# How is that possible that simple arithmetic mean works well even for strongly skewed distribution?

I was taught, that the arithmetic mean is sensitive to outliers and skewness. This was natural to me - the observations lying far from the "central point" of the distribution "pull" the measure towards them. Then I was advised to use the arithmetic mean only for symmetric distributions, ideally normal, but symmetry was found as "good enough". When it came to asymmetric distributions, left or right, I was told to use the median. In case of log-normal distributed data, which - as I was told - have multiplicative nature (whatever it means), I was advised to use the geometric mean, as better measure of the central tendency. But then I was told, that the arithmetic means is "valid for any distribution", as it is the BLUE (best linear unbiased) estimator of the population expected value (still - the arithmetic mean).

Now I'm lost. If the arithmetic mean is so good measure even for so strongly skewed distributions as the log-normal, then why telling about "robust measures", like geometric mean, winsorized mean, trimmed mean, mid-mean? According to what I was told, only the arithmetic mean is the BEST estimator. Moreover, the geometric mean is not an estimator of the population expected value, so why even bother with? Is this because the expected value, the first raw moment, not always is the best measure characterizing the central tendency? And that is because, even if the arithmetic mean is best estimator, we instead propose the geometric mean as the best measure of the central tendency instead (not expectance) instead?

In other words: is it that however every distribution (for which the first raw moment is deifned) has the expected value, which is the arithmetic mean, but the expected value is "useful" only for the normal (or symmetric) distribution, and for any other skewed ones we may prefer different measure of central tendency instead of the expectance (which plays important role in theoretical considerations anyway)?

• useful for what purpose? moments are always useful for something Jan 31, 2020 at 20:58
• You might want to define "well" when you ask if it works well. Jan 31, 2020 at 20:59
• The sample mean is a good estimator of the true mean (if it exists) but the true mean is not always a good measure of central tendency. Jan 31, 2020 at 21:31
• Thank you, Michael M.! That is it! So the dozens of central tendency measures (Wikipedia lists more than ten of them, including the Winsorized mean) are invented to better describe the "typical observations". For symmetric distributions, including the normal one, the central tendency also coincides with the arithmetic mean, the first raw moment, right? Jan 31, 2020 at 21:56
• 1. The sample mean can be an excellent estimator of the population mean when the distribution is moderately skew (consider the exponential, for example; it's maximum likelihood in that case). 2. One thing I tend to find when people try things with the lognormal is that they restrict attention to quite "nice" cases. You might look at how things go when the skewness is actually high - like say $\sigma=4$. Feb 1, 2020 at 6:16

What is your definition about "good"? I assume you want to say the bias of sample mean is zero, it is consistent, if so, in 1947, Hsu and Robbins proved that the arithmetic mean converges completely to the population mean provided the second moment is finite. That means, for any distribution, no matter skewed or symmetry, the arithematic mean is a consistent estimator, just assuming a regularity condition.

If you want to say variance, or the overall error (often, bias+variance+contamination), the sample mean is often not the optimal choice in a skewed distribution.

The Hodges-Lehmann estimator or median of means are often better choices, as proven by Bickel that the Hodges-Lehmann estimator is the best choice among all current nonparametric robust location estimators for a symmetric distribution. Peter J. Bickel. "On Some Robust Estimates of Location." Ann. Math. Statist. 36 (3) 847 - 858, June, 1965. On Some Robust Estimates of Location (On Some Robust Estimates of Location (https://doi.org/10.1214/aoms/1177700058)) and median of means nears the optimal of nonparametric mean estimation when the distribution has a heavy tail, proven by Luc Devroye. Matthieu Lerasle. Gabor Lugosi. Roberto I. Oliveira. "Sub-Gaussian mean estimators." Ann. Statist. 44 (6) 2695 - 2725, December 2016. Sub-Gaussian mean estimators (https://doi.org/10.1214/16-AOS1440) ).

If you are confident that the percentage of outliers is smaller than the breakdown point of Hodges-Lehmann estimator (29%), then, you can use median Hodges-Lehmann mean, as proposed in my papers. Robust estimations for semiparametric models: Moments (Robust estimations for semiparametric models: Moments (https://zenodo.org/records/8127703)(https://www.researchgate.net/publication/377974264_Robust_estimations_from_distribution_structures_Central_Moments) (https://www.researchgate.net/publication/377974419_Robust_estimations_from_distribution_structures_Invariant_Moments) Robust estimations for semiparametric models: Mean (Robust estimations for semiparametric models: Mean (https://zenodo.org/records/6629988)https://www.researchgate.net/publication/377973944_Robust_estimations_from_distribution_structures_Mean

Median Hodges-Lehamnn mean is an extension of the Hodges-Lehamnn estimator, instead of the pairwise means, the Hodges-Lehmann kernel proposed by Serfling in 1984, $$hl_{k}\left( x_1,\ldots,x_k\right)=\frac{1}{k}\sum_{i=1}^{k}x_i$$, where $$k\in\mathbb{N}$$, is used (Serfling, R. J. (1984). Generalized L-, M-, and R-Statistics. The Annals of Statistics, 12(1), 76–86. http://www.jstor.org/stable/2241035). In my paper, I further proposed a solution for $$k\notin\mathbb{N}$$ case, so the breakdown point of the median Hodges-Lehmann mean can be any value from 0 to 29%.

I also attached a picture here to compare the asymptotic biases of all current robust location estimators and new estimators proposed in my papers for four unimodal distributions with different kurtosis (linear dependent on the skewness).

• This is actually a pretty comprehensive answer coming from a new user, here. Welcome to CV. Feb 10 at 4:50
• Thank you! I will continue post more answers here. Feb 10 at 5:12
• would you be able to add a graph of the overall error for mean and other estimators of population mean? Feb 10 at 16:12
• I have related data, but don't have a graph, because the overall errors are estimated using Monte Carlo simulation, this graph is not as elegent as this graph, instead, it is very complex, for different kurtosis, the optimal choice is very different. I dicussed this issue in my YouTube video. youtube.com/watch?v=9fkcTgDHTeI Feb 10 at 16:22
• I deposited the codes in GitHub, if you are interested, you can also run the codes to get the data and then draw the graph. github.com/tubanlee/REDS_Invariant_Moments Feb 10 at 16:25

Whether the arithmetic mean is a good measure for a strongly skewed distribution depends on what you are using it for -- what does "good" mean?

One variable that is usually very right skew is income and, usually, you see median income rather than mean income. But why?

Because, usually, when we look at some measure of central tendency, we are looking for something like a "typical" value. So if we wanted to compare incomes in different countries, it would be natural to compare medians (or maybe other quantiles) rather than arithmetic means, because it is closer to what we are interested in. On the other hand, if we were interested in the total capital available in different countries, the arithmetic mean would be better.

Sometimes, though, the arithmetic mean makes very little sense. If we are trying to average numbers on different scales then the arithmetic mean is still BLUE, but it's a BLUE of a number that isn't useful. For instance, suppose you are averaging students' SAT scores (0 to 1600) and their high school grades (0 to 4). Then you won't want the arithmetic mean. Here you want the geometric mean.

Or, if you are averaging rates that have different denominators, the arithmetic mean will again be BLUE, but of a number that isn't useful. For instance, if you make a trip of 300 km at 100 kph and return at 150 kph, then the arithmetic mean is 125 kph, but this is not very useful because you took 5 hours to go 600 km, which is 120 kph. Here you probably want the harmonic mean.

For more details on this, see my blog post the geometric and harmonic means.