Quick answer
A normal probability is P(X < x) = Φ((x − μ) ÷ σ). With μ = 100 and σ = 15, P(X < 115) = 0.8413 (z = 1).
Enter the mean, standard deviation, a probability type and the x values. The calculator returns the probability and z-score. Use this normal distribution calculator to check your own figures.
Related tools: Binomial Distribution Calculator: Binomial probabilities for n trials. Poisson Distribution Calculator: Poisson probabilities for a given mean rate. Sampling Distribution of the Mean Calculator: Probability for a sample mean using the central limit theorem.
How to use the Normal Distribution Calculator
- Enter mean (μ). It starts at 0.
- Enter standard deviation (σ) (above 0). It starts at 1.
- Choose probability: P(X < x₁), P(X > x₁) and P(x₁ < X < x₂).
- Enter x₁.
- Optionally, enter x₂.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Mean (μ) | A number | starts at 0 |
| Standard deviation (σ) | A number | > 0, starts at 1 |
| Probability | Choose one: P(X < x₁); P(X > x₁); P(x₁ < X < x₂) | Starts at P(X < x₁) |
| x₁ | A number | — |
| x₂ (optional) | A number | — |
Formula and method
Formula
P(X < x) = Φ((x − μ) ÷ σ)
IEEE-754 double precision; shown to at most 6 decimal places.
Φ is the standard normal CDF evaluated to double precision (series near zero, continued fraction in the tails).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Mean (μ) = 100; Standard deviation (σ) = 15; Probability = P(X < x₁); x₁ = 115.
| Result | Value |
|---|---|
| P(X < x₁) | 0.841345 |
| z-score | 1 |
Example 2
Inputs: Mean (μ) = 100; Standard deviation (σ) = 15; Probability = P(X > x₁); x₁ = 130.
| Result | Value |
|---|---|
| P(X > x₁) | 0.02275 |
| z-score | 2 |
Example 3
Inputs: Mean (μ) = 100; Standard deviation (σ) = 15; Probability = P(x₁ < X < x₂); x₁ = 85; x₂ = 115.
| Result | Value |
|---|---|
| P(x₁ < X < x₂) | 0.682689 |
| z₁ | -1 |
| z₂ | 1 |
The first line is the headline result (p(X < x₁)). Below it the normal Distribution Calculator lists z-score, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Between two values
P(85 < X < 115) = Φ(1) − Φ(−1) = 0.6827, the 68% of the empirical rule. 'Between' needs x₂ as well.
Requirements
σ must be positive. Use it for heights, test scores and measurement errors, where data are bell-shaped. For other shapes the result is only approximate.
How the result changes with mean
Holding every other input at the first worked example, this table changes only mean. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Mean | P(X < x₁) | z-score |
|---|---|---|
| 50 | 0.999993 | 4.333333 |
| 100 | 0.841345 | 1 |
| 200 | 0 | -5.666667 |
| 400 | 0 | -19 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Standard deviation (σ): entering "0" shows "Standard deviation (σ) must be greater than 0." beside that field and no result.
- x₂: left empty shows "Enter x₂ for a between-probability." beside that field and no result.
Choosing between the Normal Distribution Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Binomial Distribution Calculator](/statistics/binomial-distribution/) | trials, success probability and successes | P(X = 5) |
| [Poisson Distribution Calculator](/statistics/poisson-distribution-calculator/) | average rate and events | P(X = 3) |
| [Sampling Distribution of the Mean Calculator](/statistics/normal-probability-sampling-distributions/) | population mean, population SD and sample size | P(x̄ < value) |
| [IQ Percentile Calculator](/statistics/iq-percentile/) | iQ score, mean and standard deviation | Percentile |
| [Empirical Rule Calculator](/statistics/empirical-rule/) | mean and standard deviation | ≈68% (μ ± σ) |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- x₂ required for 'between'
- σ > 0
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for standard deviation (σ) and x₂ to confirm this.
Frequently asked questions
What is P(X > 115)?
P(X > 115) = 1 − 0.8413 = 0.1587 when μ = 100 and σ = 15.
What is the total area under the curve?
The total area under a probability density curve is 1, representing 100% of outcomes.
How do I find x from a probability?
Use x = μ + zσ: the 90th percentile z of 1.2816 with μ = 100, σ = 15 gives x ≈ 119.2.
What does the Normal Distribution Calculator calculate?
Probabilities for a normal distribution. It returns P(X < x₁) and z-score.
What formula does the Normal Distribution Calculator use?
P(X < x) = Φ((x − μ) ÷ σ). Φ is the standard normal CDF evaluated to double precision (series near zero, continued fraction in the tails).
What do I need to enter in the normal Distribution Calculator?
You enter Mean (μ), Standard deviation (σ), Probability and x₁. Optional: x₂.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. x₂ required for 'between'.
What are the limits of this calculator?
x₂ required for 'between'. σ > 0.
What does the “Probability” option change?
It sets how the normal Distribution is worked out. The choices are P(X < x₁), P(X > x₁) and P(x₁ < X < x₂).
What is an example of a normal Distribution result?
With mean (μ) 100, standard deviation (σ) 15, probability P(X < x₁) and x₁ 115, the p(X < x₁) is 0.841345.
Related calculators
- Binomial Distribution Calculator — Binomial probabilities for n trials.
- Poisson Distribution Calculator — Poisson probabilities for a given mean rate.
- Sampling Distribution of the Mean Calculator — Probability for a sample mean using the central limit theorem.
- IQ Percentile Calculator — Percentile of an IQ score on a normal scale.
- Empirical Rule Calculator — 68–95–99.7 intervals for a normal distribution.
- Geometric Probability Calculator — Probability the first success occurs on trial k.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.