Bell curves show up throughout statistics. Diverse measurements such as diameters of seeds, lengths of fish fins, scores on the SAT, and weights of individual sheets of a ream of paper all form bell curves when they are graphed. The general shape of all of these curves is the same. But all of these curves are different because it is highly unlikely that any of them share the same mean or standard deviation. Bell curves with large standard deviations are wide, and bell curves with small standard deviations are skinny. Bell curves with larger means are shifted more to the right than those with smaller means.

## An Example

To make this a little more concrete, let’s pretend that we measure the diameters of 500 kernels of corn. Then we record, analyze, and graph that data. It is found that the data set is shaped like a bell curve and has a mean of 1.2 cm with a standard deviation of .4 cm. Now suppose that we do the same thing with 500 beans, and we find that they have a mean diameter of .8 cm with a standard deviation of .04 cm.

The bell curves from both of these data sets are plotted above. The red curve corresponds to the corn data and the green curve corresponds to the bean data. As we can see, the centers and spreads of these two curves are different.

These are clearly two different bell curves. They are different because their means and standard deviations don’t match. Since any interesting data sets we come across can have any positive number as a standard deviation, and any number for a mean, we’re really just scratching the surface of an *infinite* number of bell curves. That’s a lot of curves and far too many to deal with. What’s the solution?

## A Very Special Bell Curve

One goal of mathematics is to generalize things whenever possible. Sometimes several individual problems are special cases of a single problem. This situation involving bell curves is a great illustration of that. Rather than deal with an infinite number of bell curves, we can relate all of them to a single curve. This special bell curve is called the standard bell curve or standard normal distribution.

The standard bell curve has a mean of zero and a standard deviation of one. Any other bell curve can be compared to this standard by means of a straightforward calculation.

## Features of the Standard Normal Distribution

All of the properties of any bell curve hold for the standard normal distribution.

- The standard normal distribution not only has a mean of zero but also a median and mode of zero. This is the center of the curve.
- The standard normal distribution shows mirror symmetry at zero. Half of the curve is to the left of zero and half of the curve is to the right. If the curve were folded along a vertical line at zero, both halves would match up perfectly.
- The standard normal distribution follows the 68-95-99.7 rule, which gives us an easy way to estimate the following:
- Approximately 68% of all of the data is between -1 and 1.
- Approximately 95% of all of the data is between -2 and 2.
- Approximately 99.7% of all of the data is between -3 and 3.

## Why We Care

At this point, we may be asking, “Why bother with a standard bell curve?“ It may seem like a needless complication, but the standard bell curve will be beneficial as we continue on in statistics.

We will find that one type of problem in statistics requires us to find areas underneath portions of any bell curve that we encounter. The bell curve is not a nice shape for areas. It’s not like a rectangle or right triangle that have easy area formulas. Finding areas of parts of a bell curve can be tricky, so hard, in fact, that we would need to use some calculus. If we don’t standardize our bell curves, we would need to do some calculus every time we want to find an area. If we standardize our curves, all the work of calculating areas has been done for us.