ZoomStat 12

13. 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 13

14. Displaying Quantitative Data: Histograms
The choice of intervals in a histogram can affect the appearance of a distribution. The figure below shows two different histograms of the foreign-resident data.
The one on the left uses the intervals of width 5 from the previous example.
The one on the right uses intervals half as wide: 0 to <2.5, 2.5 to <5, and so on.
Histograms with more intervals show more detail but may have a less clear overall pattern.

15. 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 15

16. Using the TI-83 to make histograms
Voila!
Of course, you can still change the ranges if you don’t like the results. 16

17. 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. 

18. Shape, Center, and Spread
When describing a distribution, make sure to always tell about three things: shape, center, and spread…
What is the Shape of the Histogram?
Does the histogram have a single, central mountain-like bump or several separated bumps?
Is the histogram symmetric?
Do any unusual features stick out?

19. Modes of a Histogram
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.

20. Humps and Bumps (cont.)
A bimodal histogram has two apparent peaks:

21. 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:

22. Symmetry 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.

23. 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.

24. Anything Unusual? 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.

25. Anything Unusual? (cont.)
The following histogram has outliers—there are three cities in the leftmost bar:

26. Displaying Quantitative Data: Histograms
Histograms can be used to compare the distribution of a quantitative variable in two or more groups.
It’s a good idea to use relative frequencies (percents or proportions) when comparing, especially if the groups have different sizes.
Be sure to use the same intervals when making comparative histograms so the graphs can be drawn using a common horizontal axis scale.
Is it true that students who graduate from high school earn more money than students who do not graduate from high school?

27. How old are U.S. presidents?
LESSON APP 1.5
How old are U.S. presidents?
The table gives the ages of the first 44 U.S. presidents when they took office.
President
Age
Washington 57
Taylor 64
B. Harrison 55
Eisenhower 61
J. Adams
Fillmore 50
Cleveland
Kennedy 43
Jefferson
Pierce 48
McKinley 54
L. B. Johnson
Madison
Buchanan 65
T. Roosevelt 42
Nixon 56
Monroe 58
Lincoln 52
Taft 51
Ford
J. Q. Adams
A. Johnson
Wilson
Carter
Jackson
Grant 46
Harding
Reagan 69
Van Buren
Hayes
Coolidge
G. H. W. Bush
W. H. Harrison 68
Garfield 49
Hoover
Clinton
Tyler
Arthur
F. D. Roosevelt
G. W. Bush
Polk 47
Truman 60
Obama
Make a frequency histogram of the data using intervals of width 4 starting at age 40.
Describe the shape of the distribution.
What percent of presidents took office before the age of 60?

28. Displaying Quantitative Data: Histograms
Learning Targets
After this lesson, you should be able to:
Make histograms of quantitative data.
Interpret histograms.
Compare distributions of quantitative data with histograms.