LOG#078. Averages.


I am going to speak a little bit about Statistics. The topic today are “averages”. Suppose you have a set of “measurements” x_i where i=1,...,n. Then you can define the following quantities:

Arithemtic mean.

    \[ \boxed{\overline{X}_{AM}=\dfrac{1}{n}\displaystyle{\sum_{i=1}^n}x_i=\dfrac{\displaystyle{\sum_{i=1}^n x_i}}{n}=\dfrac{x_1+x_2+\cdots+x_n}{n}}\]

Geometric mean.

    \[ \boxed{\displaystyle{\overline{X}_{GM}=\sqrt[n]{\prod_{i=1}^n x_i}=\sqrt[n]{x_1x_2\cdots x_n}=\left(\prod_{i=1}^n x_i\right)^{1/n}}}\]

Harmonic mean.

    \[ \boxed{\displaystyle{\overline{X}_{HM}=\dfrac{1}{\displaystyle{\dfrac{1}{n}\sum_{i=1}^{n}\dfrac{1}{x_i}}}=\dfrac{n}{\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}}}}\]

Remark: In the harmonic mean we need that every measurement is not null, i.e., x_i\neq \forall i=1,...,n

Remark (II): \overline{X}_{AM}\geq\overline{X}_{GM}\geq\overline{X}_{HM}

There are some other interesting “averages”/”means”:

Quadratic mean.

    \[ \boxed{\displaystyle{\overline{X}_{QM}=\sqrt{\dfrac{1}{n}\sum_{i=1}^{n}x_i^2}}}\]

Generalized p-th mean.

    \[ \boxed{\displaystyle{\overline{X}_{GEN}=\sqrt[p]{\dfrac{1}{n}\sum_{i=1}^{n}x_i^p}}}\]

Weighted mean/average.

    \[ \boxed{\displaystyle{\overline{X}_{WM}=\dfrac{\displaystyle{\sum_{i=1}^n w_i x_i}}{\displaystyle{\sum_{i=1}^n w_i}}}}\]

where w_i are the weight functions and they satisfy

    \[ \displaystyle{\sum_{i=1}^n w_i}=1\]

A particularly important case occurs when the weight equals to inverse of the so-called variance of a population with finite size (generally denoted by \sigma^2), i.e.,  when w_i=1/\sigma^2_i, the weighted mean yields:

    \[ \boxed{\displaystyle{\overline{X}_{WM}=\dfrac{\displaystyle{\sum_{i=1}^n \dfrac{x_i}{\sigma^2_i}}}{\displaystyle{\sum_{i=1}^n \dfrac{1}{\sigma_i^2}}}}}\]


Finally, a “naive” and usually bad statistical measure for a sample or data set is the midrange. Really, it is a mere measure of central tendency and no much more:

    \[ \boxed{\displaystyle{\overline{X}_{MR}=\dfrac{max(x)+min(x)}{2}}}\]

Here, max(x), min(x) refer to the maximum and minimum value of the sampled variable x in the full data set x_i.

Many of the above “averages” have their own relative importance in the theory of Statistics. But that will be the topic of a future blog post handling statistics and its applications.

What average do you like the most? Are you “on the average”? Are you “normal”? 😉 Of course, you can consult your students, friends or family if they prefer some particular mean/average over any other in their grades/cash sharing, or alike :).

See you soon in other blog post!

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