Comment by srean | Hacker News Reader

Comment by srean | Hacker News Reader

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> when people recommend the median over the mean to describe center. The median, on its own, does not describe what is "typical" any more than the mean does; it just has a small advantage

I think this swings the other extreme in selling the 'median' short. As long as we agree that it is only strictly meaningful to talk about the 'center' for symmetric distributions, median does a fine job.

In fact in many realistic scenarios a far better job than the mean. The main trouble is the normal or the Gaussian distribution is no where close to being as ubiquitous as it seems, neither is CLT as universal as it is made out to be. Gauss sort of got away with it, Gauss did not discover the distribution nor the associated CLT.

Many real data of day to day consequence have heavy tails, and mean is a pathetic measure of 'central tendency' for these. Mean is particularly sensitive to outliers. Median does significantly better than the mean in this non-academic situation. Although one could do better than median for symmetric heavy tailed data (for example trimmed means), but its mean that I find guilty of entirely disproportionate fame. If its needed to exaggerate median a bit to get people to grow up beyond the pervasive Normal / Gaussian fetish, I am all for it.

No single number is going to characterize what is 'typical'. One really needs the CDF here, and yes avoid estimating densities as much as possible.

BTW mapping to a real observation is not true, you only get a 50% chance of that.

The median is worse than the mean in a skewed distribution if you want to take the skew into account. In fact one of the strengths of the mean is that it is sensitive to changes in the entire sample, rather than only part of it. To repeat my previous example, if the lowest 20% of household incomes drop because of changes to the tax code, that's something I'd generally want reflected in my "household income" statistic. We can't get too hung up on the semantics of central tendency here--there are many good ways to measure center, and it's clear that all of these statistics actually measure very different things even though they're all grouped into the same category of statistic.

When people say the median is "better" than the mean, they often can't explain exactly why (other than to cite skew, which I've addressed). It's really just a gut feeling based on how they felt when they first learned that the median was "more realistic" for some specific set of data that was skewed. They never thought through exactly why or when you'd want to disregard the skew in the first place. I would be insane to argue that the median is never the right statistic to use; my point is merely that it shouldn't be given preference over the mean, and that people need to think about what they're actually trying to show.

The mean is not only useful for the normal distribution, and since the article specifically avoids jumping into inference territory talking about specific distributions at all is kind of putting the cart before the horse. The "disproportionate fame" of the mean is inherited from its extreme importance throughout all of statistical inference. It's true that if you ignore inference, the mean isn't particularly important as a descriptive statistic, but my argument is that it's certainly no worse than the median.

>one of the strengths of the mean is that it is sensitive to changes in the entire sample, rather than only part of it.

Yes but this is exactly what is bad with it. At least for the majority of practical applications that I can think of. For instance, if you want to measure the "average" income of the population. If there are a handful of uber-rich people, they will really raise the mean. Even if the vast majority of people make much less than the mean. The median will do a much better job of telling you what a "typical" person makes.

I really believe that when most people read or talk about "average", they mentally interpret it as the median. And that the median is generally less misleading.

Sure, there are applications where sensitivity to changes at the tails is important. But even then, mean is misleading and difficult to interpret. In your earlier example, if you raised the income of the bottom 10% by 100%, then just use that as a statistic. "The poorest 10% of people now make twice as much money" is much more informative and impressive than "the average income increased by 0.1%."

> When people say the median is "better" than the mean, they often can't explain exactly why (other than to cite skew, which I've addressed).

Well, 40~50 years of literature on robust statistics happens to disagree with the claim 'people' don't know what they mean (pun unintended) when the say the median is better. Furthermore skew has less to do with that argument than heavy tails. heavy tails are extremely (ok sorry, now its an insider pun) common.

i agree with the article in that: if you want to take the screw into account, you look at the histogram. as a single number, I dont think the mean tells you anything about the skew/tail of the data any better than the median.

It's a better than 50% chance - if the sample size is odd, you always get a real observation; if the sample size is even, you may still get a real observation (if the two median observations are equal).

You are indeed correct, I was incorrectly assuming a continuous density, in which case the probability of a tie would have been zero.

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