David M. Lane and Heidi Ziemer
- Define "distribution"
- Interpret a frequency distribution
- Distinguish between a frequency distribution and a probability distribution
- Construct a grouped frequency distribution for a continuous
- Identify the skew of a distribution
- Identify bimodal, leptokurtic, and platykurtic distributions
Distributions of Discrete Variables
I recently purchased a bag of Plain M&M's.
The M&M's were in six different colors. A quick count showed
that there were 55 M&M's: 17 brown, 18 red, 7 yellow, 7 green,
2 blue, and 4 orange. These counts are shown below in Table 1.
Table 1. Frequencies in the
Bag of M&M's
This table is called a frequency
table and it describes the distribution of M&M color frequencies.
Not surprisingly, this kind of distribution is called a frequency
distribution. Often a frequency distribution is shown graphically
as in Figure 1.
Figure 1. Distribution of 55 M&M's.
The distribution shown in Figure 1 concerns just
my one bag of M&M's. You might be wondering about the distribution
of colors for all M&M's. The manufacturer of M&M's provides
some information about this matter, but they do not tell us exactly
how many M&M's of each color they have ever produced. Instead,
they report proportions rather than frequencies. Figure 2 shows
these proportions. Since every M&M is one of the six familiar
colors, the six proportions shown in the figure add to one. We
call Figure 2 a probability
distribution because if you choose an M&M at random, the
probability of getting, say, a brown M&M is equal to the proportion
of M&M's that are brown (0.30).
Figure 2. Distribution of all M&M's.
Notice that the distributions in Figures 1 and
2 are not identical. Figure 1 portrays the distribution in a
sample of 55 M&M's. Figure 2 shows the proportions for all
Chance factors involving the machines used by the manufacturer
introduce random variation into the different bags produced.
Some bags will have a distribution of colors that is close to
Figure 2; others will be further away.
The variable "color of M&M" used in
this example is a discrete
variable, and its distribution is also called discrete.
Let us now extend the concept of a distribution to continuous
The data shown in Table 2 are the times it took one
of us (DL) to move the cursor over a small target in a series
of 20 trials. The times are sorted from shortest to longest.
"time to respond" is a continuous variable. With time
measured accurately (to many decimal places), no two response
times would be expected to be the same. Measuring time in milliseconds
(thousandths of a second) is often precise enough to approximate
a continuous variable in psychology. As you can see in Table
2, measuring DL's responses this way produced times no two of
which were the same. As a result, a frequency distribution would
be uninformative: it would consist of the 20 times in the experiment,
each with a frequency of 1.
Table 2. Response Times
The solution to this problem is to create a grouped
frequency distribution. In a grouped frequency
distribution, scores falling within various ranges are tabulated.
Table 3 shows a grouped frequency distribution for these
Table 3. Grouped frequency distribution
Grouped frequency distributions can be portrayed
graphically. Figure 3 shows a graphical representation of the
frequency distribution in Table 3. This kind of graph is called
A later chapter contains an entire section devoted to histograms.
Figure 3. A histogram of the grouped frequency
distribution shown in Table 3. The labels on the X-axis
are the middle values of the range they represent.
The histogram in Figure 3 portrays just DL's 20
times in the one experiment he performed. To represent the probability
associated with an arbitrary movement (which can take any positive
amount of time), we must represent all these potential times
at once. For this purpose, we plot the distribution for the
continuous variable of time. Distributions for continuous variables
continuous distributions. They also
carry the fancier name probability
density. Some probability densities have particular
importance in statistics. A very important one is shaped like
a bell, and called the normal
distribution. Many naturally-occurring phenomena
can be approximated surprisingly well by this distribution.
It will serve to illustrate some features of all continuous
An example of a normal distribution is shown in
Figure 4. Do you see the "bell"? The normal distribution
doesn't represent a real bell, however, since the left and
right tips extend indefinitely (we can't draw them any further
so they look like they've stopped in our diagram). The Y-axis
in the normal distribution represents the "density of
Intuitively, it shows the chance of obtaining values near corresponding
points on the X-axis. In Figure 4, for example, the probability
of an observation with value near 40 is about half of the probability
of an observation with value near 50. (For more information, please see the chapter on normal distributions.)
Although this text does
not discuss the concept of probability density in detail, you
should keep the following ideas in mind about the curve that
describes a continuous distribution (like the normal distribution).
First, the area under the curve equals 1. Second, the probability
of any exact value of X is 0. Finally, the area under the curve
and bounded between two given points on the X-axis is the probability
that a number chosen at random will fall between the two points.
Let us illustrate with DL's hand movements. First, the probability
that his movement takes some amount of time is one! (We exclude
the possibility of him never finishing his gesture.) Second,
the probability that his movement takes exactly 598.956432342346576
milliseconds is essentially zero. (We can make the probability
as close as we like to zero by making the time measurement more
and more precise.) Finally, suppose that the probability of
DL's movement taking between 600 and 700 milliseconds is one
tenth. Then the continuous distribution for DL's possible times
would have a shape that places 10% of the area below the curve
in the region bounded by 600 and 700 on the X-axis.
Figure 4. A normal distribution.
Shapes of Distributions
Distributions have different shapes; they don't
all look like the normal distribution in Figure 4. For example,
the normal probability density is higher in the middle compared
to its two tails. Other distributions need not have this feature.
There is even variation among the distributions that we call "normal."
For example, some normal distributions are more spread out than
the one shown in Figure 4 (their tails begin to hit the X-axis
further from the middle of the curve -- for example, at 10 and
90 if drawn in place of Figure 4). Others are less spread out
(their tails might approach the X-axis at 30 and 70). More information
on the normal distribution can be found in a later chapter
completely devoted to them.
The distribution shown in Figure 4 is symmetric;
if you folded it in the middle, the two sides would match perfectly.
Figure 5 shows the discrete distribution of scores on a psychology
test. This distribution is not symmetric: the tail in the positive
direction extends further than the tail in the negative direction.
A distribution with the longer tail extending in the positive
direction is said to have a positive
skew. It is also described as "skewed to the
Figure 5. A distribution with a positive
Figure 6 shows the salaries of major league
baseball players in 1974 (in thousands of dollars). This distribution
has an extreme positive skew.
Figure 6. A distribution with a very large
A continuous distribution with a positive skew
is shown in Figure 7.
Figure 7. A continuous distribution
with a positive skew.
Although less common, some distributions have a negative
skew. Figure 8 shows the scores on a 20-point problem
on a statistics exam. Since the tail of the distribution
extends to the left, this distribution is skewed
to the left.
Figure 8. A distribution with negative
skew. This histogram shows the frequencies of various
scores on a 20-point question on a statistics test.
A continuous distribution with a negative skew
is shown in Figure 9.
Figure 9. A continuous distribution
with a negative skew.
The distributions shown so far all have one distinct
high point or peak. The distribution in Figure 10 has two distinct
peaks. A distribution with two peaks is called a bimodal
Figure 10. Frequencies of times between
eruptions of the
Old Faithful geyser. Notice the two distinct
peaks: one at
1.75 and the other at 4.25.
Distributions also differ from each other in
terms of how large or "fat" their tails are. Figure
11 shows two distributions that differ in this respect. The
upper distribution has relatively more scores in its tails; its
shape is called leptokurtic.
The lower distribution has relatively fewer scores in its tails;
its shape is called platykurtic.
Figure 11. Distributions differing in
kurtosis. The top
distribution has long tails. It is called "leptokurtic."
The bottom distribution has short tails. It is called
Please answer the questions: