basic statistics review

Frequency Distributions:
Characteristics

Frequency distribution charts have many different shapes. We shall discuss some of the more common configurations here.  

Unimodal, Bimodal, and Multimodal Distributions

Unimodal distributions have one peak or "mode." The chart in Figure 4.1 shows a unimodal distribution.

unimodal distribution
Figure 4.1: Unimodal Distribution

Bimodal distributions have two modes as in Figure 4.2.  

bimodal distribution-2
Figure 4.2: Bimodal Distribution

Here, the term "mode" is used to describe a local maximum in a chart. It does not refer to the most frequently appearing score (as in the "central tendency mode"). Rather, a "local maximum" is a "high" point in a chart. The local maxima are the points where the score frequencies stop increasing and begin decreasing. (A local minimum is a point on a chart where the score frequencies stop decreasing and begin increasing.)

Figure 4.1 has one local maximum, so it is a unimodal distribution. Figure 4.2 has two local maxima, so it is a bimodal distribution. Similarly, Figure 4.3 has two local maxima.

bimodal distribution-2
Figure 4.3: Bimodal Distribution

Even though the local maximum on the left has a lower frequency than the one on the right, the chart in Figure 4.3 shows a bimodal distribution.

Frequency distributions with three or more local maxima are called multimodal distributions. 

Symmetric and Asymmetric Distributions

A symmetric distribution is one in which the shape of left side of the distribution is a "mirror image" of the right side. Figure 5.1 is a symmetric distribution.

symmetric distribution
Figure 5.1: Unimodal Symmetric Distribution

Symmetric distributions may be unimodal, bimodal, or multimodal. Figure 5.2 shows a bimodal symmetric distribution.

bimodal symmetric distribution
Figure 5.2: Bimodal Symmetric Distribution

An asymmetric distribution is not equally balanced. Figure 5.3 shows an asymmetric distribution. Run your mouse ponter over the chart to "fold" the right half over the left. Since the two halves are not congruent, the distribution is asymmetric.  

asymmetry
Figure 5.3: Asymmetric Distribution

Asymmetric distributions may also be unimodal, bimodal, or multimodal. 

Positive and Negative Skew

Asymmetric distributions are positively skewed or negatively skewed. A positively skewed distribution is one in which the right (positive) tail of the distribution is the long one (see Figure 6.1). A negatively skewed distribution is one in which the left (negative) tail of the distribution is the long one (see Figure 6.2).

positive skew
Figure 6.1: Positively Skewed Distribution

The distribution of personal income is positively skewed. Also, raw scores on most measures of psychopathology are positively skewed.

negative skew
Figure 6.2: Negatively Skewed Distribution

It is not uncommon to obtain distributions that are skewed and multimodal (see Figure 6.3).

negative skew
Figure 6.3: Bimodal Negatively Skewed Distribution

Measures of skill test scores are often in the shape of a bimodal (or multimodal) skewed distribution. Such distributions suggest that we may be looking at two groups of individuals.


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©2003 by J.T. (Tim) Stocks, Ph.D.
stocks@msu.edu