3 ms·

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 sen

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.

autokad 10y ago
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.
Houshalter 10y ago
>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%."
srean 10y ago
> 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.

autokad 10y ago

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.

Houshalter 10y ago

>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%."

srean 10y ago

> 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.