# Standard Deviation Defined

TRY to find out what the “standard deviation” of a bell curve distribution means by doing a simple web search (say, “standard deviation defined”), and whoa, you’re in for a trip. Lots about bell curves, 68%, and 95%, and something about the standard deviation being some kind of “measure,” but not a lot of crystal-clear *definition.* So let’s do it, right here:

**Definition of e**

e is defined as the number such that the curve y = e^{x} is its own derivative — which simply means that the slope of the curve is the same as its Y value.

For example, e^{0} = 1, and at that point (0,1) the curve is rising on the Y-axis at the same speed that it is progressing on the X-axis, so its slope is also 1. Another example: e^{1} = e, and the slope of the curve at (1,e) is e.

Use any other positive number besides e (say, y = 5^{x}) and the Y value is *proportionate* to the slope, but not *equal* to it.

(e happens to be about 2.718, but it has an infinite series of digits after the decimal point, with no repeating pattern.)

**Definition of A Bell Curve**

A standard bell curve is defined by the formula: y = e^{-(x2)/2}

It looks like this:

You can modify the formula in various ways to move the curve to the left or right, or to stretch it on the horizontal or vertical, but the graph depicted above is the standard version. For example, if you wanted the curve to be centered around .5, and to be 1/3 as wide as the standard version, the formula would be:

y = e^{-((3(x-.5))2)/2}

and it would look like this:

The bell curve represents the distribution of many semi-random statistics, such as the height of humans. If you graph the height of the human population such that the X coordinate represents the height of each person, and the Y coordinate represents the number of people who have are approximately that height (within a certain tolerance), you get the shape of a bell curve.

**Definition of A Standard Deviation**

In the standard bell curve depicted above, the phrase “within one standard deviation” means, *by definition,* everything with an X coordinate between -1 and 1. It so happens that about (not exactly) 68% of the area under the graph is represented by this portion.

The phrase “within two standard deviations” means (again, by definition) everything with an X coordinate between -2 and 2. It so happens that about (not exactly) 95% of the area under the graph is represented by this portion.

You can move and/or stretch the graph, but the standard-deviation markers move and/or stretch proportionately with it, such that “within one standard deviation” is always about 68% of the area under the graph. For example, you would have to move and stretch the bell curve considerably to make it accurately represent the human population’s height distribution, but no matter — about 68% of persons will be “within one standard deviation” of the average height.

**Now You Know**

Wasn’t that satisfying?

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