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Normal CDF Calculator

Compute the cumulative probability under a normal distribution.

Introduction

Normal CDF Calculator — Compute the cumulative probability under a normal distribution. Enter Value (x), Mean (μ), Std. deviation (σ) to get an instant, accurate result.

Formula

Cumulative probability P(X ≤ x) for a normal distribution, computed by converting to a z-score, z = (x−mean)/σ, then evaluating the standard normal CDF using the Abramowitz & Stegun (1964) numerical approximation — no closed-form formula exists for the normal CDF.

Step-by-step

  1. Enter the Value (x).
  2. Enter the Mean (μ).
  3. Enter the Std. deviation (σ).
  4. Click Calculate to see your result instantly.

Real-world example

Example: With Value (x) = 0, Mean (μ) = 0, Std. deviation (σ) = 1, the Normal CDF Calculator gives Cdf: 0.5.

Frequently Asked Questions

Why is an approximation used instead of an exact formula?
The normal distribution's cumulative probability has no closed-form algebraic expression, so it must be evaluated numerically; this calculator uses a well-established, highly accurate polynomial approximation.
How accurate is the result?
The Abramowitz & Stegun approximation used here is accurate to within about 7.5×10⁻⁸, more than sufficient for virtually all practical purposes.
What inputs does the Normal CDF Calculator need?
You'll need to provide: Value (x), Mean (μ), Std. deviation (σ). All fields use sensible defaults, so you can see a working example immediately and then adjust the values to match your own numbers.
How accurate is the Normal CDF Calculator?
The Normal CDF Calculator applies the formula exactly as calculated — Cumulative probability P(X ≤ x) for a normal distribution, computed by converting to a z-score, z = (x−mean)/σ, then evaluating the standard normal CDF using the Abramowitz & Stegun (1964) numerical approximation — no closed-form formula exists for the normal CDF. — so results are precise for the inputs you provide. Accuracy depends on entering correct, realistic input values.
Is the Normal CDF Calculator free to use?
Yes, the Normal CDF Calculator is completely free, requires no signup, and runs instantly in your browser.

About the Normal CDF Calculator

The Normal CDF Calculator uses a real, verifiable formula — Cumulative probability P(X ≤ x) for a normal distribution, computed by converting to a z-score, z = (x−mean)/σ, then evaluating the standard normal CDF using the Abramowitz & Stegun (1964) numerical approximation — no closed-form formula exists for the normal CDF. — so results are accurate every time, not an approximation.