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Unlearning descriptive statistics

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Re: Unlearning descriptive statistics

#41

Post by abhgh »

andy_wrote wrote: post #8

For readers who are OK with some math, I recommend John Myles White's eye-opening post about means, medians, and modes: http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medi... He describes these summary descriptive stats in terms of what penalty function they minimize: mean minimizes L2, median minimizes L1, mode minimizes L0. A single-number statistic is _going_ to leave things out, so if you must boil thing…

This is a great and oft-forgotten point. I like to think all summary numbers are lossy, you are only free to pick your poison.

Re: Unlearning descriptive statistics

#42

Post by mziel »

amelius wrote: post #33
somestag wrote: post #15

I think the message of the article is great: move beyond the "standard" descriptions and pay more attention to what you're trying to show and who your audience is. That said, it's a slight pet peeve of mine 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 in that it will always map…

Is it possible to sensibly generalize the concept of median to "higher orders"? E.g., I can imagine the difference between the 25th and 75th percentile to be descriptive of spread (like standard deviation), but those numbers seem arbitrary.

Good intuition :) https://en.wikipedia.org/wiki/Interquartile_range

Re: Unlearning descriptive statistics

#43

Post by sevensor »

FabHK wrote: post #28

To be fair the article should not be called "Unlearning descriptive statistics", but "Learning just a bit more (and often better) descriptive statistics", but granted, it's not that snappy. Certainly better than "I took a statistics course, and you won't believe what happened next!!!1!!"

Agreed, but I've got to say from experience that quantile statistics like the author proposes are often way more informative. The only downside is that they can be considerably more expensive to compute for bigger data sets, whereas mean and standard deviation are cheap and can be updated without taking another pass over the data.

Re: Unlearning descriptive statistics

#44

Post by andy_wrote »

johnmyleswhite wrote: post #13
andy_wrote wrote: post #8

For readers who are OK with some math, I recommend John Myles White's eye-opening post about means, medians, and modes: http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medi... He describes these summary descriptive stats in terms of what penalty function they minimize: mean minimizes L2, median minimizes L1, mode minimizes L0. A single-number statistic is _going_ to leave things out, so if you must boil thing…

Glad you enjoyed that post so much. It really is a shame that we do such a bad job of teaching students about the inherent subjectivity of descriptive statistics and let students leave their courses with dangerous ideas about the existence of a Holy Grail statistic that will solve all of their problems.

Thanks for writing it! It's one of my favorite math blog posts floating out there.

Re: Unlearning descriptive statistics

#45

Post by CalChris »

andy_wrote wrote: post #8

For readers who are OK with some math, I recommend John Myles White's eye-opening post about means, medians, and modes: http://www.johnmyleswhite.com/notebook/2013/03/22/modes-medi... He describes these summary descriptive stats in terms of what penalty function they minimize: mean minimizes L2, median minimizes L1, mode minimizes L0. A single-number statistic is _going_ to leave things out, so if you must boil thing…

Statistics are reductionist!

Well yes, that is their purpose.

Re: Unlearning descriptive statistics

#46

Post by smallnamespace »

zeteo wrote: post #40
somestag wrote: post #15

I think the message of the article is great: move beyond the "standard" descriptions and pay more attention to what you're trying to show and who your audience is. That said, it's a slight pet peeve of mine 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 in that it will always map…

>The median [...] just has a small advantage in that it will always map a real observation, so for discrete data you don't end up with things like "1.9 legs." If you listen to someone like Taleb, the main advantage they mention for the median is that it's a more robust statistic. For fat-tailed distributions the average can jump all over as new data come in.

Whether ignoring the tails is an advantage or not depends very much on the situation.

For example, geometric mean of returns is the right number to look at for stocks because you are indeed exposed to the tails.

Re: Unlearning descriptive statistics

#47

Post by yomritoyj »

I think that the article is unfair to statisticians in saying that they ignore descriptive statistics. The problem is that one cannot choose systematically between different descriptive statistics without clearly specifying our objectives and what we believe about the data-generating process. Otherwise on what grounds can you say that the median is 'better' than the mean or the other way round? But once you specify these things you are doing statistical estimation, which is a major branch of mathematical statistics.

Re: Unlearning descriptive statistics

#48

Post by stdbrouw »

somestag wrote: post #15

I think the message of the article is great: move beyond the "standard" descriptions and pay more attention to what you're trying to show and who your audience is. That said, it's a slight pet peeve of mine 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 in that it will always map…

> In general, the median conveys remarkably little information. It's great at showing the center, but it gives zero indication of anything else.

The median also has zero ability to make me coffee in the mornings, but I don't think I can hold that against it.

I can imagine situations where getting people to track or listen to even a single number is tough and using the mean as a measure of both central tendency and the stability of the distribution over time might be the least worst option. But is that really a common problem? Have you often encountered situations where it was impossible to communicate something like "the typical customer buys five widgets but more than 1 in 4 of our customers only buy one" because it contains two statistics and management insists on being briefed with just one?

Re: Unlearning descriptive statistics

#50

Post by somestag »

srean wrote: post #31
somestag wrote: post #15

I think the message of the article is great: move beyond the "standard" descriptions and pay more attention to what you're trying to show and who your audience is. That said, it's a slight pet peeve of mine 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 in that it will always map…

> 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 s…

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.

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