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What are False Positives and False Negatives?

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University Course Reader · STEM

The screening programme accepts a high rate of false positives because a false negative means a missed cancer.

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Overview

A false positive is a test saying something is there when it is not. A false negative is a test saying nothing is there when it is. Statisticians call them Type I and Type II errors. Every test that decides from imperfect evidence makes both, and the two trade off. Tune a test to catch more real cases and it raises more false alarms; tune it the other way and it misses more real cases. Which mistake matters more depends on what each one costs.
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Overview

A false positive is the test crying wolf: it says yes, and the answer was no. A false negative is the test dozing off: it says no, and the answer was yes. Textbooks label them Type I and Type II, which is what happens when statisticians are asked to name things. Every test that works from clues makes both. Push it to catch more and it cries wolf more; push it to cry wolf less and it dozes more. You get to pick your poison, and you have to pick. 😎

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Detail

A false positive is a test result that says yes when the true answer is no: the alarm sounds with no fire, the scan flags a tumour that is not there. A false negative is the reverse: the answer is yes and the test says no. In statistics these are Type I and Type II errors. In the language of the null hypothesis, a false positive rejects a null that was true and a false negative keeps one that was false. Neither means the test was run badly. Any test that decides from evidence short of certainty makes both, and the useful question is how often. The two errors pull against each other. Every test sets a threshold, a point where the evidence counts as enough, and moving that threshold lowers one error by raising the other. Set a screening test to flag anything faintly suspicious and it catches nearly every real case, at the price of sending many healthy people for further checks. Set it to flag only the obvious and it spares those people, at the price of missing early disease. So the threshold is set by asking what each mistake costs. A missed cancer costs more than an unnecessary follow-up, so screening leans towards false positives. A wrongful conviction costs more than an acquittal, so criminal courts lean the other way.
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Detail

A false positive is the test saying yes to something that isn't there; a false negative is the test saying no to something that is. Type I and Type II if you want to sound like the friend who corrects everyone at dinner. Both are built into any test that decides from clues instead of certainty, and every test does. So the question is never whether it makes mistakes but which kind it makes more of. The two are on a seesaw. A pregnancy test made so sensitive that it never misses a pregnancy will also light up for someone who is not pregnant. Make it strict enough that it never lights up wrongly and it will miss someone who is. There is no free move on the seesaw, so the honest question is which side hurts more. Missing something dangerous usually costs more than a scare that turns out fine. That is why medical screens are tuned to be jumpy and then confirmed with a slower, better test. Convicting someone innocent costs more than letting a guilty person go, which is why courts sit the other way. And there is a third cost. A car alarm that goes off in the wind is a false-positive machine everyone learned to ignore: raise enough false alarms and nobody comes to the real one. 😎

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Analogy

A smoke alarm makes both mistakes, and its sensitivity dial decides the mix. Turn the dial up and burnt toast sets it off: a false positive, an alarm with no fire behind it. Turn the dial down and a slow smoulder in the next room never trips it: a false negative, a fire the alarm never called. No setting removes both. Every notch away from one error is a notch towards the other. Manufacturers set the dial towards false positives on purpose. A household that hears the occasional toast alarm loses a minute. A household that sleeps through a fire loses everything.
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Analogy

A bouncer at a club door is running one test on everyone: are you old enough. Turn away someone who is 24 because the photo on their ID looks nothing like them, and that is a false positive: flagged as underage, was not. Wave through a 17-year-old with a good fake, and that is a false negative: was underage, not flagged. A bouncer who wants zero fakes inside will turn away a few grumpy adults. A bouncer who never turns away an adult lets some fakes through. The club sets the strictness by what it fears more, a fine or an empty room, and whichever way it leans, one of the two errors goes up. 😎
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Formal definition — The same term, explained the usual way

A false positive (Type I error) occurs when a test or hypothesis procedure indicates the presence of a condition or effect that is in fact absent; in null hypothesis testing it is the rejection of a true null hypothesis, and its long-run rate is the significance level (α). A false negative (Type II error) occurs when the procedure fails to indicate a condition or effect that is present; it is the retention of a false null hypothesis, with rate β, and 1 − β is the test's statistical power. For a fixed test and sample, lowering one error rate raises the other, so the decision threshold is chosen according to the relative cost of each error in context.

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