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    Path to Science

    Thinking Dictionary

    False positives and false negatives

    What it is

    Every test can be wrong in two different ways.

    • A false positive is when a test says "yes" but the true answer is "no." Think of a smoke alarm going off because of burnt toast.
    • A false negative is when a test says "no" but the true answer is "yes." Think of a smoke alarm staying silent during a real fire.

    Two terms describe how good a test is at avoiding these mistakes.¹

    • Sensitivity is the share of real cases that the test catches. A test with high sensitivity rarely misses a real case.
    • Specificity is the share of non-cases that the test correctly clears. A test with high specificity rarely raises a false alarm.

    Why it matters

    No test is perfect, and the two kinds of mistake cost different things. A false positive can cause worry, extra tests, or treatment you don't need. A false negative can give false reassurance when something really is wrong.

    Usually you can't reduce both kinds of mistake at once. Make a test stricter, so it raises the alarm more easily, and it will catch more real cases but also give more false alarms. Make it less strict, and the opposite happens. Choosing where to set the line is a trade-off. Scientists face the same choice when they decide how to present their findings.²

    Example: rapid COVID tests

    A large review pooled 155 groups of people tested with rapid COVID tests, the kind you can do at home. The tests were very good at avoiding false alarms. Their specificity was over 99%. But they missed many real infections. In people with symptoms, they caught about 73% of cases. In people without symptoms, they caught only about 55%.¹

    So a positive result was quite convincing. A negative result was much less reassuring, especially for someone without symptoms.

    Example: breast cancer screening

    Screening means testing healthy people to catch a disease early. A review of breast cancer screening with X-rays, called mammography, estimated that for every 2,000 women invited to screening over 10 years, more than 200 would go through serious distress because of a false-positive result.³ This review's overall conclusions about screening are debated, and other experts weigh the benefits differently. But the point about false positives stands: they cause real harm.

    Why a positive test can be wrong more often than right

    Even a good test can give more false positives than true ones when a condition is rare. Counting people, instead of using percentages, makes this easy to see.⁴˒⁵

    Here are some made-up numbers to show how it works. Imagine 1,000 people, of whom 10 have a condition.

    • The test catches 9 of those 10.
    • It also wrongly flags 5% of the 990 healthy people. That is about 50 people.
    • So about 59 people test positive, but only 9 of them actually have the condition.

    That means only about 1 in 6 positive results is real. How common a condition is to begin with matters a great deal.

    In everyday life

    • Your spam filter. Spam that lands in your inbox is a false negative. A real email sent to the spam folder is a false positive.
    • Airport security. Alarms go off for belt buckles all the time. That is a deliberate choice: a missed weapon would be far worse than a false alarm.
    • Flood warnings. Imagine a system that warns a town about floods. If it warns too easily, people evacuate for nothing, and they may ignore the next warning. If it warns too late, people are in danger. Someone has to decide which mistake is worse.

    The common trap

    The first trap is thinking "a positive test means I have it." It depends on how common the condition is.

    The second trap is thinking "the test is 99% accurate, so there's only a 1% chance it's wrong for me." A single "accuracy" figure mixes sensitivity and specificity together, and it ignores how common the condition is.

    The third trap is thinking "negative means safe." That depends on the test's sensitivity, and on when you took the test.¹

    Some ways out:

    • Ask about both kinds of mistake, not just one "accuracy" figure.
    • Ask how common the condition is among people like you.
    • Turn percentages into counts: "out of 1,000 people like me, how many…?"

    Try it

    Watch your email spam filter for a week. Count two things: spam that got through to your inbox (false negatives), and real emails that ended up in spam (false positives).

    Which mistake annoyed you more? If you could make the filter stricter, which number would go up, and which would go down?

    Related concepts

    Probabilistic reasoning · Scientific uncertainty · Signal vs noise · Placebo effect · Accuracy vs precision · Nocebo effect

    Sources

    1. Dinnes, J., Sharma, P., Berhane, S., van Wyk, S. S., Nyaaba, N., Domen, J., Taylor, M., Cunningham, J., et al. (2022). Rapid, point-of-care antigen tests for diagnosis of SARS-CoV-2 infection. Cochrane Database of Systematic Reviews, 2022(7), CD013705. https://doi.org/10.1002/14651858.CD013705.pub3

    2. Mastrandrea, M. D., et al. (2010). Guidance note for lead authors of the IPCC Fifth Assessment Report on consistent treatment of uncertainties. IPCC. https://www.ipcc.ch/site/assets/uploads/2017/08/AR5_Uncertainty_Guidance_Note.pdf

    3. Gøtzsche, P. C., & Jørgensen, K. J. (2013). Screening for breast cancer with mammography. Cochrane Database of Systematic Reviews, 2013(6), CD001877. https://doi.org/10.1002/14651858.CD001877.pub5

    4. Hoffrage, U., Lindsey, S., Hertwig, R., & Gigerenzer, G. (2000). Communicating statistical information. Science, 290(5500), 2261–2262. https://doi.org/10.1126/science.290.5500.2261

    5. Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684–704. https://doi.org/10.1037/0033-295X.102.4.684