# Introduction to Statistics (Online Edition)/Introduction/Distributions

​Distributions by David M. Lane and Heidi Ziemer
Prerequisites • Chapter 1: Variables
Learning Objectives
* Define “distribution”
* Interpret a frequency distribution
* Distinguish between a frequency distribution and a probability distribution
* Construct a grouped frequency distribution for a continuous variable
* 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
Figure 1. Distribution of 55 M&M's.
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 M&M's. 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.
Continuous Variables
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 variables.
​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 20 times.
Table 3. Grouped frequency distribution
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.
Probability Densities
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 are called 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 distributions.
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 probability.” 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, see Chapter 7.)
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 right.”
Figure 5. A distribution with a positive skew.
Figure 6. A distribution with a very large positive skew.
A continuous distribution with a positive skew is shown in Figure 7.
​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.
​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 distribution.
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.
This work is in the public domain worldwide because it has been so released by the copyright holder.
Public domainPublic domainfalsefalse