Who the hell would consider Tukey a fringe statistician?
Lauded, sure, but just about all of his books are out of print and his work is rarely a part of the curriculum.
Unlearning descriptive statistics
61–70 of 95 posts
Re: Unlearning descriptive statistics
#62To answer the author's postscript: "why did nobody tell me this?" Because you didn't pay attention at school? Means, medians, modes and percentiles were standard fare in my high school, and again in freshman university courses. And I didn't go to particularly expensive schools or elite universities. Then again, I also taught intro stat courses to BA students for a few years, and it was easy to tell that the vast majo…
I think the author is trying to point out that mean, median, and mode are things that people think they understand because they know how to calculate them - any one of them is trivial for the majority of the high school educated population (14-18 years old). However, just because you can calculate something doesn't mean you know how to properly apply it.
When you taught your intro stats class, did you try to give the students guidance (and convey understanding) as to when and why they should apply one of the statistics? If so, good on you, but from my, obviously anecdotal experience, that has not been the case.
Re: Unlearning descriptive statistics
#63Earlier quoted context omitted.
1) it is pretty amazing that normit transformations (map the quantiles of a non-normal distribution onto a Gaussian and use that) don't seem to be on this guy's radar. We use distributions with linearly additive and affine invariant properties (normal plus normal is normal, bernoulli plus bernoulli is bitwise bernoulli) because we find linear algebra very useful. Nonparametric tests and procedures erode your power; n…
> it is pretty amazing that normit transformations (map the quantiles of a non-normal distribution onto a Gaussian and use that) don't seem to be on this guy's radar I don't mean to offend, but this is the PhD ur-response, "you didn't mention my pet theory!" :-) You've given me some interesting stuff to chew on but I very specifically wanted to write about descriptive statistics as a way to describe data, not as a wa…
re: normit/inverse transformation: Pet theory? This isn't a theory, it's a simple inverse transformation. It's used all the time. George Box showed how to do a 2D version of it in the 1950s. Normit may be a pet name for it, though. It's just a play on words (probit, logit, expit... normit).
As far as describing data, what's wrong with median + IQR for marginal distributions, or some flavor of energy statistic for joint distributions? You will always need to trade off robustness for sensitivity and bias for variance. That's simply a mathematical feature of the universe. There are plenty of ways to take advantage of this to highlight outliers, for example, which often gets you thinking in terms of "hey, this really looks more like a mixture of two completely different distributions" and seeing if that intuition holds up.
The whole point of describing data with summary statistics is that if the assumptions are met this decouples them from the underlying data. If you use the median as an estimator for the center of your distribution and the MAD as an estimator of its scale, you may choose dimensions along which it's not very good at partitioning your observations. If you want a resistant way to describe the expected center of your data, the median and MAD are very useful. Sometimes it's even more useful to plot everything and point out "our results fall into K obvious clusters" each of which will have their own center & spread.
What I'm saying is that there's no silver bullet. Most of the time we take descriptions of the data, see if there are interesting inferences to be drawn, lather, rinse, repeat. "Get a lot more samples" often thrown into the mix. Are there strong clusters in the data? (Usually a plot will show this, whether via projection or in the raw data) Are there continuums that are interesting in relationship to things we care about? (Usually we'll turn around and model their relationship to said thing-we-care-about, conditioned upon a bunch of other items... multivariate regression, which if you're doing it right, will get you plotting the residuals, themselves descriptive of the model fit)
You simply can't do responsible statistical inference without exploring your data to see what's going on. In order to explore complex datasets, there are plenty of techniques, and most all of them demand tradeoffs (see MAD vs. SD or other metrics of "interestingness" for clustering). A number of descriptive statistics ("extremality" for example) rely upon limit behavior of specific distributions and are case-by-case.
I don't think you'll find many silver bullets for either descriptive or inferential statistics. You have to choose your tradeoffs based on what you want to accomplish.
Re: Unlearning descriptive statistics
#64For 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…
Okay, so that leads to a very obvious (imo) question I haven't seen anyone ask:
What happens when you minimize Ln, with n > 2? Why don't we use any of those?
Re: Unlearning descriptive statistics
#65Earlier quoted context omitted.
> 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…
>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%."
Re: Unlearning descriptive statistics
#66Earlier quoted context omitted.
Great points. Discrete data with few categories are a good case where the median might not cut it. As a contrived example.. data: 0 0 0 1 1 mean: 2 / 5 median: 0
With a small number of categories a full summary of the data easily fits into text. It might still be desirable to report a summary statistic, but omitting the totals for 2 or 3 categories and reporting a summary statistic is pretty sloppy.
> With a small number of categories a full summary of the data easily fits into text.
That's fair. In my experience, summary measures are often calculated at many levels of an analysis, not just the final reporting.
Re: Unlearning descriptive statistics
#67I wrote an article for the ACM Queue (also published in CACM) about this very issue called "Statistics for Engineers" http://queue.acm.org/detail.cfm?id=2903468
I took a "Statistics for Engineers" course while at university. It was mostly focused around ANOVA and hypothesis testing. By the end of the course I was sick of hearing the phrase "Null hypothesis". I wish the course had introduced some more practical applications instead it was very high level and theoretical.
Working as an engineer now (Materials Engineering) here is a short list from the top of my head of what I feel are useful statistics applications that were not covered in uni.
- Alternative probability distributions (Rosin-Ramler) - Harmonic and Geometric means - Multivariate regression - Statistical Sampling techniques (how to obtain a representative sample and avoid sampling bias) - Statistical process control - Time Series analysis
Re: Unlearning descriptive statistics
#68For 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…
Okay, so that leads to a very obvious (imo) question I haven't seen anyone ask: What happens when you minimize Ln, with n > 2? Why don't we use any of those?
Increasing n gives increasing weight to outliers, to the point where L-infinity is just the single maximum value.
I would guess this is less useful when attempting to understand how existing data can make conclusions about future aggregate/typical cases vs. analyzing specifically the outliers.
Re: Unlearning descriptive statistics
#69I would like to advertise again my favourite measurement of skewness, the medcouple:
https://en.wikipedia.org/wiki/Medcouple
It's robust (25% breakdown point) and is kind of the "optimal" measurement of skewness, as it's computed as the median of all possible interquartile range-like statistics.
I hope someone implements the faster algorithm for Python's statsmodels.
Re: Unlearning descriptive statistics
#70Variance and standard deviation gets a lot of hate here, that I don't think it deserves. It really does have nice properties, generally nicer than the MAD. Squaring feels to me much less of a hack than the absolute value. The problem that most people don't quite understand it is certainly true, but that's as much a problem with the people as the method.