What next?
The same reasoning, in the other places it decides something.
What your result means
The big figure is the positive predictive value: of everyone who tests positive, the share who actually have the condition. It is not the test’s accuracy, and it is the number a person who has just been told "positive" actually wants. Sensitivity and specificity are properties of the test; the predictive value depends just as much on prevalence — how common the condition is in the group being tested. Change nothing about the test, test a different group, and the meaning of the same result changes.
Why this one is different
The visual is the tool. Ten thousand icons redraw live on canvas as you move the prevalence, sensitivity and specificity sliders, so you can see the false positives outnumber the true positives instead of being told they do. It works on a phone.
A small error rate on a large group beats a high catch rate on a small one
Screen 10,000 people for something one in a thousand has, with a test that catches 99% of cases and clears 95% of healthy people. Ten people have it and the test finds almost all of them. But 5% of the 9,990 healthy people — around 500 — also test positive.
So roughly 510 people get a positive result and about 10 of them have the condition. That is under 2%. Nothing about the test is faulty; the arithmetic simply favours the enormous healthy group. This is the single most common error in reading a medical result, and it is why doctors retest rather than diagnose from one screen.
How it works
The grid is 10,000 people, the size of group where whole-percentage effects become countable. Prevalence splits them into those who have the condition and those who do not. Sensitivity decides how many of the first group the test catches, leaving the rest as false negatives. Specificity decides how many of the second group it correctly clears, leaving the rest as false positives. The tool then adds the two positive groups together and asks what share of them are real — which is Bayes’ theorem, done by counting instead of by algebra. Rounding is done once, with the largest group absorbing the remainder, so the four blocks always total exactly 10,000.
How to use this calculator
- Start with a prevalence that matches the group being tested, not the whole country. A symptomatic patient in a clinic has a far higher prior than a random person invited for screening — that difference is usually bigger than any difference between tests.
- Set sensitivity and specificity from the test’s own documentation. They are two separate numbers, and a single "accuracy" figure hides which one is weak.
- Watch the red block. When it dwarfs the green one, the test is not failing — the population is doing it.
- Drag prevalence up and down with the test fixed. This is the point of the tool: the same test changes meaning with the group you point it at.
- Compare specificity 95% with 99.5% at low prevalence. A half-percent change in specificity moves the false alarms more than a large change in sensitivity does.
Formula
The top is the true positives. The bottom is every positive result: true positives plus false ones. Sensitivity is the true positive rate among people who have the condition; specificity is the true negative rate among people who do not, so (1 − specificity) is the false alarm rate. Prevalence is the prior — the probability before the test told you anything.
Example calculation
The default settings: prevalence 0.1%, sensitivity 99%, specificity 95%, across 10,000 people:
Caught: 10 × 0.99 = 9.9, so 10 true positives and 0 missed
Healthy: 9,990 people
False alarms: 9,990 × 0.05 = 500 false positives
Positive results altogether: 9.9 + 499.5 = 509.4
Chance a positive is real: 9.9 ÷ 509.4 = 1.9%
About 51.5 positive results for every real case found
Frequently asked questions
Does this mean screening is pointless?＋
No, and that is the wrong conclusion to draw from it. A screening programme is judged on whether finding cases early does more good than the harm of investigating the false alarms — which is why screening is usually offered to groups where prevalence is high enough to shift the arithmetic, and why a positive screen is followed by a better, more expensive test rather than treatment. What this tool argues against is reading a single screening result as a diagnosis.
Why is a second test so much more informative?＋
Because it starts from a different prior. After a positive screen your probability is no longer the population prevalence, it is the predictive value you just calculated — so the second test operates on a group where the condition is far more common, and its own positives are much more likely to be real. Feed the result back in as the new prevalence and watch what happens. The caveat: this only holds if the two tests fail independently. Two tests that share a mechanism can be wrong together in the same person.
Where do sensitivity and specificity come from?＋
From studies of the test against a reference standard, published in its documentation or evaluation. They are not fixed constants: they vary with the equipment, the threshold chosen for "positive", the stage of the condition and, for anything interpreted by a person, the reader. Any single pair of numbers is a summary of a range. That is exactly why this tool ships illustrative shapes rather than claiming figures for named real-world tests.
Related calculators
Assumptions & limitations
Every figure here comes from a simplified model. Keep these limits in mind when reading your result:
- This is an educational tool about probability. It is not diagnostic, it is not medical advice, and no result here says anything about you. If you have had a test result you are worried about, the person to talk to is the clinician who ordered it.
- The presets are illustrative shapes chosen to show the arithmetic, not the performance figures of any real test or screening programme. Real sensitivity and specificity must come from the test’s own documentation.
- One test, one threshold, one moment. Real testing is a sequence, and the threshold for calling a result positive is a choice that trades sensitivity against specificity.
- The model assumes the tested group all share the same prevalence. In practice risk varies enormously within any group by age, family history and symptoms, and a personalised prior would be more accurate than a population one.
- Sensitivity and specificity are treated as exact. They are estimates from studies, with confidence intervals, and they shift with the population studied.
- Nothing here weighs consequences. A false negative and a false positive are one cell each on the grid, but they are not remotely equal in what they cost a person.