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Tuesday, August 16, 2011

The Shape of a Quantitative Distribution

When you graph quantitative data, you can often see some kind of shape emerge. Here are some typical block structures that illustrate some possible shapes a display might take. We call the set of values "distributions."

The first thing you need to determine is if there is any symmetry to the graph. If you were to visualize a vertical line going down the center, does each side look like a mirror image of the other? No real-life distribution will be perfectly symmetrical, but if it's close, it's worth mentioning.

(Self-Test): Which three of the above graphs look symmetric?
(Answer): B, D, and E.

The next thing you might notice is that some graphs have peaks whereas others look pretty level. We call the peaks "modes." If there's one peak, we say the graph has a unimodal distribution. If there are two peaks, the graph has a bimodal distribution

(Self-Test): Which three of the above graphs look unimodal? Which is bimodal?
(Answer): A, B, and C are unimodal; D is bimodal.

FYI, a graph that's mostly level-looking, like graph E, is called uniform.

Now take a look at distributions A and C. Do you see that each has a "tail" at one end? When a distribution is off-center (compared to graph B), we say it is skewed. The direction of the tail is the direction of the skewness. For example, graph A is skewed left because the tail is on the left side of the graph. graph C is skewed right.

When we describe a distribution we try to describe its shape, center, and spread, plus anything unusual about it, such as outliers. This will be the subject of my next post.

(Self-Test): Which of the distributions pictured above might have outliers?
(Answer): Skewed graphs, like A and C, depending on the length of their tails. The longer the tail, the more likely there are outliers.

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