What does this calculator do?

Compute statistical results from your inputs.

Enter values in the input fields and view results.

What are the formulas?

See the educational content section for formulas.

For statistical analysis and computation.

Check the related calculators section below.

Results follow standard statistical methods.

4 more

STATISTICSProbability TheoryStatistics Calculator

📊

False Positive Paradox Calculator

False Positive Paradox: PPV, NPV from sensitivity, specificity, prevalence. Why screening fails for

Run CalculatorExplore data analysis and statistical calculations

Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

📊

BASE RATE MATTERSWhy 99% accurate test → mostly false positives

False Positive Paradox — Why "Positive" Is Usually Wrong

For rare conditions, even 99% accurate tests produce mostly false positives. PPV depends on prevalence. Master Bayes, natural frequencies, and base rate neglect.

Bayes' Theorem →Conditional Prob →

Examples — Click to Load

Inputs

Sensitivity (%)

Specificity (%)

Prevalence (%)

ppv.sh

CALCULATED

Positive Predictive Value (PPV)

16.10%

PPV

16.10%

NPV

99.95%

FDR

83.90%

Prevalence

1.00%

Natural Frequency (out of 10000)

TP: 95

FP: 495

TN: 9405

FN: 5

False Positive Paradox

PPV — Positive Predictive Value

16.10%

NPV: 99.9%FDR: 83.9%Prevalence: 1.00%

numbervibe.com/calculators/statistics/false-positive-paradox-calculator

Confusion Matrix (out of 10000)

Test +

Test −

Disease +

TP
95

FN
5

Disease −

FP
495

TN
9405

Calculation Breakdown

COMPUTATION

Numerator (TP)

0.009500

Sens × Prev = 0.9500 × 0.0100

COMPUTATION

Denominator (TP+FP)

0.059000

TP + (1-Spec)×(1-Prev)

RESULT

PPV

16.10%

PPV = TP / (TP+FP)

NOTE

FDR

83.90%

ext{FDR} = 1 - ext{PPV}

Natural Frequency (out of 10,000)

PPV vs Prevalence

Of All Positive Tests

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

• Base rate neglect: People ignore prevalence and overestimate a positive test
• PPV = (Sensitivity × Prevalence) / [(Sens×Prev) + ((1-Spec)×(1-Prev))]
• When prevalence is low, false positives from the healthy majority swamp true positives
• A "99% accurate" test for a 1-in-10,000 disease gives mostly false positives — PPV < 10%

Did You Know?

🩺Mammography (0.5% prevalence, ~90% sens/spec) yields PPV ≈ 4% — most positives are false alarmsSource: BMJ

🔒Airport security (0.001% threat) has PPV under 1% — vast majority of alerts are false positivesSource: Security research

🧬Gerd Gigerenzer: natural frequencies dramatically improve Bayesian reasoning vs percentagesSource: Gigerenzer

📧Spam filters work well because spam prevalence is high (30%+) — PPV exceeds 95%Source: ML practice

How It Works

1. Sensitivity

Of those with the condition, what fraction tests positive? TP / (TP + FN).

2. Specificity

Of those without the condition, what fraction tests negative? TN / (TN + FP).

3. Prevalence

Fraction of population with the condition. Often ignored but critical for PPV.

4. PPV

Given a positive test, what is P(actually have condition)? PPV answers this.

Frequently Asked Questions

What is the false positive paradox?

When a condition is rare, even a highly accurate test produces more false positives than true positives. A positive result is usually wrong.

Why does PPV depend on prevalence?

PPV = TP / (TP + FP). When prevalence is low, the healthy population is huge, so FP = (1-Spec)×(1-Prev)×N dominates.

When is screening useful?

When prevalence is moderate to high, or when targeting high-risk populations. COVID at 5% works; cancer in general population (0.5%) yields many false positives.