Math and Arithmetic
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Is an average of averages accurate?


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Mx = M

M is the greek letter mu, by the way.

Statistically speaking, an average of averages within a sample is said to very closely approximate the population average. However, where the sample size is under 30, there needs to be an adjustment for size. If the sample is over 30 (thus you are averaging over 30 averages), then this should reflect very closely the population average. It is not perfect but statistically it is incredibly close.

However, it is important not to compare apples and oranges in general so while giving mu of mu, be sure to include the standard deviation and any notes you find relevant.

Average of averages

You're living dangerously.

If the averages don't have equal weight, your average of averages will yield skewed results. Here's why.

Let's say you're trying to calculate your overall average grade for all the tests you've taken in school, and let's say you are taking only math, English, and science. Let's say your average in math is 80, English is 85, and science is 90. The average of those averages is, of course, 85. But is that your true average?

Not if you took ten math tests, five English tests, and only two science tests!! If all those tests have equal value, then your average is closer to 80, because the ten math tests dominate the true average. In other words, your true average is the WEIGHTED average of the three averages. To weight them properly, you must think of the math average as not one data point but ten data points, the English average as not one data point but five, and the science average as not one but two data points.

Here's how the weighted average math looks:

10 x 80 = 800
5 x 85 = 425
2 x 90 = 180
__ ___
17 1405

Now, divide 1405 by 17. So, 1405/17 = 82.7. (The 17 is the sum of the weighting factors, which, as you can see, is equal to the total number of tests you took.)

So, you can see that taking the average of averages may give erroneous results -- unless you find the weighted average.