Histograms2.42001/06/292003/06/20 10:44:20.327 GMT-5DavidLanelane@rice.eduDavidLanelane@rice.eduAdanGalvanjago@rice.eduhistogramsclass intervalsIntroduction to histograms.
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. Frequency Table: A table containing the number
of occurrences in each class of data; for example, the number of
each color of M&Ms in a bag. Frequency tables often used to
create histograms and frequency polygons. When a frequency
table is created for a quantitative variable, a grouped
frequency table is generally used. Unfortunately a simple
frequency table would be too big, containing over 100 rows. To
simplify the table, we group scores together as shown in .
Grouped Frequency Distribution of Psychology Test ScoresInterval's Lower LimitInterval's Upper LimitClass Frequency39.549.5349.559.51059.569.55369.579.510779.589.514789.599.513099.5109.578109.5119.559119.5129.536129.5139.511139.5149.56149.5159.51159.5169.51
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 is
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 .
Histogram of scores on a psychology test.
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 on the left.
The distribution is therefore said to be skewedA distribution is skewed if one tail
extends out further than the other.. (We'll have more to
say about the shape of distributions in our discussion on
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.)
Sturgis' rule is to set the number of intervals as
close as possible to
12N, where
2N
is the base 2 log of the number of observations. The formula
can also be written as
13.310N
where
10N
is the log base 10 of the number of observations. According to
Sturgis' rule, 1000 observations would be graphed with 11 class
intervals since 10 is the closest integer to
21000. We prefer the Rice 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 the
Sturgis' rule. For the psychology test example used above,
Sturgis' rule recommends 10 intervals while the Rice rule
recommends 17. In the end, we compromised and chose 13
intervals for 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.
To provide experience in constructing histograms, we have
developed an interactive demonstration. The demonstration
reveals the consequences of different choices of bin width and
of lower boundary for the first interval.
Interactive histogram