Describing Distributions

• Features of Distributions of Quantitative Variables
  • Shape (Symmetry/Skewness, Modality)
  • Center
  • Spread
  • Outliers
• Let’s Summarize

Features of Distributions of Quantitative Variables

Once the distribution has been displayed graphically, we can describe the overall pattern of the distribution and mention any striking deviations from that pattern.

Shape

When describing the shape of a distribution, we should consider:

• Symmetry/skewness of the distribution.

• Peakedness (modality) — the number of peaks (modes) the distribution has.

We distinguish between:

Symmetric Distributions

Note that all three distributions are symmetric, but are different in their modality (peakedness).

• The first distribution is unimodal — it has one mode (roughly at 10) around which the observations are concentrated.
• The second distribution is bimodal — it has two modes (roughly at 10 and 20) around which the observations are concentrated.
• The third distribution is kind of flat, or uniform. The distribution has no modes, or no value around which the observations are concentrated. Rather, we see that the observations are roughly uniformly distributed among the different values.

Skewed Right Distributions

Note that in a skewed right distribution, the bulk of the observations are small/medium, with a few observations that are much larger than the rest.

• An example of a real-life variable that has a skewed right distribution is salary. Most people earn in the low/medium range of salaries, with a few exceptions (CEOs, professional athletes etc.) that are distributed along a large range (long “tail”) of higher values.

Skewed Left Distributions

Note that in a skewed left distribution, the bulk of the observations are medium/large, with a few observations that are much smaller than the rest.

• An example of a real life variable that has a skewed left distribution is age of death from natural causes (heart disease, cancer etc.). Most such deaths happen at older ages, with fewer cases happening at younger ages.

Comments:

1. Distributions with more than two peaks are generally called multimodal.
   

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2. Bimodal or multimodal distributions can be evidence that two distinct groups are represented.
   

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3. Unimodal, Bimodal, and multimodal distributions may or may not be symmetric.

Here is an example. A medium size neighborhood 24-hour convenience store collected data from 537 customers on the amount of money spent in a single visit to the store. The following histogram displays the data.

Note that the overall shape of the distribution is skewed to the right with a clear mode around $25. In addition, it has another (smaller) “peak” (mode) around $50-55.

The majority of the customers spend around $25 but there is a cluster of customers who enter the store and spend around $50-55.

Center

One way to define the center is as the value that divides the distribution so that approximately half the observations take smaller values, and approximately half the observations take larger values.

Another common way to measure the center of a distribution is to use the average value.

From looking at the histogram we can get only a rough estimate for the center of the distribution. More exact ways of finding measures of center will be discussed in the next section.

Spread

From looking at the histogram, we can approximate the smallest observation (min), and the largest observation (max), and thus approximate the range. (More exact ways of finding measures of spread will be discussed soon.)

Outliers

For example, the following histogram represents a distribution with a highly probable outlier:

Let’s look at a new example.

The following exercises provide more practice with shapes of distributions for one quantitative variable.

Let’s Summarize

• When examining the distribution of a quantitative variable, one should describe the overall pattern of the data (shape, center, spread), and any deviations from the pattern (outliers).

• When describing the shape of a distribution, one should consider:
  • Symmetry/skewness of the distribution
  • Peakedness (modality) — the number of peaks (modes) the distribution has.
  • Not all distributions have a simple, recognizable shape.

• Outliers are data points that fall outside the overall pattern of the distribution and need further research before continuing the analysis.

• It is always important to interpret what the features of the distribution mean in the context of the data.