Two kinds of accuracy
A screening test is usually described by two numbers. The first is how often it correctly picks up the condition when it is present, often called sensitivity. The second is how often it correctly stays negative when the condition is absent, called specificity. The false positive rate is the flip side of the second: the share of people without the condition who still get a positive result.
A test can score well on both and still produce a surprising pattern of results. The reason is not a flaw in the test. It is the arithmetic of applying a test to a large group where few people have what is being looked for.
Everything below is a generic illustration with invented numbers. It is not medical information about any real test, and nobody should interpret a real result from it. Anyone with a real result should talk to a qualified health professional.
Counting through an example
Suppose a condition affects 1 person in 100. A screening test catches 90% of people who have it, and gives a false positive for 10% of people who do not. Take 1,000 people:
- With the condition: 1% of 1,000 = 10 people. The test is positive for 90% of them, so 9 true positives and 1 missed case.
- Without the condition: 990 people. The test is positive for 10%, so 99 false positives and 891 correct negatives.
Everyone who tests positive: 9 + 99 = 108 people. Only 9 of them have the condition. So the chance that a positive result is a true one is 9 / 108, which is about 8.3%.
That is the core result. The test is right about people 90% of the time in one sense, and yet more than 9 out of 10 positives are false alarms. The two statements do not conflict, because they answer different questions: “of people with the condition, how many test positive?” versus “of people who test positive, how many have the condition?”
Why the second number is so low
Look at the sizes of the two groups behind the positive results. There are 9 true positives and 99 false ones. The false positives are many because the group without the condition is 99 times larger than the group with it, even though the test only errs on one in ten of them.
This is base rate neglect in its cleanest form: forgetting how rare the condition is. It is also a direct application of Bayes theorem, where the positive result updates a low starting chance to a higher, but still modest, one.
What a better test changes
Keep everything the same but improve the false positive rate to 1%. Using 10,000 people so the counts stay whole:
- With the condition: 100 people, of whom 90% = 90 test positive.
- Without the condition: 9,900 people, of whom 1% = 99 test positive.
Total positives: 90 + 99 = 189. True positives: 90. So 90 / 189, about 47.6%, of positive results are true.
A tenfold drop in the false positive rate lifted the chance that a positive is real from roughly 8% to roughly 48%. It is better, yet still below half, because the condition is still rare. That is why accurate tests used on whole populations are often followed by a second, more specific check rather than treated as final.
What a second test can do
A positive result changes the starting point for the next step. In the first example, the group of 108 positives is now much richer in true cases than the general population: 9 in 108 instead of 10 in 1,000. A follow-up test applied to that group works from the higher starting chance.
Suppose the follow-up has the same rates and its result is independent of the first given the person’s true status. Of the 9 true positives, 90% = 8.1 would test positive again; of the 99 false positives, 10% = 9.9 would. The chance that a double positive is real is then 8.1 / (8.1 + 9.9) = 8.1 / 18 = 45%. The independence assumption is a big one. If the same factor causes both tests to err for the same person, repeated testing helps less than this simple calculation suggests.
Reading numbers carefully
Some habits make screening statistics easier to read:
- Ask what the percentage is a percentage of. “90% accurate” might describe people with the condition, people without it, or all results.
- Ask how common the condition is. The same test means something different in a group where the condition is common.
- Prefer counts to percentages. Phrasing like “9 of the 108 people who test positive have it” is harder to misread. This is the case for natural frequencies.
- Watch for changes in percentage points versus percentages. Moving from 10% to 1% is a drop of nine percentage points, but a tenfold reduction in the rate. Both descriptions are true and feel quite different.
Why this matters beyond medicine
The same structure appears anywhere a rare thing is searched for with an imperfect detector: fraud alerts, anti-cheat systems, spam filters, and automated checks of any kind. Any time an alarm is rare in truth but the system raises many alerts, most alerts will be false even though the system is “good”. Understanding that avoids two opposite mistakes: panicking at every alert, and distrusting an entire system because many alerts turn out to be nothing.
For games of chance the lesson is narrower. Results come from independent random events, a casino game carries a house edge, and no pattern-detecting method removes that. If gambling is causing difficulty, contacting a local support service is a sensible step.