Bell Curve Details

The bell curve, or normal distribution, is an extremely useful predictor of probable outcomes of a large number of events. An event, in this case, is any single instance that is measured. It is predictive and mathematically perfect, distinguishing it from actual distributions of real-world outcomes. Essentially, the bell curve describes how a very large number of random events should be distributed, in contrast with what may have actually occurred on any given instance.

The bell curve is derived from a formal mathematical function, and it's not dependent on real-world events. Its shape characteristics derive from the fact that peak of the bell is the average outcome of all events described. Values nearer the average are much more common and form the bulk of the bell, and values further away from the average become the much thinner rim. The width of the bell is related to a value called the standard deviation, a formal measure of how varied the data are.

In addition, there are other characteristics of the bell curve worth noting:

  • The mean, or average, of the distribution being described is the center of the bell.
  • The distribution is symmetrical around the mean, just like a bell is.
  • Both the median and the mode of the distribution, which are the 50th percentile and the most common individual measure are the same value as the mean. This isn't true of other distributions, or curves, which aren't symmetrical.
  • If the total area under the bell curve adds up to 100%, then 68% of it lies between -1 and +1 standard deviation (SD), 95% between -2 and +2 SD, and 99.7% between -3 and +3 SD. This allows a data scientist or statistician to be able to predict how many items or people would fall into a particular range of the distribution.

Although many legitimate and proper examples of the use of the bell curve exist, it is not universally applicable, so it needs to be used appropriately.

Real-World Examples of a Bell Curve

1. Mass Production

One popular example of the use of a bell curve involves the quality assurance in the mass production of products. Mass-produced parts are designed to be uniform, but random variance does occur. This variance, if truly random, will be distributed as a bell curve. Most parts will be in the middle, and a few at the limits. The very popular 6-sigma ® program is a statistical quality control system that seeks to control tolerances to such that only parts that are more than 6 SD away from the mean would fail, and only a few per million parts would fall into that category.

These programs steadfastly measure parts to constantly update the ‘bell curve’ of each day and sometimes hour to assure that they are conforming. But underlying these efforts is that the random variations indeed fall in a pattern of the bell curve.

2. IQ Tests

A second example is IQ testing. IQ tests were defined as having a mean of 100 and an SD of 15. Based upon this, and the fact that scores are normally distributed, 68% of all people would fall between 85 and 115. If we know, for example, that we have a job that needs to have at least 115 IQ, we immediately know that only about 16% of all people would qualify.

4. Inventory

A third example would be related to inventory management. Imagine that we know shoe sizes are normally distributed with a mean of 10.5 and have an SD of 1.5. We would then know the relative number of each size to put in inventory, assuming that our market is normal. We’d order the most of 10.5. Because the bell curve is symmetrical, we order slightly fewer of sizes 10 and 11. There would be fewer still of sizes 9.5 and 11.5, and so on until we fulfilled our inventory.

If the actual buyers fall into a bell curve, we would run out of all sizes exactly at the same time. Because our actual shoppers have are not mathematically perfect, we would likely end up with a few odd sizes for our clearance rack.