Normal Distribution Calculator

Q: What is the normal distribution?

A symmetric bell-shaped distribution described by μ and σ.

Q: What is Φ(z)?

The cumulative distribution function (CDF) of the standard normal.

Q: 68-95-99.7 rule?

68% within 1σ, 95% within 2σ, 99.7% within 3σ of the mean.

Q: What is the inverse CDF?

Finds x such that P(X < x) = p.

Q: What is a z-score?

The number of standard deviations from mean: z = (x-μ)/σ.

Use our free normal distribution calculator to find P(X < x), P(X > x), P(a < X < b), or find x for a given cumulative probability.

Math

Calculate

Mean (μ)

Std Dev (σ)

Value (x)

Lower bound (a)

Upper bound (b)

Probability (0-1)

Formula

The normal (Gaussian) distribution is the most important probability distribution in statistics. Compute cumulative probabilities, tail probabilities, interval probabilities, or find x for a given probability (inverse CDF).

P(a < X < b) = Φ((b-μ)/σ) - Φ((a-μ)/σ)

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Select the calculation mode.
2Enter the mean (μ) and standard deviation (σ).
3Enter the relevant value(s).
4Click Calculate.

Examples

P(X < 1.96) for μ=0, σ=1

1.z=1.96
2.Φ(1.96)≈0.9750
3.P(X<1.96)=97.50%

Result: 0.975000

P(-1 < X < 1) for μ=0, σ=1

1.z₁=-1, z₂=1
2.Φ(1)-Φ(-1)=0.8413-0.1587=0.6827

Result: 0.682689

x for p=0.975, μ=0, σ=1

1.z=Φ⁻¹(0.975)≈1.96
2.x=0+1.96×1=1.96

Result: 1.959964

Frequently Asked Questions

A symmetric bell-shaped distribution described by μ and σ.

The cumulative distribution function (CDF) of the standard normal.

68% within 1σ, 95% within 2σ, 99.7% within 3σ of the mean.

Finds x such that P(X < x) = p.

The number of standard deviations from mean: z = (x-μ)/σ.

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CalculateMe Team

Last updated: 2026-07-14