Standard Deviation

Statisticians often refer to data that are “normally distributed” (that is, where most values are close to the mean and fewer are at the extremes). For example, a community’s weekly calorie consumption would typically be normally distributed, with a few “outliers” consuming far more or far fewer.

When graphed, normally distributed data form the classic bell curve. Per the above example, this horizontal axis would show calories consumed while the vertical axis would show how many people eat x calories).

Of course, not every data set’s curve looks this perfect. Some are flatter, some are steeper, and some have means that lean to either side. But all normally distributed data resemble this basic shape.

What the standard deviation tells us is how tightly data are clustered around the mean. When values are tightly clustered in a steep bell curve, the standard deviation is small. When values are spread apart in a flatter curve, the standard deviation is larger.

Graphically, one standard deviation (the red area) away from the mean (the center vertical line) represents about 68 percent of the people. Two standard deviations away (the red area plus the green area) accounts for about 95 percent. And three standard deviations away (the red, green, blue areas) accounts for about 99 percent.

If the above curve were flatter and more spread out, the standard deviation would have to be larger to account for 68 percent of the people. That’s how the standard deviation tells us how spread out the values are from the mean.

If you were comparing test scores across school districts, for example, the standard deviation would tell you how diverse each district’s scores are. Let’s say District-A has a higher mean test score than District-B. Does this mean that kids in District-A are really smarter? Perhaps not.

Because a bigger standard deviation means that more kids scored toward one extreme or the other, a few follow-up questions might determine that District-A’s mean scores skewed higher because the State sends “gifted and talented” kids there. Or that District-B’s mean scores skewed lower because “mainstreamed” special education students were sent there. As you can see, the standard deviation can reveal a less obvious but highly relevant layer of information.

The standard deviation also can help you to evaluate the merit of highly publicized research studies. For example, in a study that claims to show a relationship between eating spinach and building muscle mass, a large standard deviation might suggest that such claims are not valid as they first appear.