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Histograms
Author(s)
David M. Lane and David Scott
Prerequisites
Distributions,
Graphing Qualitative Data
Learning Objectives
- Create a grouped frequency distribution
- Create a histogram based on a grouped frequency distribution
- Determine an appropriate bin width
A histogram is a graphical method for displaying
the shape of a distribution. It is particularly useful when there
are a large number of observations. We begin with an example consisting
of the scores of 642 students on a psychology test. The test consists
of 197 items, each graded as "correct" or "incorrect."
The students' scores ranged from 46 to 167.
The first step is to create a frequency
table. Unfortunately, a simple frequency table would be too
big, containing over 100 rows. To simplify the table, we group
scores together as shown in Table 1.
Table 1. Grouped Frequency Distribution
of Psychology Test Scores
| Interval's Lower Limit |
Interval's Upper Limit |
Class Frequency |
| 39.5 |
49.5 |
3 |
| 49.5 |
59.5 |
10 |
| 59.5 |
69.5 |
53 |
| 69.5 |
79.5 |
107 |
| 79.5 |
89.5 |
147 |
| 89.5 |
99.5 |
130 |
| 99.5 |
109.5 |
78 |
| 109.5 |
119.5 |
59 |
| 119.5 |
129.5 |
36 |
| 129.5 |
139.5 |
11 |
| 139.5 |
149.5 |
6 |
| 149.5 |
159.5 |
1 |
| 159.5 |
169.5 |
1 |
To create this table, the range of scores was broken
into intervals, called
class
intervals.
The first interval is from 39.5 to 49.5,
the second from 49.5 to 59.5, etc. Next, the number of scores
falling into each interval was counted to obtain the
class
frequencies. There are three scores in the first interval,
10 in the second, etc.
Class intervals of width 10 provide enough detail
about the distribution to be revealing without making the graph
too "choppy." More information on choosing the widths
of class intervals is presented later in this section. Placing
the limits of the class intervals midway between two numbers (e.g.,
49.5) ensures that every score will fall in an interval rather
than on the boundary between intervals.
In a histogram, the class frequencies are represented
by bars. The height of each bar corresponds to its class frequency.
A histogram of these data is shown in Figure 1.
The histogram makes it plain that most of the scores
are in the middle of the distribution, with fewer scores in the
extremes. You can also see that the distribution is not symmetric:
the scores extend to the right farther than they do to the left.
The distribution is therefore said to be
skewed. (We'll have more to say about shapes of
distributions in the chapter " Summarizing Distributions.")
In our example, the observations are whole numbers.
Histograms can also be used when the scores are measured on a
more continuous scale such as the length of time (in milliseconds)
required to perform a task. In this case, there is no need to
worry about fence-sitters since they are improbable. (It would
be quite a coincidence for a task to require exactly 7 seconds,
measured to the nearest thousandth of a second.) We are therefore
free to choose whole numbers as boundaries for our class intervals,
for example, 4000, 5000, etc. The class frequency is then the
number of observations that are greater than or equal to the lower
bound, and strictly less than the upper bound. For example, one
interval might hold times from 4000 to 4999 milliseconds. Using
whole numbers as boundaries avoids a cluttered appearance, and
is the practice of many computer programs that create histograms.
Note also that some computer programs label the middle of each
interval rather than the end points.
Histograms can be based on relative
frequencies instead of actual frequencies. Histograms based
on relative frequencies show the proportion of scores in each
interval rather than the number of scores. In this case, the Y-axis runs from 0 to 1 (or somewhere in between if there are no
extreme proportions). You can change a histogram based on frequencies
to one based on relative frequencies by (a) dividing each class
frequency by the total number of observations, and then (b) plotting
the quotients on the Y-axis (labeled as proportion).
There is more to be said about the widths of the class
intervals, sometimes called bin widths.
Your choice of bin width determines the number of class intervals.
This decision, along with the choice of starting point for the
first interval, affects the shape of the histogram. There are
some "rules of thumb" that can help you choose an appropriate
width. (But keep in mind that none of the rules is perfect.) Sturges'
rule is to set the number of intervals as close as possible
to 1 + Log2(N), where Log2(N) is the base
2 log of the number of
observations. The formula can also be written as 1 + 3.3 Log10(N),
where Log10(N) is the log base 10 of the number of
observations. According to Sturges' rule, 1000 observations would
be graphed with 11 class intervals since 10 is the closest integer
to Log2(1000). We prefer the Rice rule (also known as Scott's rule), which is to
set the number of intervals to twice the cube root of the number
of observations. In the case of 1000 observations, the Rice rule
yields 20 intervals instead of the 11 recommended by Sturges'
rule. For the psychology test example used above, Sturges' rule
recommends 10 intervals while the Rice rule recommends 17. In
the end, we compromised and chose 13 intervals for Figure 1 to
create a histogram that seemed clearest.
The best advice is to experiment with different choices of width,
and to choose a histogram according to how well it communicates
the shape of the distribution.
References
Scott, D.W. On Optimal and Data-Based Histograms. Biometrika, 66:605-610, 1979.
Scott, D.W. (2010). Scott's Rule. Wiley Interdisciplinary
Reviews: Computational Statistics, 2, 497--502.
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