Wikipedia:Reference desk/Archives/Mathematics/2023 January 1

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January 1

What is the name of average type that works with powers of 2?

What is the average type that works like this:

1-To each number find the result X of this formula 2^x= number.

2-Sum all those results and divide by the amount of numbers, the value is Y.

3-Get the result of 2^Y. This is the average I am talking about.

177.207.98.160 (talk) 21:15, 1 January 2023 (UTC)

Apparently we are working in the realm of nonnegative numbers. One of the defining characteristics of the concept of average is that when you have a bag of values, all equal, their average is again that value. For example, if all students in a class are 17 years old, their average age is 17. In particular, the average of a bag containing just a single value is that value. Your procedure, as described, does not have that property. If you start with the singleton bag $\{0\}$, the result is $1.$ So this is not an average. I suppose, though, you meant to write $x^{2}$ in the final step. Then this is indeed a kind of average, but I have never seen it used, and if it has a name, it is not one that is commonly known. If, however, you apply the two transformations in the reverse order – take the squares in Step 1 and the square root in Step 3, the result is called the root mean square. It is a very common procedure, and often abbreviated as RMS.  --Lambiam 22:19, 1 January 2023 (UTC)

Thanks I fixed the entire thing, all the steps were wrong.177.207.98.160 (talk) 23:12, 1 January 2023 (UTC)

What you've got now is the Geometric mean. NadVolum (talk) 23:30, 1 January 2023 (UTC)

Basically the log of geometric mean equals the arithmetic mean of the logs. Note that the statement is true for any log base. This is a special case of the Generalized mean. --RDBury (talk) 23:50, 1 January 2023 (UTC)

And the outcome of the computation is insensitive to the choice of base. Arithmetic mean can be written as $\textstyle {(\sum _{i}x_{i}})\times (1/n).$ Geometric mean can be written as $\textstyle {(\prod _{i}x_{i}})^{1/n}.$ Look ma, no base!  --Lambiam 03:07, 2 January 2023 (UTC)