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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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  • $\begingroup$ 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. $\endgroup$
    – user126
    Commented Jul 22, 2010 at 1:13
  • $\begingroup$ 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. $\endgroup$
    – Justin L.
    Commented Jul 22, 2010 at 1:26
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    $\begingroup$ @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. $\endgroup$
    – Larry Wang
    Commented Jul 22, 2010 at 1:35
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    $\begingroup$ 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. $\endgroup$
    – Jamie Banks
    Commented Jul 22, 2010 at 1:54
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    $\begingroup$ 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. $\endgroup$
    – Larry Wang
    Commented Jul 22, 2010 at 1:57

4 Answers 4

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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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    $\begingroup$ observations like this should probably be left as comments. $\endgroup$
    – Jamie Banks
    Commented Jul 22, 2010 at 2:10
  • $\begingroup$ @Katie Noted, edited so answer comes before observation. $\endgroup$
    – Adam C.
    Commented Jul 22, 2010 at 2:19
  • $\begingroup$ Why can't I conclude that? All I am saying is that there is association! Isn't that what correlation means? $\endgroup$
    – Larry Wang
    Commented Jul 22, 2010 at 2:31
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    $\begingroup$ @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. $\endgroup$
    – Adam C.
    Commented Jul 22, 2010 at 2:35
  • $\begingroup$ It was directly stated that there is a negative correlation. AKA association. Isn't that what my conclusion says? $\endgroup$
    – Larry Wang
    Commented Jul 22, 2010 at 10:50
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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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  • $\begingroup$ 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? $\endgroup$
    – Larry Wang
    Commented Jul 22, 2010 at 2:29
  • $\begingroup$ 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 en.wikipedia.org/wiki/Paul_the_octopus ? $\endgroup$
    – Wilhelm
    Commented Jul 22, 2010 at 3:13
  • $\begingroup$ I didn't say that getting older made them right handed, just that they happen to show up together. $\endgroup$
    – Larry Wang
    Commented Jul 22, 2010 at 9:28
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Without getting much into conclusions that can be drawn from correlation coefficients, I wanted to point out a few things about sample size...

10% of people are left-handed. https://www.huffingtonpost.com/2012/10/29/left-handed-facts-lefties_n_2005864.html

That means, unless you have specific data stating otherwise, it is likely that this hypothetical study sampled 36 right-handed people and 4 left-handed people. The sampled ages of 4 left-handed people will tell you basically nothing about their mortality rate. The difference in longevity between left-handed and right-handed people is probably not very high, so accurately approximating the difference would require quite a high sample size of each population (probably about 100-300 of each). For further reading, see how to select sample size based on standard error (if it is known or can be reasonably estimated) and confidence level.

There is also an experimental design error. Taking a survey of the ages of a few living people doesn't give a very good estimation of the life expectancy of these populations. If you went to the morgue and surveyed the age of the last N people who died, you would be somewhat closer to having an accurate measure of life expectancy.

You also made a serious error in how you worded your conclusion, which others have pointed out.

Also, this is far beyond the scope of the pure math that you're being tested on, but it happens that the left-handed portion of the population declined to about 3% during the Victorian era, before rising back up to 10% in the present. This could be largely because people of the time were pressured to become right-handed, and could also correlate to the design of industrial tools of the time. Correlation doesn't necessarily imply causation, and the most common reason why is that there might be a third variable that is unnaturally forcing the correlation. In this case, the life expectancy of people living in the present may not vary greatly between the left-handed and right-handed; but lack of knowledge about historical context might increase the likelihood of making a correlation = causation mistake, or historical data might imply a difference in life expectancy that once existed but now does not. https://www.snopes.com/fact-check/left-hand-right-hand-death/

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