# What are the different types of "averages" and when to use each one?

So, the concept of an average truly is somewhat abstract. Most statisticians define it as a "measure of central tendency." Others say it is the "center of gravity" for a set of numbers.

I personally prefer a slightly more concrete explanation: A statistic that describes the "typical", or better yet, "representative" value of a data set. We humans get to decide what "representative" actually means. Is it the most common number? The number that falls "in the middle" of a set of numbers? Etc.

I am going to list a few types of averages and am wondering if someone could provide a data set where that particular average is the best choice for describing a "typical" or "representative" value for that data set.

## Types of averages:

• Arithmetic Mean: The number that you could use in place of each of the values of a data set, and still have the same sum. Formula: $$\overline{X}=\frac1N\sum\limits_{i=1}^{N} X_i$$

• Mode: The most common value in a data set. No formula that I know of.

• Median: The literal middle number of a data set where the values are listed in ascending order. No formula that I know of.

• Root Mean Square: Don't really know how to describe this average in a physical sense. Maybe the average that gives larger numbers in the data set more "weight" or "significance?" Formula: $$X_{rms}=\sqrt{\frac1N\sum\limits_{i=1}^{N} X_i^2}$$

• Mean Root Square: I just made this one up, but it seems to work well when I applied it to some random data sets. It seems to do the opposite of the RMS and gives smaller numbers in the data set more "weight" and "significance." Formula: $$X_{mrs}=\frac1N\sqrt{\sum\limits_{i=1}^{N} X_i^2}$$ EDIT: Turns out this should not be considered an average because it fails to describe a data set that consists of only one number, unlike all the other averages.

• Geometric mean: The number that you could use in place of each of the values of a data set, and still have the same product. Formula: $$GM=\sqrt[N]{\prod\limits_{i=1}^{N} X_i}$$

Feel free to add in any other popular or useful types of averages and when to use them!

• The "mean root square" of the dataset 1,1,1,1 is 1/2, so it is not an average in any sense. You should try the "square mean root" instead.
– user856
Apr 13, 2017 at 18:53
• @Rahul Hmm, but I could make a similar argument for other averages as well. In this data set, 1, 1, 20, 21, 22, 23, 24, 25, the mode is 1, but certainly fails to describe the "representative" value and thus also not an average in any sense.
– Nova
Apr 13, 2017 at 19:17
• @Rahul I don't think there is any one average that works universally. Maybe my MRS idea fails to work for some data sets, but is effective in many others.
– Nova
Apr 13, 2017 at 19:19
• There is no complete axiomatization that covers all the existing averaging methods as far as I know, but certainly all averages I know of have the property that if all values in the dataset are identical then the average is the same as that value. Anyway, your mean root square is just the root mean square divided by $\sqrt N$.
– user856
Apr 13, 2017 at 19:43
• The harmonic mean is the reciprocal of the mean of the data reciprocals. It occurs in classic problems like "if you drive for some distance at 30 mph and then return home at 20mph, what is your average speed for the whole trip."
– Ned
Apr 13, 2017 at 20:19

An average should be no less than the least of the numbers being averaged, and no greater than the greatest of the numbers being averaged.

An average should depend only on the numbers being averaged, and not on the order in which the numbers are listed. E.g., if we denote the average by $$M$$, then for all $$a,b$$ we need $$M(a,b)=M(b,a)$$.

An average should be weakly monotone increasing, that is, if $$a_j\le b_j$$ for all $$j$$, then $$M(a_1,\dotsc,a_n)\le M(b_1,\dotsc,b_n)$$.

Any function that obeys these rules can be considered an average.

A large class of averages can be constructed as follows. Let $$f$$ be any continuous monotone increasing function. Then let $$M(a_1,\dotsc,a_n)=f^{-1}(n^{-1}\sum_1^nf(a_i))$$. Most of the averages described elsewhere on this page fall under this heading. The arithmetic mean is what you get if you take $$f(x)=x$$. You get the geometric mean if you take $$f(x)=\log x$$. You get the harmonic mean if you take $$f(x)=x^{-1}$$. You get the root mean square if you take $$f(x)=x^2$$.

As to which mean to use in a given situation, you find out what number your boss wants the average to be, and choose $$f$$ so as to get that number.

• Upvote for the last sentence which reminds me of my time in the financial sector Feb 1 at 15:16

I did some testing using what you cal "MSR" but I call Square Mean Root (SMR). It looks useful for dealing with data where large measurements indicate the opposite of an effect. For example in reaction time measurements, large reaction times indicate the participant was not paying attention. Unfortunately the "tyranny of large numbers" means those large reaction times have an inaapropriate effect on the results.

OTHER AVERAGES 1. Normalized Averages. a) Convert each data set to relative values. b) find the mean relative value c) restore the mean to actual value by multiplying and adding back the means of the subtracted constants and mean fo the divisors. NB this method wcan also produce realistic standard deviations.

1. Kalman Averages These weight means inversely according to their standard deviations. Thus less reliable measurements have less influence. This type of average can be found on the net and in some statistics books.

2. Winsorized Means These set limits to outliers, but do not eliminate them. The 5% and95% values of a set of measurments are taken as maximum values. Any measurments above or below these values are set to these max and minimum values. A mean is calculated from these modified measurements in the usual way. This is less questionable than simply eliminating the "outliers".

When using any "average" suh as mean, median, RMS, .., it is important to keep in mind what has been measured and what the size of the measurement numbers mean in terms of the effect producing those numerical values.

In the data set (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) "6' is the median.

There are 5 numbers before it, (1, 2, 3, 4, 5), and 5 numbers after it (6, 7, 8, 9, 10, 11)

In the data set, (1, 22, 37, 45, 51, 66, 79, 84, 90, 107, 3035), "66" is the 'median'. This doens't tell you much about the OTHER members of this set of numbers, but it is still the 'median'.

A good question. You are right, for statistician, the major focus is estimating the population mean. In statistics, there are three major errors: bias, variance, and contaminations. Sample mean is a consistent estimator, however, its variance and robustness is not desired in many scenarios, so historically, many attempts have made to reduce the overall errors of mean estimation. For example, trimmed mean, Winsorized mean, Hodges-Lehmann estimator, Huber M-estimator, and median of means.

Comparing the performance of these estimators is still a hot topic in statistics research. Recent advances suggested that the bias bound of Winsorized mean is better than that of the trimmed mean (Mariusz Bieniek (2016) Comparison of the bias of trimmed and Winsorized means, Communications in Statistics - Theory and Methods, 45:22, 6641-6650, DOI: 10.1080/03610926.2014.963620 )

Also, the concentration bound of median of means nears the optimum of sub-Gaussian mean estimator (Luc Devroye. Matthieu Lerasle. Gabor Lugosi. Roberto I. Oliveira. "Sub-Gaussian mean estimators." Ann. Statist. 44 (6) 2695 - 2725, December 2016. https://doi.org/10.1214/16-AOS1440 )

Statistics is undergoing a trend from nonparametrics to semiparametrics, because these research are new, so they are not emphasizing in current textbooks.

In my paper, I defined two new classes of semiparametric distributions, introduced several new mean estimators and further explain why the Winsorized mean is better than the trimmed mean in most cases. Also, the median Hodges-Lehmann mean is proposed as the optimun nonparametric robust mean estimator. If you are interested, you can watch my youtube videos or read my paper https://www.youtube.com/playlist?list=PLv12WMZUyCNCxgQdS8wguSWs60uKttHaM .

Any usual average of some numbers $$x_1$$, ..., $$x_n$$ has form $$M(x_1, ..., x_n) = f^{-1}\left(\frac{f(x_1) + \cdots + f(x_n)}{n}\right)$$ for some suitable function $$f$$.

Such an average for a given function $$f$$ is called the $$f$$-mean or a quasi-arithmetic mean or a Kolmogorov mean. See this Wikipedia entry for more information