The other night, I was chatting with my brother-in-law, who recently qualified as a doctor. He is a very conscientious and hard-working fellow who spends a lot of time reading up on the latest research and findings in medical journals (which, I have to say, are surprisingly far more palatable than economics journals).

One thing he was complaining about was the nature of statistics, and how hard it was to correctly interpret findings from medical tests and experiments, and referred to a recent article that looked at how many doctors got the correct answer to a statistical problem. Most of them failed.

In part, I think this reflects the fact that statistical theory is not really a core component of learning to become a doctor; it’s generally more important to know how the body works, etc. But I thought it would be interesting to ask a variant of the question here, and see what answers people came up with. Please use the comments box below to submit your views – if necessary, I’ll come back in a bit with the answer.

The question: A test for cancer is 90% accurate in both directions – 90% of the time, it returns a positive result amongÂ those people with cancer, and 90% of the time it correctly says that those people without cancer don’t have it. So, if your patient is tested for cancer, and the result is positive, what is the likelihood that the patient has cancer? (The incidence of cancer in the population as a whole is 1%).

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comments

This one's caused a calculator melt down!

More like a ruddy brain meltdown! I can't work out if it's meant to be more than 90% or less!

Is it not 90%? Because it's a 90% accurate positive test.

Answer 1%

you dont need to be a MENSA candidate to know that most doctors are dummies who have just been taught to regurgitate information rather than use their brains.

How can it be 1% davidh? The result would be obtained by a test that is 90% accurate for those with cancer that are tested. If it was 1% there'd be no point in having the test would there?

99% of population don't have cancer.If you tested everyone 9.9% would have a positve result (10% error in tests)

1% have cancer, so ).9% would have positive result

So if you tested whole population 10.8% would have a positve result but only (0.9/10.8) would have cancer – an 8.333…. % chance.

However this is based on random selection, in fact you are probably having a test for a reason (ill health, family history, etc).

Like that answer jonzo. But surely it should be 1/10.8?

If you have 10,000 people and 1% (100) have cancer, then 9,900 people don't have cancer.

So 90 of the first 100 correctly test positive but 990 of the second group test positive when they are negative. So that means that 1080 people test positive, which is 10.8% of the 10,000, but as only 1% actually has cancer then it must be 1%/10.8% X 100% = 9.26%?

Jonzo has this spot on (and also raises a good point about the assumed randomness here, which is unrealistic in real life)

Across 1,000 people, 10 have cancer and 990 don't.

Of the 10 with cancer, 9 will be correctly diagnosed (ie get a positive test result)

Of the 990 people without cancer, 99 will be incorrectly diagnosed (ie get a positive test result)

In total, this means 108 people get positive test results – of whom only 9 actually have cancer. So the probability is 9/108 = 8.333

Jonzo, have you thought about a career in medicine?

Suppose we take a sample of 1000 patients. Since cancer affects 1% of the population, it means that 10 of them, by the law of averages, will have cancer.

If all the patients are subjected to the tests, out of the 10 of them that have cancer, 9 of them will get a positive result, because the test is 90% effective.

Of the 990 patients that don't have cancer, 99 of them will get a misleading result because the test is 10% inaccurate.

So if you were a patient in these tests and got a positive test result, don't panic – you are still 11 times likely NOT to have cancer as to have it.