Normal CDF Calculator
Normal CDF Calculator quickly computes probabilities and percentiles using z-scores, standardization, tail areas, step-by-step examples, and clarity.
Equation Preview
Helping Notes
- Area between bounds: \(P(a \le X \le b) = \Phi(z_b)-\Phi(z_a)\), where \(z=(x-\mu)/\sigma\).
- Standard normal CDF: \(\Phi(z)=\tfrac12\big[1+\operatorname{erf}(z/\sqrt2)\big]\). Use σ > 0; any real μ, a, b are valid.
- Common checks: \([-1,1]\) ≈ 0.6827; \([-2,2]\) ≈ 0.9545 for μ=0, σ=1.
Results
Probability \(P(a \le X \le b)\)
z-Scores & CDF Values
Error
Steps
What is Normal CDF Calculator?
A Normal CDF Calculator evaluates cumulative probabilities for the normal distribution. For a standard normal variable \(Z\sim\mathcal N(0,1)\), the cumulative distribution function (CDF) gives the probability that \(Z\) is less than or equal to a threshold \(z\). For a general normal variable \(X\sim\mathcal N(\mu,\sigma^2)\), values are first standardized to the standard normal. Core definitions are shown below.
With the CDF you can compute left‑tail, right‑tail, and interval probabilities, as well as convert probabilities to percentiles.
About the Normal CDF Calculator
This tool accepts \(\mu\), \(\sigma\), and a value (or bounds) and returns probabilities with clear steps: standardization, CDF lookups, and complement rules. It supports left tail \(\Pr(X\le x)\), right tail \(\Pr(X\ge x)\), and two‑sided intervals \(\Pr(a\le X\le b)\). It also computes percentiles by inverting the CDF. For teaching and auditability, each formula is displayed in a responsive block and intermediate results are shown alongside concise interpretations.
How to Use this Normal CDF Calculator
- Enter the mean \(\mu\) and standard deviation \(\sigma>0\).
- Choose a mode: single bound (left/right tail), two bounds (interval), or percentile (inverse CDF).
- Provide the value(s) (e.g., \(x\) or \(a,b\)) and, if needed, desired percentile \(p\in(0,1)\).
- Click calculate to see standardization, CDF evaluations, and the final probability or percentile.
- Review steps and copy the formatted formulas for homework, labs, or reports.
Examples
Example 1: Left tail for standard normal
\(Z\sim\mathcal N(0,1)\). Compute \(\Pr(Z\le1.96)\).
Example 2: Interval probability
\(X\sim\mathcal N(100,15^2)\). Find \(\Pr(85\le X\le115)\).
Example 3: Upper tail
\(X\sim\mathcal N(100,15^2)\). Compute \(\Pr(X\ge130)\).
Example 4: Percentile
\(X\sim\mathcal N(70,8^2)\). Find the 95th percentile.
FAQs
What is the difference between CDF and PDF?
The PDF gives density; the CDF accumulates probability up to a point. Probabilities come from CDF differences or tails.
How do I standardize a normal variable?
Subtract the mean and divide by the standard deviation: \(z=(x-\mu)/\sigma\).
Can probabilities be computed for any \(\mu\) and \(\sigma\)?
Yes, provided \(\sigma>0\). Standardization converts to the standard normal and uses the same CDF.
How do I get right‑tail probabilities?
Use the complement rule: \(\Pr(X\ge x)=1-\Phi((x-\mu)/\sigma)\).
What about two‑tailed areas?
Compute each tail and add, or double a symmetric tail when \(\mu\) is the center.
Can the calculator return percentiles?
Yes. It uses the inverse CDF: \(x_p=\mu+\sigma\,\Phi^{-1}(p)\).
Are results exact?
They are numeric approximations accurate to specified precision; the integral has no elementary closed form.
How is continuity correction used?
For approximating discrete distributions (e.g., binomial), shift by \(\pm0.5\) before standardizing.
What if \(z\) is very large in magnitude?
Extreme tails may underflow numerically; specialized approximations are used to maintain stability.
Does the CDF handle negative values?
Yes. Symmetry holds: \(\Phi(-z)=1-\Phi(z)\).
Is the normal model always appropriate?
No. Check assumptions; skewed or heavy‑tailed data may require different models.
How many decimals should I report?
Use context‑appropriate precision (often 3–4 decimals) and be consistent across reported probabilities.