Sensitivity vs specificity: read the right denominator
A 90% sensitivity does not mean a positive result is 90% certain. One table shows why the denominator changes the question.
A diagnostic test has 90% sensitivity. Does that mean a person with a positive result has a 90% chance of having the condition? No. Those statements start from different groups of people.
The short film above makes the distinction visually. Work through the same fictional numbers below, then test whether you can choose the denominator before reaching for a formula.
Start with the reference groups
Imagine 100 people. A reference standard identifies 20 with the condition and 80 without it. The test being evaluated returns these results:
| Test result | Condition present | Condition absent | Row total |
|---|---|---|---|
| Positive | 18 true positives | 16 false positives | 34 |
| Negative | 2 false negatives | 64 true negatives | 66 |
| Column total | 20 | 80 | 100 |
These are invented counts for learning the calculation, not performance data for a real diagnostic test. The row labels describe the test result; the column labels describe condition status according to the reference standard.
Sensitivity: start with people who have the condition
The denominator is the 20 people in the condition-present column. Eighteen test positive; two are missed.
Sensitivity = true positives / (true positives + false negatives) = 18 / 20 = 90%.
Read that as: in this example, the test correctly identifies 90% of those who have the condition. It describes detection within that reference group.
Specificity: start with people who do not have it
Now use the condition-absent column. Of these 80 people, 64 test negative and 16 test positive.
Specificity = true negatives / (true negatives + false positives) = 64 / 80 = 80%.
Here, the test correctly returns a negative result for 80% of those without the condition. The 100-person grand total is not the denominator for either calculation.
A positive result asks a different question
If the starting point is a positive result, use the positive row. There are 34 positive results: 18 true positives plus 16 false positives.
Positive predictive value (PPV) = true positives / all positive results = 18 / 34 ≈ 52.9%.
The same table therefore gives 90% sensitivity and approximately 52.9% PPV. There is no contradiction: the calculations answer different conditional questions.
Our fictional group has a condition prevalence of 20%. Predictive values depend on prevalence as well as test sensitivity and specificity. Do not transfer the 52.9% figure to another population or to a real patient.
A routine for the next question
- Label which dimension is the test result and which is condition status.
- Ask what the question gives you first: condition present, condition absent or a test result.
- Circle that reference group. Its total is your denominator.
- Choose the matching numerator, calculate, then write the answer in words.
Quick check: keep 20 people in the condition-present group, but change the results to 16 true positives and four false negatives. Sensitivity becomes 16/20 = 80%. If the condition-absent results stay unchanged, specificity stays 80%.
Keep learning
Risk ratio vs odds ratio: compare the right fractions →
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