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I just got my statistics test back and I am totally confused about one of the questions!

A study was done that took a simple random sample of 40 people and measured whether the subjects were right-handed or left-handed, as well as their ages. The study showed that the proportion of left-handed people and the ages had a strong negative correlation. What can we conclude? Explain your answer.

I know that we can't conclude that getting older causes people to become right-handed. Something else might be causing it, not the age. If two things are correlated, we can only conclude association, not causation. So I wrote:

We can conclude that many people become right-handed as they grow older, but we cannot tell why.

That's exactly what association means, but my teacher marked me wrong! What mistake did I make? Is 40 too small of a sample size to make any conclusions?

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We had a discussion over on mathoverflow about whether or not this exact question has any mathematical merit. We concluded that it does not and is therefore off topic. – user126 Jul 22 '10 at 1:13
It might be off-topic on MO, but I'm not sure if there is a consensus yet on MU whether or not to allow statistics-related questions. However, there is already a Statistics StackExchange 2.0 Site in Beta that will be public before this one, so this question might be more proper for that. – Justin L. Jul 22 '10 at 1:26
@Harry: I am not at all surprised that this question was deemed inappropriate for MathOverflow. However, that doesn't necessarily makes it off-topic here. @Justin: stats.stackexchange is intended for professionals in statistics-related fields. I think that such an elementary question would be a much better fit here. – Larry Wang Jul 22 '10 at 1:35
Maybe there's a difference between "math-related" and "of interest and direct relevance to those studying/using math or related things." I dislike this question as asked, as I did the corresponding one on MO, but there were several useful answers about mathematical measures of causation versus correlation on MO, and I think several people benefitted from the discussion. Part of learning math is dealing personally with the choices and philosophical issues it raises, so within reason, I think questions addressing those can be valuable. – Jamie Banks Jul 22 '10 at 1:54
It's certainly not a math question. The 'correct' answer, as far as I'm concerned, should address the difference between longitudinal and cross-sectional studies, which is clearly in the realm of statistics. Whether we consider statistics questions to be math-related is less clear. Personally, I would consider anything covered in a freshman statistics course (regression, probability, box models, confidence intervals) to be fair game for math.stackexchange, and that is the context in which I get asked this question. – Larry Wang Jul 22 '10 at 1:57
up vote 3 down vote accepted

This is wrong: "We can conclude that many people become right-handed as they grow older." We cannot conclude this at all from the given data.

For one, the study only takes a sample at one point in time, rather than selecting a sample and monitoring their progress through many decades. This is what would be needed for us to even entertain the possibility that aging causes a change in handedness.

Other possible causes include that left handed people might have a shorter life expectancy, or perhaps there was a spike in the birth rate of right handed people in the past. There are many other possibilities that have been mentioned in others answers which would also account for the skewed proportions without requiring people to change handedness with age, which is what you falsely concluded in the test.

Also, just an observation, but it appears the "study" was conducted under false pretenses. Handedness is a false dichotomy, people can also be ambidextrous.

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observations like this should probably be left as comments. – Jamie Banks Jul 22 '10 at 2:10
@Katie Noted, edited so answer comes before observation. – Adam C. Jul 22 '10 at 2:19
Why can't I conclude that? All I am saying is that there is association! Isn't that what correlation means? – Larry Wang Jul 22 '10 at 2:31
@Kaestur, you cannot conclude that people become right handed, only that the proportion of right handed people to left handed people appears larger. Sure, left handed people learning to use their right hands is one possibility, but definitely not the only one, and I give two other possibilities in my answer. – Adam C. Jul 22 '10 at 2:35
It was directly stated that there is a negative correlation. AKA association. Isn't that what my conclusion says? – Larry Wang Jul 22 '10 at 10:50

Here is one example of a plausible explanation that disagrees with your analysis:

Cultural expectations for left- and right-handedness have changed over time. Older people may have gone to school at a time where left-handedness was discouraged and students were forced to write with their right hands, training children never to use the left hand instead of the right. Younger participants in the study were in school more recently and learned to write at a time where left-handedness was not discouraged, creating a positive correlation between left-handedness and youth.

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We can only conclude that the result is interesting and it deserves more research.

Without further study we can not say if right-handedness causes age (ie. being left handed causes biological alterations that shorten the life span), age causes right-handedness (ie. as people age they become right handed), they are correlated because they are caused by another variable (ie. older people became educated in a different system that discouraged left-handedness), the study had bad luck selecting its sample, the study sample was bad designed, the study was bad designed, etc.

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So strong correlation only means "interesting?" I understand that we need an experiment before we decide there is causation, but I thought correlation meant there was association! Isn't that what I wrote? – Larry Wang Jul 22 '10 at 2:29
Not really what you wrote. You wrote that "people become right handed as they grow older" which either means age -> right-handedness or age <-> right-handedness. But if there is another factor that influences both, those statements may be false. With the question as stated nothing can be concluded, more information is needed. Or do you believe in ? – Wilhelm Jul 22 '10 at 3:13
I didn't say that getting older made them right handed, just that they happen to show up together. – Larry Wang Jul 22 '10 at 9:28

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