LOG#078. Averages.
Posted: 2013/02/24 Filed under: Experimental H.E.P., Physmatics, Statistics | Tags: arithmetic mean, average, data set, generalized p-th mean, geometric mean, harmonic mean, mean, measurement, midrange, quadratic mean, random variable, statatistics, variance, weighted mean Leave a commentI am going to speak a little bit about Statistics. The topic today are “averages”. Suppose you have a set of “measurements” where . Then you can define the following quantities:
Arithemtic mean.
Geometric mean.
Harmonic mean.
Remark: In the harmonic mean we need that every measurement is not null, i.e.,
Remark (II):
There are some other interesting “averages”/”means”:
Quadratic mean.
Generalized p-th mean.
Weighted mean/average.
where are the weight functions and they satisfy
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 ), i.e., when , the weighted mean yields:
Midrange.
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:
Here, refer to the maximum and minimum value of the sampled variable x in the full data set .
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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