# Examples for teaching: Correlation does not mean causation

There is an old saying: "Correlation does not mean causation". When I teach, I tend to use the following standard examples to illustrate this point:

1. number of storks and birth rate in Denmark;
2. number of priests in America and alcoholism;
3. in the start of the 20th century it was noted that there was a strong correlation between 'Number of radios' and 'Number of people in Insane Asylums'
4. and my favorite: pirates cause global warming.

However, I do not have any references for these examples and whilst amusing, they are obviously false.

Does anyone have any other good examples?

• Flip through Freakonomics for some great examples. Their bibliography is chock full of references. – Stephen Turner Jul 19 '10 at 19:36
• xkcd.com/552 – Ami Jul 19 '10 at 19:48
• That pirates / global warming chart is clearly cooked up by conspiracy theorists - anyone can see they have deliberately plotted even spacing for unequal time periods to avoid showing the recent sharp increase in temperature as pirates are almost entirely wiped out. We all know that as temperatures rise it makes the rum evaporate and pirates cannot survive those conditions. ;-) – AdamV Jul 22 '10 at 16:08
• WTF is up with the x-axis on that pirate graph? – naught101 Mar 31 '12 at 10:15
• Or pretty much anything you put into Google Correlate, come to that. – conjugateprior Mar 31 '12 at 13:44

It might be useful to explain that "causes" is an asymmetric relation (X causes Y is different from Y causes X), whereas "is correlated with" is a symmetric relation.

For instance, homeless population and crime rate might be correlated, in that both tend to be high or low in the same locations. It is equally valid to say that homelesss population is correlated with crime rate, or crime rate is correlated with homeless population. To say that crime causes homelessness, or homeless populations cause crime are different statements. And correlation does not imply that either is true. For instance, the underlying cause could be a 3rd variable such as drug abuse, or unemployment.

The mathematics of statistics is not good at identifying underlying causes, which requires some other form of judgement.

• Judgement is a good word, since all we can ever observe is correlation. All that experiments and/or clever statistics can do is allow us to exclude some alternative explanations for what could have caused an effect. – Jonas Aug 14 '10 at 18:18
• Very good comment about the symmetric/asymmetric relations. One also might claim that global warming causes piracy to increase. – Andre Holzner Aug 16 '10 at 11:02

My favorites:

1) The more firemen are sent to a fire, the more damage is done.

2) Children who get tutored get worse grades than children who do not get tutored

and (this is my top one)

3) In the early elementary school years, astrological sign is correlated with IQ, but this correlation weakens with age and disappears by adulthood.

• (@xmjx Supplied the first example last year.) I love the astrology example. – whuber Nov 8 '11 at 15:09
• Can you explain the sample with astrological sign please? – Eugene D. Gubenkov Mar 18 '17 at 8:21
• Never mind, I got it. That has to do with age difference between ones born at the beginning of the year and ones born at the end. Nice. – Eugene D. Gubenkov Mar 18 '17 at 8:26

I've always liked this one:

• Nice, but I can't see anyone trying to draw a conclusion of causality there. Or are mexican lemon-truck drivers notoriously dangerous once they get over the border? – AdamV Jul 27 '10 at 12:57
• Obviously an unforseen side-effect of the profusion of lemon laws in the US. For example see: en.wikipedia.org/wiki/Lemon_law – Thylacoleo Aug 7 '10 at 10:03
• A colleague of mine looked at the data for this in the post-2000 period, and found that the relationship held fairly well 'out-of-sample', which is even more disturbing... – shabbychef Oct 4 '10 at 4:12
• pbs.twimg.com/media/BBBMyXnCIAIw1ED.jpg:large – Džuris Jan 21 '13 at 21:24
• A simple rationalization would be that both are decreasing with time. Does the post-2000 data support that? PS, Box Hunter and Hunter (see below) explain the storks example the same way: both increased with time over the period in question. – Emil Friedman Feb 18 '14 at 22:16
1. Sometimes correlation is enough. For example, in car insurance, male drivers are correlated with more accidents, so insurance companies charge them more. There is no way you could actually test this for causation. You cannot change the genders of the drivers experimentally. Google has made hundreds of billions of dollars not caring about causation.

2. To find causation, you generally need experimental data, not observational data. Though, in economics, they often use observed "shocks" to the system to test for causation, like if a CEO dies suddenly and the stock price goes up, you can assume causation.

3. Correlation is a necessary but not sufficient condition for causation. To show causation requires a counter-factual.

• I like the first example you give. That will certainly get the students talking ;) – csgillespie Jul 19 '10 at 19:56
• There's an interesting discussion by Steve Steinberg on his blog here: blog.steinberg.org/?p=11 about some of the implications of 1 and where it might lead in terms of Weak AI. – Amos Jul 19 '10 at 20:01
• Could someone expand on the last sentence a little? – naught101 Mar 31 '12 at 10:41
• Just a quick clarification: Correlation is not necessary for causation (depending on what is mean by correlation): if the correlation is linear correlation (which quite a few people with a little statistics will assume by default when the term is used) but the causation is nonlinear. For example, if $X$ in $(-1,1)$ directly causes $Y$ (which takes values in $(0,1)$), but $Y = \sqrt{1-X^2}$. If the $X's$ are symmetrically distributed, $X$ and $Y$ will be uncorrelated even though perfectly dependent. – Glen_b Feb 3 '14 at 22:06

I have a few examples I like to use.

1. When investigating the cause of crime in New York City in the 80s, when they were trying to clean up the city, an academic found a strong correlation between the amount of serious crime committed and the amount of ice cream sold by street vendors! (Which is the cause and which is the effect?) Obviously, there was an unobserved variable causing both. Summers are when crime is the greatest and when the most ice cream is sold.

2. The size of your palm is negatively correlated with how long you will live (really!). In fact, women tend to have smaller palms and live longer.

3. [My favorite] I heard of a study a few years ago that found the amount of soda a person drinks is positively correlated to the likelihood of obesity. (I said to myself - that makes sense since it must be due to people drinking the sugary soda and getting all those empty calories.) A few days later more details came out. Almost all the correlation was due to an increased consumption of diet soft drinks. (That blew my theory!) So, which way is the causation? Do the diet soft drinks cause one to gain weight, or does a gain in weight cause an increased consumption in diet soft drinks? (Before you conclude it is the latter, see the study where a controlled experiments with rats showed the group that was fed a yogurt with artificial sweetener gained more weight than the group that was fed the normal yogurt.) Two references: Drink More Diet Soda, Gain More Weight?; Diet sodas linked to obesity . I think they are still trying to sort this one out.

• The last one is slightly more complicated than you present it, but I agree much of the observational associations found between soda/diet soda and obesity should be looked at with a critical eye. Theoretically some have posited that the fake sugar/fat substitutes have other physiological effects beyond the simple calorie intake. See for example this experiment on rats and synthetic fats (taken from the Freakonomics blog). – Andy W Nov 8 '11 at 16:55

The number of Nobel prizes won by a country (adjusting for population) correlates well with per capita chocolate consumption. (New England Journal of Medicine)

• +1 I was very disappointed with NEJM when they published this – MattBagg Dec 14 '12 at 15:37
• Seems to also correlate quite well with proximity to Sweden.. – naught101 Dec 15 '12 at 0:46
• Chocolate consumption (per capita) also significantly correlates with the per capita number of serial murderers. replicatedtypo.com/… – Harvey Motulsky Dec 15 '12 at 15:57
• I asked three Nobel prize winners I (vaguely) know, and all three said they have eaten way more chocolate than most of their colleagues. Of course, these answers came after they read the NEJM paper! – Harvey Motulsky Dec 15 '12 at 18:20
• @MattBagg This was published as "Occasional Notes" and obviously to not be taken seriously. – Pascal Aug 22 '14 at 18:48

Although it's more of an illustration of the problem of multiple comparisons, it is also a good example of misattributed causation:

Rugby (the religion of Wales) and its influence on the Catholic church: should Pope Benedict XVI be worried?

"every time Wales win the rugby grand slam, a Pope dies, except for 1978 when Wales were really good, and two Popes died."

There's two aspects to this post hoc ergo propter hoc problem that I like to cover: (i) reverse causality and (ii) endogeneity

An example of "possible" reverse causality: Social drinking and earnings - drinkers earn more money according to Bethany L. Peters & Edward Stringham (2006. "No Booze? You May Lose: Why Drinkers Earn More Money Than Nondrinkers," Journal of Labor Research, Transaction Publishers, vol. 27(3), pages 411-421, June). Or do people who earn more money drink more either because they have a greater disposable income or due to stress? This is a great paper to discuss for all sorts of reasons including measurement error, response bias, causality, etc.

An example of "possible" endogeneity: The Mincer Equation explains log earnings by education, experience and experience squared. There is a long literature on this topic. Labour economists want to estimate the causal relationship of education on earnings but perhaps education is endogenous because "ability" could increase the amount of education an individual has (by lowering the cost of obtaining it) and could lead to an increase in earnings, irrespective of the level of education. A potential solution to this could be an instrumental variable. Angrist and Pischke's book, Mostly Harmless Econometrics covers this and relates topics in great detail and clarity.

Other silly examples that I have no support for include: - Number of televisions per capita and the numbers of mortality rate. So let's send TVs to developing countries. Obviously both are endogenous to something like GDP. - Number of shark attacks and ice cream sales. Both are endogenous to the temperature perhaps?

I also like to tell the terrible joke about the lunatic and the spider. A lunatic is wandering the corridors of an asylum with a spider he's carrying in the palm of his hand. He sees the doctor and says, "Look Doc, I can talk to spiders. Watch this. "Spider, go left!" The spider duly moves to the left. He continues, "Spider, go right." The spider shuffles to the right of his palm. The doctor replies, "Interesting, maybe we should talk about this in the next group session." The lunatic retorts, "That's nothing Doc. Watch this." He pulls off each of the spider's legs one by one and then shouts, "Spider, go left!" The spider lies motionless on his palm and the lunatic turns to the doctor and concludes, "If you pull off a spider's legs he'll go deaf."

The best one I've been taught has been the number of drownings and sales of ice creams may be highly correlated but that doesnt imply one causes the other. Drownings and sales of ice cream are obviously higher in the summer months when the weather is good. Third variable aka good weather causes them.

As a generalization of 'pirates cause global warming': Pick any two quantities which are (monotonically) increasing or decreasing with time and you should see some correlation.

You can spend a few minutes on Google Correlate, and come up with all kinds of spurious correlations.

• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. – gung Jun 4 '14 at 22:15
• @gung are you serious? The link is to an application, not to a simple page describing an answer. The answer would become invalid if the linked page changes anyway, as the tool would become unavailable (in the current form). – Jerome Baum Nov 21 '14 at 20:35

I work with students in teaching correlation vs causation in my Algebra One classes. We examine a lot of possible examples. I found the article Bundled-Up Babies and Dangerous Ice Cream: Correlation Puzzlers from the February 2013 Mathematics Teacher to be useful. I like the idea of talking about "lurking variables". Also this cartoon is a cute conversation starter:

We identify the independent and dependent variable in the cartoon and talk about whether this is an example of causation, if not why not.

I read (a long time ago) of an interesting example about a decline in birth rates (or fertility rates if you prefer that measure) especially in the US, starting in the early 1960's, as nuclear weapons testing was at an all-time high (in 1961 the biggest nuclear bomb ever detonated was tested in the USSR). Rates continued to deline until towards the end of the twentieth century when most countried finally stopped doing this.

I can't find a reference which combines these figures now, but this Wikipedia article has figures on nuclear weapons test numbers by country.

Of course, it might make better sense to look at the correlation of birth rate with the introduction and legalisation of the contraceptive pill 'coincidentally' starting in the early 1960s. (In only some states first, then all states for married women only, then some for unmarried, then across the board), But even that could only be part of the cause; lots of other aspects of equality, economic changes and other factors play a significant part.

• Interesting example, because it looks, at first glance, like a probable cause-and-effect relationship, unlike many of the silliest examples. – Bossykena Jul 22 '10 at 19:10
• What I like is that you can provoke lots of discussion about whether the "effect" was to actually impact fertility (in a medical sense of ability to conceive) or was it social ("I don't want to bring a child into this bad world"). Then drop the bombshell about the Pill if no-one else has brought it up. And then point out that even this can only be one possible factor and discuss some of the others. – AdamV Jul 23 '10 at 13:32

A correlation on its own can never establish a causal link. David Hume (1771-1776) argued quite effectively that we can not obtain certain knowlege of cauasality by purely empirical means. Kant attempted to address this, the Wikipedia page for Kant seems to sum it up quite nicely:

Kant believed himself to be creating a compromise between the empiricists and the rationalists. The empiricists believed that knowledge is acquired through experience alone, but the rationalists maintained that such knowledge is open to Cartesian doubt and that reason alone provides us with knowledge. Kant argues, however, that using reason without applying it to experience will only lead to illusions, while experience will be purely subjective without first being subsumed under pure reason.

In otherwords, Hume tells us that we can never know a causal relationship exists just by observing a correlation, but Kant suggests that we may be able to use our reason to distinguish between correlations that do imply a causal link from those who don't. I don't think Hume would have disagreed, as long as Kant were writing in terms of plausibility rather than certain knowledge.

In short, a correlation provides circumstantial evidence implying a causal link, but the weight of the evidence depends greatly on the particular circumstances involved, and we can never be absolutely sure. The ability to predict the effects of interventions is one way to gain confidence (we can't prove anything, but we can disprove by observational evidence, so we have then at least attempted to falsify the theory of a causal link). Having a simple model that explains why we should observed a correlation that also explains other forms of evidence is another way we can apply our reasoning as Kant suggests.

Caveat emptor: It is entirely possible I have misunderstood the philosophy, however it remains the case that a correlation can never provide proof of a causal link.

• For what it's worth, in current terminology I think one should read Kant as asserting, e.g. in the Second Analogy, that whatever correlations you observe, there is some causal graph generating them. As far as I am aware he had no particular method for identifying the structure but did assume that it must be fully connected (because 'every event has a cause'). In this sense he is contemporary: causal inference requires a mixture of causal assumptions, e.g. expressed via a graph, and observed regularities in the data. And you can typically neither avoid the first part nor induce it from data – conjugateprior Dec 16 '12 at 23:33
• +1 well explained! Maybe I am too Bayesian, but I am not too bothered with the idea that we can have no certain knowledge of any causal relationship. – Dikran Marsupial Dec 17 '12 at 12:55

when $x_t$; $y_t$ are stationary time series, then correlation between $y_t$ and $x_{t-1}$ implies causation of $y_t$ by $x_{t-1}$. For some reason it's not mentioned here.

The standard citation pointing out the correlation between the number of newborn babies and breeding-pairs of storks in West Germany is A new parameter for sex education, Nature 332, 495 (07 April 1988); doi:10.1038/332495a0

Sperm count in males in Slovene villages and the number of bears (also in Slovenia) show a negative correlation. Some people find this very worrying. I'll try and get the study that did this.

I've recently been to a conference and one of the speakers gave this very interesting example (although the point was to illustrate something else):

• Americans and English eat a lot of fat food. There is a high rate of cardiovascular diseases in US and UK.

• French eat a lot of fat food, but they have a low(er) rate of cardiovascular diseases.

• Americans and English drink a lot of alcohol. There is a high rate of cardiovascular diseases in US and UK.

• Italians drink a lot of alcohol but, again, they have a low(er) rate of cardiovascular diseases.

The conclusion? Eat and drink what you want. And you have a higher chance of getting a heart attack if you speak English!

• It's also a good example of the ecological fallacy (i.e., making inferences about the individual-level from group-level data). – Jeromy Anglim Aug 16 '10 at 11:50

This cartoon rom XKCD is also posted elsewhere at CrossValidated.

Another example of correlation I've used is the large increase in the number of people eating organic food and the increase in the number of children diagnosed with autism in the U.S. There's a parody graph on the web -

http://tylervigen.com/

This shows a ton of correlations that obviously have nothing to do with causation - Or do you have any good idea what's the causation to the correlation of Age of Miss America correlates with Murders by steam, hot vapours and hot objects

??

Teaching "Correlation does not mean causation" doesn't really help anyone because at the end of the day all deductive arguments are based in part on correlation.

Human are very bad at learning not to do something.

The goal should rather be constructive: Always think about alternatives to your starting assumptions that might produce the same data.

• This doesn't respond to the question: perhaps it should be understood as a comment. – whuber Nov 8 '11 at 15:11

Well my Prof. used these in Introductory probability class:

1) Shoe size are correlated with reading ability

2) Shark attack is correlated with sale of ice cream.

The more fire engines sent to a fire, the bigger the damage.

• The only problem with this as an example is that there is clear reverse causation. – naught101 Mar 31 '12 at 11:02

I think a better paradigm might be causation requires correlation associated with a credible and preferably proven mechanism. I think the word imply should be used very sparingly in this context, as it has several meanings including that of suggestion.

The storks example is on page 8 of the first edition (1978) of Box, Hunter & Hunter's book entitled "Statistics for Experimenters..." (Wiley). I don't know whether it's in the 2nd edition. They identify the city as Oldenburg and the time period as 1930-1936.

They reference Ornithologische Monatsberichte, 44, No 2, Jahrgang, 1936, Berlin, and 48, No 1, Jahrgang, 1940, Berlin, and Statistiches Jahrbuch Deutscher Gemeinden, 27-33, 1932-1938, Gustav Fischer, Jena.

I saw a funny one in an article.

Butter production in Bangladesh has one of the highest correlation with the S&P 500 over a ten year period.

• Huh? The graph shows the S&P over time. The title talks about butter and cheese production, which are not visible on the graph. ??? – Harvey Motulsky Dec 15 '12 at 15:27
• nerdsonwallstreet.typepad.com/my_weblog/files/… – ThiS Dec 15 '12 at 19:19
• OK, now I see. The graph shows the prediction of a multiple regression model, showing that including three silly variables does a pretty good job of making the model predict change in the SP500 over time. This is a good example of overfitting in multiple regression, and indirectly shows that correlation (or improved goodness-of-fit of a fancy model) does not imply causation. – Harvey Motulsky Dec 16 '12 at 0:12

Someone said, correlation may not mean causation but it surely can be a good hint :)

Ok leaving aside the fun part, what exactly is causation? Are we really sure that pirates do not cause global warming?

Counter-intuitive, but what is taken as cause and what as effect (in a correlation study is not that clear). Of course many times both can just be effects of common cause (and thus correlated)

It all boils down to the method of determining causation.

This is the cause (pun intended) of the saying :

There are petty lies. There are big lies And there are statistics.