Talk:Simpson's paradox

1915:"intuition" to suggest a natural choice. Examples like the "Bertrand paradox" show that this it not unproblematic. I would argue that here, there is not enough context or "intuition" to give those without any precise understanding of statistics/probability any sense of what it means to fill in a 2x2x2 table "at random" according to the distribution assumed in the Perlman paper, and it is potentially misleading to present the context-free assertion as if it has enough context to determine an intuitive meaning. (I should point out that the paper itself makes no claims that this distribution is "the only one" worth considering--- nor does it argue, for example, that actual statistical practice in filling in 2x2x2 tables is at all comparable to the model they assume when they calculate the .0166 figure cited here. It just computes various probabilities in a model.) 1374:

investigated for a proper analysis. But in the Bart-Lisa case, the underlying groups are 'week 1' and 'week 2'. Why are the success rates of editing being divided into weeks? The only reason for doing so would be that the success rates are changing consistently across weeks for both Bart and Lisa. But I can obvious see no reason why 'week' would be an appropriate grouping factor. This example makes the impression that you should divide your data into different groups for no reason and assess across those meaningless groups - perhaps doing so until you get the answer you want (e.g. Bart should be better than Lisa. We don't see in across both weeks, so we divide into weeks and aha! there we see it. If we hadn't seen it within weeks, maybe we should divide into days...)

2605:

in each case, making it very easy to understand what is happening. In addition, one editor had the good idea of adapting two of my figures (the weighing scales and the vector interpretation) to this actual example. I think the result is quite nice, and actually adds to the whole article. However, everything from "Here are some notations:" still seems long and clumsy to me -- not to say that it adds little to what is in the tables. I would at least remove this part of the section, but the rest of the example is useful. (ideally, we should find a real examples using such low values, and use that one instead...).

2131:"This arose because regional affiliation is a very strong indicator of how a congressman or senator voted, but party affiliation is a weak indicator." This statement is obvious from the chart, and can even be made more formal. Let's say I pick a Senator or Congressman that voted on the Civil Rights Act of 1964 in a uniformly random way, and you have to guess whether they voted for or against the Civil Rights Act. You get to ask one of two questions: "Do they represent a formerly Confederate State?", or "Are they a Republican or Democrat?". Which question do you ask? 208: 187: 3323:

men favour and lower to the programs women favour. But thinking systemically, one should ask why Berkeley enables programs that men favour to admit more students? Similarly, looking at the death penalty sentencing example, one could ask why the death penalty is given more often when the victim is white vs. black. In other words, the paradox provides an explanation but adding a new variable isn't necessarily the end of the investigation. Perhaps there could be a new section after implications for decision making?

113: 1935:"quality" that is different than the common use, where quality is usually contrasted quantity. Here's the sentence: "Also when the two tests are combined using a weighted average, overall, Lisa has improved a much higher percentage than Bart because the quality modifier had a significantly higher percentage. Therefore, like other paradoxes, it only appears to be a paradox because of incorrect assumptions, incomplete or misguided information, or a lack of understanding a particular concept."-- 103: 82: 490:

management. Then, of the women in the management, 8/10=80% are highly competent, and of the sub-management women, 9/90=10% are highly competent. Of the men, only 14/20=70% of those in the management group are highly competent, and only 4/80=5% in the sub-management group are highly competent. So, in both groups, more of the women than of the men are highly competent, but combined, only 17/100=17% of the women are highly competent, while 18/100=18% of the men are.

3341:

because admission rates are higher to the programs men favor and lower to the programs women favor. The example illustrates that discrimination can arise from the allocation of resources, instead of (or as well as) from individual admission decisions. Thinking systemically, one should ask why Berkeley allows programs that men favor (e.g., engineering) to admit a higher proportion of students than the programs women favor (e.g., English)?

2168:

In fact, the Knowledge article on the Civil Right Act itself links back to this article. This example should not have been removed without updating that page as well. The example is not original research since the data is taken directly from the CRA Knowledge article. The text in the example could be improved but the example should be restored. I will update the text and restore the example unless there is a reasonable objection. -

3302:

entirely. For example, the Kidney example fits a causal model where A is indeed better in all cases, but biased assignment of treatments led to misleading overall estimates. The UC Berkley example fits a model where it does indeed have a problem with rejecting women applicants more often than men, but the fault lies in the under-availability of positions in highly-demanded departments rather than in the admissions process.

1403:

situation to occur. The underlying cause in the Bart-Lisa case is not systematic (a lurking variable). The cause is random variability. The case is very plausible, since the sample sizes are so low. But does it matter what the source of the paradox is? (Random variability vs. lurking variable?). It seems to me that the paradox is the paradox regardless of the underlying causality.

51: 1955:

different readers will "get it" from different ways of presenting it, it would be clearer to present THE SAME example in different formats. My suggestion would be to present the various examples in more or less the same way. Personally, I prefer the presentation of the kidney stone example with the "group 1, 2, 3, 4" and the two effects listed. What do you think?--

21: 3296:

proportions in the study. If we wanted to estimate the population-level effects of using treatment A over treatment B we would need to weight our samples to adjust them to population proportions of small and large stones (and for other potentially confounding factors). The population-level estimates would then agree with the indication-specific estimates.

3002:

stars of the first product more than the 4.5 of the second. So it's entirely possible that you'd similarly prefer treatment A to B on small stones. Can't prove it on the actual number set without math I don't know, but logically it seems plausible. At least enough to demand further evidence and sourcing for the assertion to the contrary

1628:

smokers overall have a much higher mortality rate than infants of non-smokers. This is (of course) because many more infants of smokers are low birth weight, and low birth weight babies have a much higher mortality rate than normal birth weight babies. The reference does explicitly state that it is an example of Simpson's paradox.

965:"If Bart only edited one article (and that one edit brought about world peace), Lisa's lifetime of editing thousands of articles may statistically appear better (to friends, family, politicians, religious leaders, and others viewing the statistical view), but may be judged by history to be worth less than Bart's one edit." 3518:

counterargument could be made, it's very important that the examples be relatable in order to teach readers, and the current pandemic is probably the most widely relatable topic in statistics right now. I've dug up a few papers on the topic of COVID-19 and Simpson paradox that might be worth checking

3295:

Unfortunately, the implied interpretation of the kidney study is misleading. The aggregate success rates depend entirely upon the mismatched ratio of treatment assignment to large and small stones, and so the unweighted aggregates have no valid use in decision-making outside of the specific treatment

2438:

This paradox is related to the old joke about the man who left Scotland for England and thereby raised the average IQ of both countries. It is particularly clear in the joke that the overall average cannot change, because the two situations (before and after) are simply different partitionings of the

2351:

Less seriously, but still confusingly, the height of the bars appears to be only crude estimates. Bart's high contribution of 100% appears to be quite the same height as Lisa's high contribution of 71.4%, while Lisa's low contribution of 0% appears to be the same height as Bart's low contribution of

2347:

I believe that the 'Bart' graph (identified as the lower one) in this example is seriously misleading. The percent of articles improved by Bart (14.2% in the 1st week and 100% in the 2nd week) do not fit at all with the graph, which appears to show the opposite: a much greater percent improvement in

2167:

I think the Civil Rights Act example should be restored. A quick google search confirms that many math websites use this example when discussing Simpson's Paradox and specifically cite this Knowledge article. This suggests that math educators consider it at least as relevant an example as the others.

2134:

Which question you should ask is obvious: asking about the party affiliation is absolutely no use in making your guess, they best you can do after asking this question is to simply guess "yes", which will be correct 70% of the time. On the other hand, if you ask what region of the country they

1982:

average test score in mathematics for American 9-year-old children rose, from 1978 to 2004, by 10.0%. But the average score of white 9-year-olds rose by 10.3%, that of Hispanics by 13.3%, that of blacks by 16.7%, and that of all others by 12.8%. Thus while no racial/ethnic group experienced a gain of

1914:

Omitting explicit reference to a probability distribution in choosing an object "at random" is commonly done in elementary expositions of statistical concepts when the situation is simple enough that the distribution can be inferred from the surrounding context, or there is in some other sense enough

1756:

I removed it because it seemed redundant, unnecessary - the current table (it seems to me) shows everything that the first table showed; in fact, it contains that entire table. Listing two different tables made it seem, I thoiught, as if something changed between them, but the second table was merely

1740:

recently removed the first table in the kidney stone example, which showed only the results when no distinction is made for kidney stone sizes. As the section now stands, I don't find it satisfactory. I think it needs to be made clearer that false conclusions may be drawn when the lurking variable is

1545:

changing vector to line in one instance. First, the section it's in is called "Vector Interpretation", so referring to vectors is the expected language of that section. Second, the word change was made in only one instance, making the whole paragraph internally inconsistent as it switched from line

1362:

removed it, since the consensus above was for now to move the example rather than delete it. We all agree that we have enough real examples and do not need fake examples on top of that; however, this section is the only one that goes beyond giving an example, but also discuss the question of weighted

1279:

We have four different real-world examples now, some with statistics. Do we need the "bart/lisa" fake example to explain it any more? At the very least, I'd like to move the real examples up above the pretend one - I think lots of people stop reading when the article lurches into "explaining" mode. -

303:

As an example, suppose two people, Ann and Bob, who are let loose on Knowledge. In the first test, Ann improves 60 percent of the articles she edits while Bob improves 90 percent of the articles he edits. In the second test, Ann improves just 10 percent of the articles she edits while Bob improves 30

3340:

This paradoxical result illustrates the difference between overt and systematic discrimination. Berkeley really did admit a lower proportion of women than men. The paradox explains that the outcome is not overt sexism, since women are if anything favored in admission in most departments, but rather

3322:

The discussion of the paradox provides an entree to thinking about systemic vs. overt discrimination. For instance, Berkeley really did admit a lower proportion of women than men. The paradox explains that the outcome is not overt sexism, but rather because admission rates are higher to the programs

3001:

This section suggests that obviously Treatment A should always be preferred. I think that's wrong. Think about it. If you saw a product on Amazon with 4 stars and 1000 ratings (small stones-treatment B) vs a product with 4.5 stars and 10 ratings (small stones-treatment A), you'd probably trust the 4

2604:

Actually, now that I have reread the section in more details, I see that it has changed since I last read it. In the past, the numbers used as example were in the order of 100, making the Bart and Lisa example similar to other, real-life, examples. I see that the example now uses a total of 5 events

2390:

I agree. The examples and the graphic are quite different forms of Simpson's paradox and I don't see how one is explained by the other. So it seems odd to have a linear-trend-reversal as the most prominent graphic but then only give examples of ratio-reversal. I suggest adding (or replacing) another

2070:

An examination of racial differences in the management of localized prostate cancer in Pennsylvania simultaneously revealed that whites were more likely to receive prostate surgery than blacks, that whites and blacks were equally likely to get surgery, and that blacks were more likely to get surgery

2039:

I think it's a good real world example that seems to differ from the existing examples because it is not the actions that are changing over time but the composition of the actors. Mathematically it's the same, but I find it intuitively quite different. If space is an issue, I prefer it to the made

1243:

I suspect the analogy (the College cannot reproduce the paradox exactly since the outcome in each state is only related to the difference in votes through the sign of the difference, not magnitude. One could not lose the College if every state was won.) is that one can "win" the nationwide popular

799:

Thanks for your changes, it reads more clearly. I don't have access to the original study, but from the review and title it appears to compare surgery, ultrasound and/or using catheters. Unsurprisingly the open surgery (treatment A) is the most effective, and probably is the most the expensive with

790:

therefore primarily given to those patients who need it the most. In fact, if there are no other confounding variables involved, and if A is more expensive than B, then, within a given budget, the largest number of cures is obtained by treating as many as possible from the large-stone-group with A.--

628:

The storytelling conceit, complete with sly reference to those other Simpsons, "Bart" and "Lisa," works well for me. This kind of explanation helps me in explaining a concept to others, even as I work to fully grasp it myself. The inclusion of the Knowledge within the definition does not seem overly

528:

It all makes sense in the end, but it's still initially surprising for most people who are not aware of the explanation or suspect it. If you only know the partial percentages, then the total percentages would come as a surprise to most people. Obviously, once the weights are introduced, the initial

2813:

Ouch to see this sketch you would have to click out into Twitter, because I cannot upload that sketch directly here, because indeed I cannot "attest that I own the copyright on" this image. I only know I remixed it from Twitter's video transcode of their Gif of someone's two slides, it is copyright

2688:

P.S. Also, we could easily imagine many ways to partition the 10,000 rich and 1,000 poor voters in the example (e.g. college graduates vs. others, baseball fans vs. others...) where the poor voters end up also less likely to vote for Trump. I can think of no obvious way to decide which partition is

1860:

Although this situation is called Simpson's paradox, this article is very useful in illustrating a fallacy in statistics that can be corrected. Of course, Simpson's paradox goes away when one properly accounts for external variables. For example in the Male/Female admissions lawsuit, the statistics

1580:

I agree. It looks like the example states that, given that a child is low birth weight, it has a lower infant mortality rate if born to a smoking mother. It would only be an example of Simpson's paradox if, given the child is born to a smoking mother, it has a lower infant mortality rate if it were

1402:

And, it was by far the easiest one for me to understand because one didn't need to understand or even know anything beyond the simple data that were presented. After I thought about the Bart-Lisa case for a long time, it suddenly hit me how it is possible--even easy!--for the seemingly paradoxical

3615:

Another criticism of the apparent Simpson's paradox is that it may be a result of the specific way that data is stratified or grouped. The phenomenon may disappear or even reverse if the data is stratified differently or if different confounding variables are considered. Simpson's example actually

2020:

might be interested in this example at least as much for its intriguing policy implications, rather than as a good example of the paradox. The point of examples in a wikipedia article is to illustrate the concept, not to make an argument for it or to cover all possible bases - and five examples is

1934:

In the Bart/Lisa example, which I found the most helpful example, especially the graph, do we need to define "quality modifier" or at least provide a link to another article? Frankly, I am not sure what is meant by this phrase. How can numbers be qualitatively different? Clearly, this is a use of

1910:

I've just removed the section on "how likely" Simpson's paradox is. The reason for this is that in order to make sense of the statement you need to assume a probability distribution for the entries of a 2x2x2 table (presumably what the section's statement about "assuming certain conditions" was a

1627:

It's poorly stated, but the paradox is that normal birth weight infants of smokers have about the same mortality rate as normal birth weight infants of non-smokers, and low birth weight infants of smokers have a much lower mortality rate than low birth weight infants of non-smokers, but infants of

523:

For Ann, the time that she royally screwed up barely counts, while the time that she did poorly counts the most. For Bob, the time that he royally screwed up hugely affected his total, while the time that he did amazing barely counts at all. I don't quite see why the results are surprising. Anyone

2575:

Well, I don't know if "no wikipedia article is allowed to exist without a Simpsons reference", but given the title of this page, I see why this particular one would be a good candidate for a Simpsons reference ;) Joking aside, I've never been a big fan of this section -- "long and clumsy" sums it

2442:

You could modify the joke so that the overall average moves in the opposite direction, e.g. if two men leave Scotland and only one goes to England. But that would not only reduce the elegance of the joke; in fact, the constancy of the overall average in this case brings out the true nature of the

1997:

As for point (2), I definitely disagree with it -- just five examples are not enough in my opinion. The more examples we have, the more likely a reader is to find one that resonates with him, one that he can latch onto as, for him, a memorable example. Different people will be prone to latch onto

1519:

relationship controlling for confounding factors. The fact that a conditional relationship can have the opposite sign of an unconditional relationship is precisely Simpson's Paradox and is also precisely the reason why correlation cannot be taken to imply causation. No two concepts could be more

727:

And, continuing that thought and going back to the self reference guideline, the plan as I have understood it is to eventually do a printed Knowledge. Regardless of the form, any time this article appears outside the wikipedia.org website the chances of the reader understanding the example become

3547:

No question that current events like COVID-19 make for lively examples, but even with a ready consensus there is a pedantic effect of injecting a hot topic into an article that should be nothing but dreary mathematics. That's why trivia like baseball statistics are worthy. I thought about adding

2182:

If you must, but add a good reference and make sure it's clear, because the previous wordy example wasn't. The websites that "cite" the article seem to be mostly wikipedia scrapers - a very circular argument, at best. To be honest, I think it's a lame example of the Simpson's paradox because the

2150:

I have removed the Civil Rights example, partly because we're getting too many real-world examples - we still have four, all of which are pretty well known - and partly because it's not sourced. The data and analysis may be accurate, but they're not reported elsewhere, as the other examples are.

789:

I have not consulted the references on this case story, but perhaps someone who has (or will) can answer my question. I imagine one of two answers: (i) Before this particular investigation, they did not know that B was inferior even in the milder cases. (ii) Treatment A is more expensive, and is

723:

However, I don't believe that the act of editing Knowledge articles is a good example of much anything, because most people I know who read Knowledge have never edited anything. I've been reading for years and only today even created an account to post anything. So the example took a little more

470:

Assume a population with 50% men and women and in both groups competence is spread in the same way. Imagine a situation where women are required to have more competence to get a promotion to management. You will then notice that women on the management level are more competent than male managers

314:

The result comes about this way: In the first test, Ann edits 100 articles, improving 60 of them, while Bob edits just 10 articles, improving 9 of them. In the second test, Ann edits only 10 articles, improving 1 of them, while Bob edits 100 articles, improving 30 of them. When the two tests are

639:

Agreed. The examples are clear, well written, and logical. And the references to Bart & Lisa Simpson are not only clever and fun, they also make it EXTREMELY easy for many people to remember this phenomenon as well as its associated name. If we name them Dick & Jane it would be far less

3301:

In fact, the important question is how the data fit into our causal model, and that is only discussed in a very abstract sense in the third paragraph. If we could find good sources, I'd recommend expanding the causal model discussion with concrete examples and removing the preceding paragraphs

2551:

Somebody placed a "tone" hatnote on this article but didn't give any details about what bothered them. I hate it when editors do that and often remove such hatnotes, under the assumption that if they can't be bothered to explain their reasoning then the rest of us shouldn't be hassled by their

1954:

Except in the sex bias case (where there are more than two departments), many of the exampls presented seem to have exactly the same structure and could be presented in the same format. Presenting the various examples in different formats seems confusing to me. If the point of doing so is that

1786:

It seems to me that these sentences following the table make the point clear: "The paradoxical conclusion is that treatment A is more effective when used on small stones, and also when used on large stones, yet treatment B is more effective when considering both sizes at the same time. In this

1392:

I don't understand this comment, or maybe I misunderstand the definition of Simpson's Paradox. I thought that the term applied to any case where the two group results agreed with each other but disagreed with the aggregate result--regardless of the underlying cause. If so, then the Bart-Lisa

715:

The question was raised as to whether or not it's appropriate for this article to reference Knowledge . I believe it may be, but should certainly be discussed. The point of avoiding self references, as I read that guideline, is to not use phrases such as "elsewhere on this site" or "in another

670:

The first (graphical) example could be made a whole lot clearer if the symbols x and y and relationships between them were explicitly defined. I would be pleased to contribute to this cause, but -- well, I am still bewildered by it. The other (real life) examples work quite well and make the

2748:

It is borderline excessive. We could certainly toss the low birth-weight paradox, referencing to it in "see also" - and frankly, I've never been a fan of the Bart and Lisa item, as above discussion shows; I don't think we need a fake example when we have such detailed description of real life

1161:

Then I am at a loss. I am certain I understand Simpson's paradox, and I am certain it (in the Bart-Lisa-example) has nothing to do with distingushing between large and small improvements. The context is clear (wikipedia editing, some edits being improvements, other not). Adding more context -

489:

Suppose we have 100 men and 100 women. 18 of the men are highly competent, and 14 of them are in the management. Of the 82 less competent men, 6 are in the management. 17 of the women are highly competent, but only 8 of them are in the management. Of the 83 less competent women, 2 are in the

3476:

Since someone previously deleted my added section, I would like to open a discussion about it. I think the example shows the enormous practical relevance of the paradox as it leads to an underestimation of the vaccine efficacy and hence probably to a lower vaccination rate which has deadly

3244:

The batting average is very well known - probably the best-known example in the United States. It should not be removed. My objection to the made-up example is that it's made up, yet we have several very clear real-world examples which show that this is not a theoretical issue but an actual

1127:

Well, I believe I have made my concerns clear, where as I do not understand what your point is. Do you think your contribution is related to Simpson's paradox, or does it merely offer an alternative angle on the Lisa-and-Bart example, an angle unrelated to Simpson's paradox? Do you actually

1373:

The Bart-Lisa example is pointless and misleading. The whole point of Simpson's paradox is that differences in underlying groups may be causing changes that lead to misleading results when the groups are not taken into account - the underlying groups are important in themselves and must be

2555:

I suspect the problem is the long and clumsy section titled "Description," which gives an imaginary example of the paradox involving Bart and Lisa (because no wikipedia article is allowed to exist without a Simpsons reference) and which is, indeed, written in an unnecessarily loose tone.

3402:

The section on the racial disparity in death sentences should, I think, be removed because it ends: "Radelet found that none of the aforementioned correlations were statistically significant" - That makes it a very poor example of the paradox and we have three other real-world examples.

1299:

I agree with the removal of fake examples (as I've just done with the baseball example). This section should be moved below the examples, and then transformed into a general discussion of what may cause the paradox to appear (talking about weighted averages, confounding variables, etc).

3356:

Hi, and welcome to Knowledge. Something about what you're describing might possibly be able to be added to the article, but there are problems with this edit. Firstly, it's not really written in the kind of encyclopedic tone that Knowledge articles are meant to be. Have a quick read of

3616:

highlighted a phenomenon called noncollapsibility, which occurs when subgroups with high proportions do not make simple averages when combined together. This suggests that the paradox may not be a universal phenomenon, but rather a specific instance of a more general statistical issue.

1490:

I think the lead is fine as it stands. Correlation/causation is a much wider topic than Simpson's paradox, but it seems to me the ONLY relevance of Simpson's paradox is that it is ONE of the counterexamples that can be used to reject the intuition saying that correlation DOES imply

3522: 2135:

represent, you can be correct 91% of the time: you'll be right 90% of the time by guessing "yes" if they come from the north (which happens 75% of the time), and you'll be right 94% of the time by guessing "no" if they come from the south (which happens 25% of the time).

719:

The article could reference bowling or mowing lawns or a great host of other activities where the characters' performance can be quantified. I suspect the Knowledge reference was used simply because the author assumes that those reading it will be familiar with the process.

3556:) that uses a correlation of COVID-19 vaccines to higher death rates to demonstrate Simpson's Paradox, but even though he's a reliable source and he's correct (as far as I can tell), it's still a radioactive topic and personality that would invite an edit controversy. 3151:

I've just seen your messages today and think the Bart/Lisa example easier for a layperson to understand that the real-world examples as the numbers are smaller, also making the vector graph easier to read. I'd prefer restoring it, but await a second opinion. Cheers,

2658:"This aggregate phenomenon of poor voters being seemingly less likely than the rich to vote for Trump is driven by the following facts: (1) the majority of voters are white; (2) the white are more likely to be rich; (3) the white are more likely to vote for Trump. 1992:

Jack Jennings of the Center on Education Policy uses these data in asserting that when the composite data are used, "one important trend tends to be overlooked -- namely, the notable gains made by African American and Latino students in reading and math

1987:

His reason for reverting was that (1) it's a weak example since the composite went up, not down, and (2) we really don't need another example. As for (1), I see his point, but I propose adding the following to illustrate the importance of the example:

3067:

Reading through this article again, I'd still like to dump it and replace it with one of the real-world examples. I don't see how it explains anything better than several of the real examples, which carry more weight because they're not hand-waving.

3626:

Critics of the apparent Simpson's paradox also argue that the focus on the paradox may distract from more important statistical issues, such as the need for careful consideration of confounding variables and causal relationships when interpreting

1179:

In both the Lisa/Bart example and the kidney stones example, there is a 3x2 table with 6 entries. How can the Electoral College data be presented in this way? There are the 2 parties, so that's the "2" dimension. But what is the "3" dimension?

2311:

Now, you could certainly have a Simpson's paradox in such a case, but you would have to say something like "a treatment could benefit both males and females, yet a group receiving the treatment did worse on average than a group not receiving it"

1771:

YYes, the second table contain all info, but the way the section reads now fails to make an important point clear. The easiest way to fix that is to revert your edit, but I'm sure there are other ways (and probably better ways) to fix it. Feel

1337:

Too late :-) I'll think about the transformation, but, as you say, it requires quite a bit of thinking first. Before that, I'll add a few more references and reformat the examples, and hopefully (if I can get around to doing it), add 2 images.

2728:

Do we really need six examples (counting Bart and Lisa) for the paradox? They take up more space than all the other sections combined. I would keep only two or maybe three at most, that should be illustrative enough for anyone IMO.

2301:"Psychological interest in Simpson's paradox seeks to explain why people deem sign reversal to be impossible at first, offended by the idea that a treatment could benefit both males and females and harm the population as a whole. " 2071:

than whites. This example statistical analysis used hypothetical data. All of the above conclusions were correct, but they reflected answers to subtly different questions that relied on different parsings of the same aggregate data.

2859:

Given that Men are more likely to be affected than women by some disease, and that young people are more likely to be affected, it is reasonable to conclude that young Men are the most affected group. But it's not true, of course.

3477:

consequences. If there are too many examples, I suggest to delete the "Batting averages" example which has no practical relevance. It was criticised that the example is just speculation. While there is only one source indeed,

2374:

The graph next to the introduction seems to be misleading, as in that case the groups are distinguished by the variable in which the trend appears as opposed to the examples in which the groups are distinguished by some other

2090:

Agree to deleting this example. For one thing, the above text is a travesty of the referenced material. For another, the data in the reference do not in fact provide an example of Simpson's paradox as there is no reversal.

1787:

example the "lurking" variable (or confounding variable) of the stone size was not previously known to be important until its effects were included." But perhaps not; perhaps the matter needs to be expanded or clarified. -

3210:

If it is felt that there are too many examples, I'd rather replace the batting average one with this. Baseball statistics are not universally known about. If there is no objection, I'll remove it and restore the Bart-Lisa

2524: 1678:

sounds interesting. However, as it stand, I don't think it belongs. EITHER, it should be expanded to make it an illuminating exapmle of the paradox, OR it should be removed or boiled down to at most one sentence and a

2464:

Note that the initial populations of Scotland and England need not have any particular ratio for this to work. I haven't thought for long enough yet about the precise relationship between this and the other examples.

2062:

The article has six examples of the paradox appearing in real-world situations, which is IMHO excessive. I'd like to remove this one, because it's the least detailed and informative; in fact, it's kind of confusing:

1083:

If you are seriously suggesting changes to the article, I think you should either be bold and make those changes, or explain clearly at this talk page what you'd like to change, and why. I've no idea what your point

629:

self-referential, as one observer has worried. Entries like this are the reason I seek out Knowledge's take on things before looking to other, traditional sources. Thanks for an entertaining and elucidating entry!

1889:

True - like most paradoxes, it's only paradoxical when when described in a misleading way. You can see a paradox as a challenge to find the right way of describing the situation. - Do you suggest a change to the

924:

that you quote may be misleading for the same reason: They seem to suggest some edits not merely improve articles, but that they display particular diligence, which (though of course true) is, as I said, utterly

1801:

What made the fallacy clearer was that the "combined case" and the "obvious" conclusion was stated before the extra information was added and the refined conclusion reached. I think this was a more paedagogical

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I feel like this is a fallacy that's committed extremely commonly (e.g. by newspapers) and as such deserves mention. (I have looked in other places too but couldn't find it. Please let me know if I missed it.)

298:, in which the accomplishments of several groups seem to be reversed with the groups are combined. This seeminhgly impossible result is encountered surprisingly often in social science and medical statistics. 503:

competent. However, I like the original better, and I think someone should go ahead and add it to the article. I'm afraid it takes skills beyond mine to write it in a simple way that makes it clear that it

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fictitious, original, self-referent narrative unnecessary. Finally, if a vote gets taken, please cast mine in favor of "fallacy" -- not to exclude "paradox" but to strengthen the importance of this entry.

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I do not believe the section is necessary at all, because we have several real-world examples that provide just as much illustration. I would like to kill that section altogether. What do others think? -

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issue it examines is not very clear, a mix of geography and politics that requies a knowledge of that period of American history to seem surprising. The other examples are much more straightforward. -

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paradox: not only can the overall average and the sub-averages move in strictly opposite directions, but more generally the overall average and the sub-averages are decoupled in a surprising sense.

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If you find a source for the material you're talking about and clean up the prose a bit, it could maybe be okay, but I'm not entirely sure this is really on topic for an article about statistics. ‑‑

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It's been a week. I want to be cautious about making such a big change, deleting something that's been in the article for so long, so I'll ask again: What do others think of deleting the section? -

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reference to). My basic argument here is that without a statement of those "certain conditions" the statement is essentially meaningless, so we have to go to the paper to find out what it means.

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averages. I don't think it is very good, or that it covers everything it should, but at the moment it is better than nothing. If nothing happens with it in the near future, then it can be removed.

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One criticism is that the paradox is not really a paradox at all, but rather a failure to properly account for confounding variables or to consider causal relationships between variables.

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Two years ago I suggested dumping the whole Bart and Lisa section - the made-up example. It was improved and the only other editor who responded at the time didn't mind it, so it stayed.

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The correlation/causation issue is important in its own right, but has little to do with "Simpson's Paradox." I would suggest removing this part of the text in the extant introduction.

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Seems reasonable enough to me, although I wouldn't say "accomplishments" for "successes". "Success" in statistical jargon is not necessarily a positive thing! How about "ratings" instead?

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When you say it like this though, its not very counterintuitive, if you've read the earlier part of the article. Hence I think it's not needed and I just removed the illogical clause.

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I do not support the idea that the phenomenon is not "really" a paradox. Many good paradoxes are based on representing a situation in such a way that a false conclusion seems obvious.--

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Since we don't really need this many examples - the situation is quite clear that it crops up in reality in many different circumstances - it's no drawback to just get rid of it. -

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is very strongly connected to Simpson's Paradox. Correlation is based on the unconditional (or marginal) relationship between two variables. But causation would be based on their

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that women in sub-management are more competent than men on the same level. This seems paradoxical at first considering that, on the whole, women and men are equally competent.

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I'd like to change the first few paragraphs of this article to make it friendlier to folks afraid of math, and was wondering what other people thought. Here's a possibility:

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less than 10.3%, the children as a whole experienced a gain of only 10.0%, a result that is due to the shift over time in the percentages of the various groups in the total.

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same set. Yet the sub-averages can both move in the same direction. (They both go up if the Scotland average is higher than the England average and the man is in between.)

3252:) , and editors making up examples that they think are clearer to layfolk is classic textbook material. Finally, I don't think it IS clearer, it's just another example. - 785:

I expanded the text on the two factors at the end of the section to relate more specifically to the medical example. Reading what I've written, it seems natural to ask:

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not identified. One way to clarify this would be to put back the table (reverting half the edit in question), and I'm inclined to do that - but I'll wait and see...--

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Agreed with Niels. The key point to remember is that in the baseball batting average example, there are large differences in the number of at-bats between years.

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Both times, Bob improved a much higher percentage of articles than Ann - yet when the two tests are combined, Ann has improved a much higher percentage than Bob!

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I've removed the Electoral College example, it's not an example of Simpson's paradox. Unless, that is, someone can show how it fits the 3x2 table pattern. --

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It still seems unnecessary to me, but you're right that the "notation" section is really unnecessary. I have removed that portion, and the "tone" hatnote. -

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good edits from minor improvements; that's not what the example is about. Whether they elucidate complex subjects is utterly irrelevant. However, the words

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P.S. The discussions above imply that at least some readers find the Bart-Lisa example useful as the numbers are much smaller and can be easily visualised:

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It doesn't strike me as much different than the examples we already have - not really worth adding as yet another example to the article, in my opinion. -

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I had forgotten which paradox was this paradox of 1899 Pearson et al., 1903 Yule, 1951 Simpson, but googling your work reminded me in a moment, thank you.

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Would it not be more natural to say "tell" instead of "retell", since it is the original statement of the situation that appears to have this conclusion?--

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Conclusion: This is indeed a Simpson paradox, and the only change compared to that suggested above is that I made it a little sharper by making the women

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Its not strictly a paradox, since there is a straight forward solution. But, its widely known by that name, so we ought to keep it. --best, kevin

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The caption to the vector interpretation image should be changed so it doesn't reference an example that can no longer be found in the article.

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Like some sketch with all of four groups misleadingly reversing the true overall trend, something akin to the image held back from us here at my

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It's a nice example. In order to convince myself (and perhaps others) that it's the same paradox, I'll now assume that on average, the women are

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irrelevant to the paradox - will confuse matters by having readers trying to understand how it is relevant. Please explain, what is the point?--

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I don't understand the psychology section at all. I would rewrite it, but I have no idea what it means. Could someone who does fix it please?

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is good and useful, but it needs sourcing. Right now it's a textbook example of original research / synthesis, which wikipedia frowns on. -

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I think I understand what is trying to be said, but no sane person could describe a situation where a treatment benefits males and females

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I have recently been browsing the logic & game theory articles. This is the best I have seen so far. Congratulations to all concerned.

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competent than the men (no offence, just to sharpen the paradox and make it clearer that Simpson is involved), and I'll add some numbers:

2955:(1/1). Do you consider this a variant of Simpson's Paradox? In some sense, it's a double application of the fallacious subset principle: 3303: 3003: 2280: 1998:

different examples, but I'd bet that more people will latch onto the test scores example than, say, the kidney stone research example.

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graphic - that's not an obvious illustration of Simpon's paradox at all. It will confuse readers who know nothing about the topic. -

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I propose to add . To see how this would affect the over-all appearance of this article, view the proposed revision in my sandbox. --

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I wonder if this is the same paradox and if it could be used as an example. I find it very easy to understand — and from real life.

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added together, both edited 110 articles, yet Ann improved 69 of them (63 percent) while Bob improved only 40 of them (36 percent)!

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How is this an example of Simpson's paradox? From the information given, I see only a medical "paradox", not a statistical one.

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I was about to make an almost identical heading. It's a pretty asinine self-reference in addition to being original research.

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worth of work/Success/managed/achieved successful/worse/we feel/disappointed/accomplished/mutual friends think/better/diligent

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https://en.wikipedia.org/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_manual,_guidebook,_textbook,_or_scientific_journal

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Does Simpson's Paradox always disappear when causal relations are brought into consideration, as the text currently implies?

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I haven't checked the sources, and those critisisms may exist, but they all seem ill-informed to me. Do we need them here?

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what are the respective departments. I don't seem to able to find them in the given citation or the actual research paper.

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can be shown with a common weighting of departments (apples-to-apples-comparison). If this is done, there is no paradox.

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If so, you appear to be writing "Simpson's Paradox does not distinguish between large improvements and small improvements"

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OK, so I did it - although I kept the Vector Interpretation section, moving it down below all the real-world examples. -

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What is the critisism here? It seems to me to reiterate what is actually the point of the paradox as a cautionary tale.

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should have a link somewhere from this article - if it's not appropriate at any pther point, it should be in "See also".

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We have clarified our disagreement: It struck me as redundant, even a bit confusing. Anybody else have an opinion? -

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I don't see how that would be Simpson's paradox either. If low birth weight meant lower mortality in both smokers

1455: 3531: 2799:"I see this trend here, there, everywhere. And it reverses when I zoom out. ~~ Jabawack MethodsManMD Aug/2016 ..." 2666:

I don't like the choice of words here. Is controlling for race necessary? One could also say, once we control for

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But it is possible to have told the story in a way which would make it appear obvious that Bart is more diligent.

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In other words, I like the example used here, but a different example may be more comprehensible and practical.

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I'm just a newcomer here, but maybe these changes would be better suited to the simple.wikipedia.com version?

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I am not a frequent editor but shouldn't description come before the examples and not the other way around?

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All in all, a reader has a better chance of understanding the paradox if he/she ignores this graph entirely.

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into the 'See also' section? Apologies if this has already been covered, I don't find any references to it.

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If under 30's have a higher rate than over 30's, then Women-under-30 have a higher rate than Women-over-30.

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up Bart/Lisa, Knowledge-emphasizing example (though I like the thoughtful math put into that example). --

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I have removed it - it is original research and speculation. Plus, we have enough real-world examples. -

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In the example of Lisa and Bart, I do not see any causal relationships being brought into consideration.

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By using "it"(in the sentence above "It does not distinguish..."), I assume you mean Simpson's Paradox.

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Sure, but it's got nothing to do with Simpson's paradox. The Bart-and-Lisa example is solely about the

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Why did doctors give the inferior treatment B to the milder cases, when A is better in those cases too?

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Hence Men-under-30 have a higher rate than Women-under-30 who have a higher rate than Women-over-30.

2376: 2308:. Every member of the population is M or F. Therefore every member will benefit. End of story. 1466: 1153: 1104: 1051: 968: 896: 854: 641: 50: 3603: 3515: 3383: 2380: 2006: 1249: 630: 603: 441: 3535: 2730: 2690: 2671: 2323: 1097:

on 01March2007. Also, I hoped I had clearly explained my suggestion above (at 09:51, 2 March 2007)

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Those changes were discussed 14 years ago. The newest comments in a talk page are on the bottom -

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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But that will require thought and skill - I hoped I could get away with a nice, mindless move. -

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Again, isn't the point of the paradox to promote proper consideration of confounding variables?

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If Men have a higher rate than Women, then Men-under-30 have a higher rate than Women-under-30.

2821:"A study suggesting pasta consumption can reduce BMI is a great example of Simpson's Paradox..." 1446:

I'd also suggest that Milo Schield's fine paper "Simpson's Paradox and Cornfield's Conditions" (

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DavidWBrooks has reverted my newly added example of Simpson's Paradox, which read as follows:

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Does Simpson's Paradox always disappear when causal relations are brought into consideration?

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But it is possible to retell the story so that it appears obvious that Bart is more diligent.

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of the edits is unrelated to the paradox we're dealing with here; it's entirely about the

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self-contradictory statement anyway, so I see nothing wrong with calling this a paradox. -

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OK' I didn't put that as clearly as I should have. The point is, we need not distinguish

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an expansion. However, if others disagree, then I certainly will bow to the majority. -

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If psychologists really say this to their subjects, then it is they who are confused:

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Good poinmt. Thanks. I've changed that line to something that I think is even better:

3649: 3342: 3324: 2667: 2527: 1718: 1703: 1684: 1496: 1359: 1291: 694: 555: 2798: 2662:, we find that poor voters are in fact more likely than the rich to vote for Trump." 1711: 3549: 3374: 3370: 1547: 735: 3035:

Lesser, L. (Winter 2010). Confounded. The Mathematical Intelligencer, 32(4), 53.

3026:

https://www.causeweb.org/cause/resources/fun/songs/no-one-counted-simpsons-paradox

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I suggest to put an even simpler and clearer example at the begining like this:

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You're still welcome to do the transformation now that I have done the move :-)

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Is it a problem that the example explicitly refers to Knowledge? (I'm thinking

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Nobody seems to be terribly excited about this. I might just kill it, then. -

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Knowledge article". The point is NOT to pretend that Knowledge doesn't exist.

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Good point. It's so jargon-laden that it's hard to tell if it's gibberish. -

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effort for me to understand than many other possible analogies could have.

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I presume you are intending to leave the remaining paragraphs unchanged? --

1048:

When Simpson's Paradox occurs improvements can be difficult to distinguish.

2886:

Suppose we have the following data of people affected by a disease, say:

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Since Edward Simpson did not actually discover this statistical paradox,

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It reached me today as a retweet from Emilio Ferrara (29 July 2017).

2016:

Since I removed it, my thinking is obvious! I think (just my opinion)

529:

surprise is exchanged for comprehension, but then a paradox is only a

2398:

article as the lead since it is the prime example of ratio-reversal?

2997:

Implications for decision making needs sources and is probably wrong

712:

I'm new to commenting here, so I apologize if I'm doing this wrong.

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vote, but under certain circumstances can lose in the College.

3573:

COVID-19 vaccine misinformation and hesitancy#Claims of inefficacy

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memorable. How great it is when practicality and humor intersect!

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Then I'll do the move, and we can do the transformation later. -

1603:

non-smokers, but higher mortality in the population as a whole,

295: 2670:, we find that poor voters are less likely to vote for Trump. 476: 339:

That was my thought, yes. So I'll go ahead and do this, then.

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How is the Electoral College an example of Simpson's paradox?

3602:

Most paradoxes are failure to do something properly. Is the

3037:

http://link.springer.com/article/10.1007%2Fs00283-009-9127-x

2792:

We should start off with a far more cogent visual argument?

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Alternatively, why not use the introductory graphic of the

734:

Ha! That's funny! Thank's for putting Bart and Lisa in the

39:. It may contain ideas you can use to improve this article. 1717:

There being no objection, I moved it into the article. --

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Boys and girls applyed for physics or math scholarship

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number' that were not. It does not distinguish between

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I was attempting to provide others with some context.

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and I am very impressed! This article is brilliant! --

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http://web.augsburg.edu/~schield/MiloPapers/99ASA.pdf

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way more than enough to do that. IMHO, of course! -

1227:??? / ??? / total number of Electoral College votes 219:, a collaborative effort to improve the coverage of 130:, a collaborative effort to improve the coverage of 3337:I propose to add the following after the UCB case: 2552:concerns, but in this case he/she/it has a point. 1103:

at the end of the 'Explanation by example' section.

810:

Suggested addition to aid paradoxical comprehension

3421:Well, no immediate response so let's remove it. - 3182: 2391:example that explains that graphic in more detail. 3479:other scientists supported Prof. Morris' argument 3187: 2767:So I have removed the low birthweight paradox. - 956:aspects of the article I might better understand. 895:Are these "entirely about the number of edits" ? 3365:to the reader. Secondly, Knowledge is based off 3248:Knowledge is not a how-to manual or a textbook ( 2261:4 girls out of 10 had scholarship this is 40% 2256:3 boys out of 10 had scholarship this is 30% 1100:My original article edit can be found here: --> 449:I just read this article too, having come from 3060:Let's remove the Bart and Lisa section (redux) 1546:in the first instance to vector in all other. 3198:#Let's_remove_the_whole_Bart_and_Lisa_section 752:The Lisa-Bart example ends in this sentence: 8: 2547:Let's remove the whole Bart and Lisa section 3554:https://www.youtube.com/watch?v=XVRfBhy5vGI 1980:National Assessment of Educational Progress 1316:.. oops, never mind: somebody already did. 1182: 962:Do you agree with the following statement ? 800:the greatest post-treatment complications. 3514:I think that's an excellent idea. While a 3193:#Definition_needed_for_"quality_modifier"? 3122: 3039: 2862: 1607:would be an example of Simpson's paradox. 1149:I also believe context aids understanding. 181: 76: 1930:Definition needed for "quality modifier"? 996:of edits that were improvements, and the 949:I have tried several times to understand. 3361:– we shouldn't be, for instance, posing 2888: 880:Extracted from the article's sub-section 3472:Section "Efficacy of Covid-19 vaccines" 2855:A popular variant of Simpson's paradox? 2517: 1216:small stones / large stones / together 1146:I believe I understand Simpson Paradox. 183: 78: 48: 1393:example fits the definition perfectly. 3606:, say, "really a paradox"? Or any of 2493:"UC Berkeley gender bias" departments 7: 2576:well indeed. In my mind, it can go. 2412:Perhaps a change, but don't use the 2201:Suggesting an intial simpler example 1513:Correlation does not imply causation 1485:Correlation does not imply causation 1435:Correlation does not imply causation 213:This article is within the scope of 124:This article is within the scope of 2819:F. Perry Wilson (10 August 2016). 499:competent over all instead of just 67:It is of interest to the following 33:by Knowledge editors, which is now 3676:Mid-importance Statistics articles 3373:, rather than introducing our own 2940:rate (9/15) than over 30's (1/4). 693:Why did I not get this earlier... 14: 3666:Mid-priority mathematics articles 3318:Systemic vs. overt discrimination 2306:but not the population as a whole 2231:8 boys applyed for physics =: --> 1511:I would emphatically stress that 1358:I have readded the example after 946:I do not understand your meaning. 144:Knowledge:WikiProject Mathematics 3575:. Feel free to edit it further. 3278:Implications for decision making 3272:Implications for decision making 2343:Graph of Bart & Lisa Example 2244:1 girl applyed for physcis=: --> 233:Knowledge:WikiProject Statistics 206: 185: 147:Template:WikiProject Mathematics 111: 101: 80: 49: 19: 3681:WikiProject Statistics articles 2933:rate (9/13) than Women (1/6), 2788:Cogent visual argument up front 2238:9 girls applyed for math =: --> 2109:Let's do it, then. *POOF!!!!*- 952:If you could avoid criticising 276:Changes to first few paragraphs 253:This article has been rated as 236:Template:WikiProject Statistics 164:This article has been rated as 3466:12:23, 20 September 2021 (UTC) 3451:07:34, 19 September 2021 (UTC) 3183:#One_of_the_finer_Wiki_entries 2936:and under 30's (total) have a 2526:for the racial breakdown, and 2225:2 boys applyed for math =: --> 2119:18:10, 23 September 2011 (UTC) 2101:18:01, 23 September 2011 (UTC) 2085:17:38, 23 September 2011 (UTC) 1689:08:00, 24 September 2009 (UTC) 1132:Simpson's paradox, or are you 1093:Thankyou for the advice, but, 570:04:19, September 2, 2005 (UTC) 294:described by E. H. Simpson in 1: 3583:13:51, 23 February 2022 (UTC) 3543:17:50, 28 November 2021 (UTC) 3501:16:14, 25 November 2021 (UTC) 3188:#Do_we_need_the_fake_example? 3114:21:25, 30 November 2018 (UTC) 3096:14:01, 13 November 2018 (UTC) 3081:22:17, 10 November 2018 (UTC) 3012:05:36, 26 February 2018 (UTC) 2817:It tumbled across my desk as 2797:Pat LaVarre (29 July 2017). 2718:01:02, 14 November 2016 (UTC) 2703:01:00, 14 November 2016 (UTC) 2684:00:56, 14 November 2016 (UTC) 2641:13:05, 4 September 2016 (UTC) 2615:08:54, 2 September 2016 (UTC) 2600:23:47, 1 September 2016 (UTC) 2385:06:15, 9 September 2013 (UTC) 2336:19:10, 16 February 2013 (UTC) 2058:How many examples do we need? 1906:on my removal of "how likely" 1900:11:04, 14 November 2010 (UTC) 1882:06:38, 14 November 2010 (UTC) 1727:14:34, 23 February 2010 (UTC) 1712:02:31, 22 February 2010 (UTC) 1660:07:52, 12 February 2014 (UTC) 1556:22:17, 26 February 2008 (UTC) 1384:14:39, 18 February 2010 (UTC) 703:14:22, 11 December 2007 (UTC) 660:13:01, 11 December 2007 (UTC) 624:One of the finer Wiki entries 538:01:09, 2 September 2005 (UTC) 428:12:08, 28 November 2010 (UTC) 227:and see a list of open tasks. 138:and see a list of open tasks. 3661:B-Class mathematics articles 3656:Old requests for peer review 3566:03:15, 2 December 2021 (UTC) 3262:19:48, 3 November 2019 (UTC) 3232:10:31, 3 November 2019 (UTC) 3163:18:02, 26 October 2019 (UTC) 2991:12:30, 3 November 2017 (UTC) 2976:12:08, 3 November 2017 (UTC) 2881:11:15, 8 November 2017 (UTC) 2459:14:26, 16 January 2014 (UTC) 2426:11:59, 8 November 2017 (UTC) 2408:11:32, 8 November 2017 (UTC) 2289:07:27, 20 October 2013 (UTC) 1925:06:28, 30 January 2011 (UTC) 1530:06:10, 18 January 2010 (UTC) 1501:08:15, 3 December 2009 (UTC) 1475:18:28, 2 December 2009 (UTC) 1275:Do we need the fake example? 805:13:36, 31 October 2006 (UTC) 795:13:29, 13 October 2006 (UTC) 458:15:48, 27 January 2007 (UTC) 3671:B-Class Statistics articles 3431:21:58, 7 January 2021 (UTC) 3416:13:22, 6 January 2021 (UTC) 3290:14:15, 25 August 2019 (UTC) 3137:23:28, 8 January 2019 (UTC) 2586:18:11, 30 August 2016 (UTC) 2570:18:28, 25 August 2016 (UTC) 2488:00:58, 16 August 2014 (UTC) 2239:4 awarded this is 44.4% 1460:08:42, 14 August 2008 (UTC) 1433:Would it be an idea to add 1421:23:46, 18 August 2014 (UTC) 775:12:13, 6 October 2006 (UTC) 761:08:18, 6 October 2006 (UTC) 743:03:02, 26 August 2006 (UTC) 681:19:46, 2 January 2011 (UTC) 586:08:18, 6 October 2006 (UTC) 3697: 3312:07:19, 7 August 2020 (UTC) 2050:15:32, 13 April 2012 (UTC) 1650:Page updated accordingly. 1141:12:57, 13 March 2007 (UTC) 1108:12:31, 13 March 2007 (UTC) 1089:22:01, 12 March 2007 (UTC) 1055:17:29, 12 March 2007 (UTC) 612:06:18, 9 August 2009 (UTC) 445:12:36, 24 April 2006 (UTC) 402:16:41, 22 March 2020 (UTC) 387:15:33, 22 March 2020 (UTC) 370:21:24, 10 March 2006 (UTC) 3333:22:25, 10 June 2020 (UTC) 2850:19:06, 29 July 2017 (UTC) 2777:01:36, 28 June 2017 (UTC) 2529:for the combined results. 2275:22:17, 31 July 2012 (UTC) 2193:13:38, 21 July 2012 (UTC) 2178:11:11, 21 July 2012 (UTC) 2161:21:33, 14 June 2012 (UTC) 2145:15:56, 14 June 2012 (UTC) 1965:15:46, 8 March 2011 (UTC) 1945:04:19, 8 March 2011 (UTC) 1842:19:28, 8 April 2010 (UTC) 1812:18:47, 8 April 2010 (UTC) 1797:13:18, 8 April 2010 (UTC) 1782:07:11, 8 April 2010 (UTC) 1767:17:19, 7 April 2010 (UTC) 1751:15:37, 7 April 2010 (UTC) 1442:08:57, 17 July 2007 (UTC) 1167:14:49, 3 April 2007 (UTC) 1157:13:50, 3 April 2007 (UTC) 1011:16:24, 8 March 2007 (UTC) 972:11:52, 8 March 2007 (UTC) 930:20:25, 4 March 2007 (UTC) 900:19:34, 4 March 2007 (UTC) 876:09:56, 2 March 2007 (UTC) 858:09:51, 2 March 2007 (UTC) 843:answer those who fail to 252: 201: 163: 96: 75: 3642:13:45, 13 May 2023 (UTC) 3406:What do people think? - 3392:18:24, 23 May 2021 (UTC) 3351:22:55, 12 May 2021 (UTC) 3054:21:09, 4 June 2018 (UTC) 2943:Yet it is not true that 2929:Thus Men (total) have a 2854: 2759:20:52, 3 June 2017 (UTC) 2743:18:38, 3 June 2017 (UTC) 2660:Once we control for race 2507:20:13, 23 May 2016 (UTC) 2365:21:51, 8 July 2013 (UTC) 2279:This is clearly sexist. 2245:0 awarded this is 0% 2232:2 awarded this is 25% 2226:1 awarded this is 50% 1674:The newly added section 1638:20:18, 8 July 2009 (UTC) 1617:13:52, 16 May 2009 (UTC) 1591:05:36, 16 May 2009 (UTC) 1575:22:23, 15 May 2009 (UTC) 1561:Low birth weight paradox 1368:07:44, 24 May 2007 (UTC) 1343:21:27, 23 May 2007 (UTC) 1333:14:00, 23 May 2007 (UTC) 1324:13:44, 23 May 2007 (UTC) 1314:10:00, 23 May 2007 (UTC) 1305:07:12, 23 May 2007 (UTC) 1295:04:32, 23 May 2007 (UTC) 1285:23:41, 22 May 2007 (UTC) 1267:12:53, 28 May 2007 (UTC) 1254:03:07, 16 May 2007 (UTC) 1238:21:46, 15 May 2007 (UTC) 1205:Week 1 / Week 2 / Total 1174: 343:13:13, 17 Feb 2004 (UTC) 170:project's priority scale 3588:The section "Criticism" 2127:Civil Right Act of 1964 2067:Health care disparities 2031:22:27, 9 May 2011 (UTC) 2011:21:08, 9 May 2011 (UTC) 1676:Health care disparities 1670:Health care disparities 1520:strongly related! -- -- 819:Explanation by example' 634:18:42, 2 May 2006 (UTC) 513:20:04, 2 May 2006 (UTC) 127:WikiProject Mathematics 2724:The number of examples 2042:Michael Scott Cuthbert 1950:Presentation of tables 1045:or, put alternatively, 524:care to enlighten me? 519:How is this a paradox? 508:a Simpson's paradox.-- 216:WikiProject Statistics 57:This article is rated 3359:WP:Encyclopedic style 2294:Psychologists section 1429:Correlation/Causation 728:greatly diminished. 3363:rhetorical questions 150:mathematics articles 3604:paradox of the heap 3571:Added some info in 2001:Comments, anyone? 1698:To where it reads, 1693: 1184: 1136:to understand it?-- 239:Statistics articles 2480:Mcamp@cinci.rr.com 2370:Introductory Graph 2348:the first week. 2075:Any objections? - 1581:low birth weight. 1542:I reverted a diff 1413:Mcamp@cinci.rr.com 1194:the "3" dimension 1183: 119:Mathematics portal 63:content assessment 3487:comment added by 3375:original research 3139: 3127:comment added by 3071:Any thoughts? - 3056: 3044:comment added by 2951:rate (9/10) than 2927: 2926: 2883: 2867:comment added by 2736: 2696: 2677: 2449:comment added by 2339: 2322:comment added by 2048: 1885: 1868:comment added by 1738:user:DavidWBrooks 1231: 1230: 1221:Electoral College 1191:the "2" dimension 1002:improvements and 736:Simpson's paradox 562: 463:The same paradox? 451:Texture filtering 418:comment added by 285:Simpson's paradox 273: 272: 269: 268: 265: 264: 180: 179: 176: 175: 43: 42: 27:Simpson's paradox 3688: 3608:Zeno's paradoxes 3552:'s short talk ( 3541: 3513: 3503: 3371:reliable sources 3150: 2889: 2832: 2810: 2735: 2695: 2676: 2540: 2536: 2530: 2522: 2461: 2338: 2316: 2044: 1884: 1862: 1452:Haruhiko Okumura 1185: 815:existing section 569: 566: 554: 552: 549: 430: 259:importance scale 241: 240: 237: 234: 231: 210: 203: 202: 197: 189: 182: 152: 151: 148: 145: 142: 121: 116: 115: 105: 98: 97: 92: 84: 77: 60: 54: 53: 45: 23: 16: 3696: 3695: 3691: 3690: 3689: 3687: 3686: 3685: 3646: 3645: 3590: 3558:Richard J Kinch 3534: 3507: 3482: 3474: 3439: 3400: 3320: 3274: 3144: 3129:157.182.151.169 3062: 3057: 3033: 3028: 3022: 2999: 2857: 2818: 2814:unknown to me. 2796: 2790: 2726: 2655: 2549: 2544: 2543: 2537: 2533: 2523: 2519: 2514: 2495: 2471: 2444: 2436: 2372: 2345: 2317: 2296: 2203: 2129: 2060: 1972: 1970:Another example 1952: 1932: 1908: 1863: 1858: 1802:presentation.-- 1735: 1696: 1672: 1563: 1540: 1538:Vector vs. Line 1431: 1277: 1177: 1006:improvements.-- 812: 783: 781:The kidney case 750: 626: 567: 564: 550: 547: 521: 465: 436: 413: 410: 278: 238: 235: 232: 229: 228: 195: 149: 146: 143: 140: 139: 117: 110: 90: 61:on Knowledge's 58: 12: 11: 5: 3694: 3692: 3684: 3683: 3678: 3673: 3668: 3663: 3658: 3648: 3647: 3630: 3629: 3619: 3618: 3600: 3599: 3589: 3586: 3569: 3568: 3545: 3473: 3470: 3469: 3468: 3438: 3435: 3434: 3433: 3399: 3398:Death sentence 3396: 3395: 3394: 3379: 3378: 3377:into articles. 3319: 3316: 3315: 3314: 3298: 3297: 3273: 3270: 3269: 3268: 3267: 3266: 3265: 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3662: 3659: 3657: 3654: 3653: 3651: 3644: 3643: 3639: 3635: 3628: 3624: 3623: 3622: 3617: 3613: 3612: 3611: 3609: 3605: 3598: 3595: 3594: 3593: 3587: 3585: 3584: 3581: 3578: 3574: 3567: 3563: 3559: 3555: 3551: 3546: 3544: 3540: 3537: 3532: 3530: 3527: 3525: 3523: 3521: 3517: 3511: 3506: 3505: 3504: 3502: 3498: 3494: 3490: 3486: 3480: 3471: 3467: 3463: 3459: 3455: 3454: 3453: 3452: 3448: 3444: 3436: 3432: 3428: 3424: 3420: 3419: 3418: 3417: 3413: 3409: 3404: 3397: 3393: 3389: 3385: 3381: 3380: 3376: 3372: 3368: 3367:verifiability 3364: 3360: 3355: 3354: 3353: 3352: 3348: 3344: 3338: 3335: 3334: 3330: 3326: 3317: 3313: 3309: 3305: 3304:72.234.243.59 3300: 3299: 3294: 3293: 3292: 3291: 3287: 3283: 3279: 3271: 3263: 3259: 3255: 3251: 3247: 3243: 3242: 3241: 3240: 3239: 3238: 3233: 3230: 3226: 3225: 3219: 3218: 3217: 3216: 3209: 3208: 3207: 3206: 3199: 3196: 3194: 3191: 3189: 3186: 3184: 3181: 3180: 3179: 3178: 3177: 3176: 3170: 3169: 3168: 3167: 3164: 3161: 3157: 3156: 3148: 3142: 3141: 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2008: 2004: 1999: 1993:achievement". 1991: 1990: 1989: 1981: 1977: 1976: 1975: 1969: 1967: 1966: 1962: 1958: 1949: 1947: 1946: 1942: 1938: 1929: 1927: 1926: 1922: 1918: 1917:173.30.19.136 1912: 1905: 1901: 1897: 1893: 1888: 1887: 1886: 1883: 1879: 1875: 1871: 1867: 1855: 1843: 1839: 1835: 1831: 1830: 1829: 1828: 1827: 1826: 1825: 1824: 1823: 1822: 1813: 1809: 1805: 1800: 1799: 1798: 1794: 1790: 1785: 1784: 1783: 1779: 1775: 1770: 1769: 1768: 1764: 1760: 1755: 1754: 1753: 1752: 1748: 1744: 1739: 1733:Kidney stones 1732: 1728: 1724: 1720: 1716: 1715: 1714: 1713: 1709: 1705: 1701: 1694:Stigler's law 1691: 1690: 1686: 1682: 1677: 1669: 1661: 1657: 1653: 1652:124.74.76.114 1649: 1648: 1647: 1646: 1645: 1644: 1639: 1635: 1631: 1626: 1625: 1624: 1623: 1618: 1614: 1610: 1606: 1602: 1598: 1597: 1596: 1595: 1592: 1588: 1584: 1579: 1578: 1577: 1576: 1572: 1568: 1560: 1558: 1557: 1553: 1549: 1544: 1537: 1531: 1527: 1523: 1518: 1514: 1510: 1509: 1508: 1507: 1502: 1498: 1494: 1489: 1486: 1482: 1481: 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