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# What the "Mean" Is Not...

The "mean" is the most commonly used "average" value. However, it may not always be the most appropriate...

## The "Mean" Is Not in the Middle!

First, it is important to remember that the "mean" is usually not in the middle. It may be close to the middle, and if the data is perfectly symmetrical it will just happen to correspond with the middle value, but in general the mean is not exactly in the middle. The middle value is actually represented by the "median" not the "mean." Because of this, some very odd statements can actually be true.

For example, consider the statement, "Most households make less money than the mean household." Could this possible be true? Yes, and here is why...

The US Census Bureau reported the following figures for 1998:

 US Household Income Median \$38,885 Mean \$51,855

It is clear that the "median" is less than the "mean." But the "median" represents the middle income. So half of all households make more than the \$38,855 and half of all households make less than \$38,885. Therefore, more than half of all households make less than \$51,855. Or, as we said before, "most households make less money than the mean household."

Note: a sneaky person may try and substitute the word "average" for "mean" and say, "Most households make less money than average." This is sneaky because the truth of the statement depends upon how "average" is defined. If they are using "average" instead of "median" the statement makes no sense. If they are using "average" instead of "mean" the statement is in fact true (as we just discussed). However, if they are using "average" instead of "mode" their statement is wrong! The most typical income ("mode") is actually between \$5000 and \$9999 (as reported by the US Census). Therefore, not only is this sneaky but it could give someone the wrong impression. Most households actually make more than the typical household.

## The "Mean" Is Not Typical!

Second, remember that the "mean" is not the most typical value. In fact, it usually happens that none of the data actually corresponds with the mean. For example, consider the numbers 0, 4, & 5. Their average is 3 [ (0+4+5)/3=> 9/3=> 3]. But "3" is not at all typical of these numbers because it does not even occur in the set of numbers.

If the data turns out to be perfectly symmetrical (unlike this example) the mean may happen to correspond with the most typical value, but in general the mean is not typical. The most typical value is better represented by the "mode." These differences can results in some very odd statements being true.

For example, consider the statement, "The typical American is older than the mean." Could this be true? Yes, and here is why...

The US Census Bureau reported the following figures for 1999:

 Age of US Population Mode 39 Median 35.6 Mean 36.4

It is clear that the "mode" is higher than the "mean." The "mode" is 39 because more people claimed to be 39 years old than any other age (that is the definition of the mode). Therefore, 39 is the most typical age in this country and "the typical American is older than the mean."

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Copyright 2000, Wayne Pafko