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Normal Distribution Calculator

Q: How do I find the probability between two values?

Compute both z-scores and subtract: P(a < X < b) = Φ(z_b) − Φ(z_a). For a standard normal, P(−1 < Z < 1) ≈ 68.27%.

Q: What is the inverse normal (invNorm)?

It goes the other way: given a probability, it finds the value. The 95th percentile is x = μ + 1.6449σ; for μ = 100 and σ = 15 that is about 124.67. For the middle 95% use μ ± 1.96σ.

Q: What is the 68-95-99.7 rule?

In a normal distribution about 68.27% of values lie within 1 standard deviation of the mean, 95.45% within 2 and 99.73% within 3. The table under the chart shows these ranges for your mean and SD.

Q: Does it matter whether I use < or ≤?

No. For a continuous distribution the probability of any single exact value is 0, so P(X < x) = P(X ≤ x).

Q: How accurate is this calculator?

The normal CDF is computed with a rational approximation of the complementary error function with an error below about 1.2 × 10⁻⁷, and the inverse uses Acklam's algorithm refined with a Halley step. That is far more precise than a printed z-table.

Q: When is the normal distribution a good model?

For quantities that are sums of many small independent effects, like measurement errors, heights or test scores. It is a poor model for strongly skewed data such as incomes or waiting times.

Enter the mean and standard deviation of a normal distribution to find the probability below, above, between or outside values, or work backwards from a probability to the value (inverse normal). The area is shaded on a bell curve.

Mean (μ)

Standard deviation (σ)

Value (x)

Lower value (a)

Upper value (b)

Probability / percentile

Area

Enter your values to see the result.

How to use the Normal Distribution Calculator

Enter the mean (μ) and standard deviation (σ); use 0 and 1 for the standard normal.
Choose what to find: below, above, between, outside, or a value from a probability.
Enter the value(s) or the probability.
Press Calculate to get the probability and z-scores.
Check the shaded bell curve and the interval table.

Normal distribution formulas

Density: f(x) = (1 / (σ√(2π))) · e^(−(x − μ)² / (2σ²))
Standardise: z = (x − μ) / σ, so Z follows N(0, 1)
Left tail: P(X < x) = Φ(z); right tail: P(X > x) = 1 − Φ(z)
Between: P(a < X < b) = Φ(z_b) − Φ(z_a)
Inverse: x = μ + σ · Φ⁻¹(p)
Symmetry: Φ(−z) = 1 − Φ(z)

Standard normal reference values

z	Φ(z) = P(Z < z)	P(−z < Z < z)	Common use
0	0.5000	0	Median
0.674	0.7500	0.5000	Quartiles
1	0.8413	0.6827	±1σ
1.282	0.9000	0.8000	90th percentile
1.645	0.9500	0.9000	95th percentile / 90% interval
1.960	0.9750	0.9500	95% confidence interval
2	0.9772	0.9545	±2σ
2.326	0.9900	0.9800	99th percentile
2.576	0.9950	0.9900	99% confidence interval
3	0.99865	0.9973	±3σ

Example

Adult heights in a group have μ = 175 cm and σ = 7 cm. The share taller than 190 cm is P(X > 190) = 1 − Φ((190 − 175)/7) = 1 − Φ(2.1429) ≈ 1.61%. The middle 95% of heights lie between 175 ± 1.96 × 7, i.e. 161.3 cm to 188.7 cm.

Frequently asked questions

How do you find a probability from a normal distribution?

Convert the value to a z-score with z = (x − μ)/σ, then use the standard normal CDF Φ(z). P(X < x) = Φ(z) and P(X > x) = 1 − Φ(z). For IQ (μ = 100, σ = 15), P(X < 120) = Φ(1.3333) ≈ 90.88%.

How do I find the probability between two values?

Compute both z-scores and subtract: P(a < X < b) = Φ(z_b) − Φ(z_a). For a standard normal, P(−1 < Z < 1) ≈ 68.27%.

What is the inverse normal (invNorm)?

It goes the other way: given a probability, it finds the value. The 95th percentile is x = μ + 1.6449σ; for μ = 100 and σ = 15 that is about 124.67. For the middle 95% use μ ± 1.96σ.

What is the 68-95-99.7 rule?

In a normal distribution about 68.27% of values lie within 1 standard deviation of the mean, 95.45% within 2 and 99.73% within 3. The table under the chart shows these ranges for your mean and SD.

Does it matter whether I use < or ≤?

No. For a continuous distribution the probability of any single exact value is 0, so P(X < x) = P(X ≤ x).

How accurate is this calculator?

The normal CDF is computed with a rational approximation of the complementary error function with an error below about 1.2 × 10⁻⁷, and the inverse uses Acklam's algorithm refined with a Halley step. That is far more precise than a printed z-table.

When is the normal distribution a good model?

For quantities that are sums of many small independent effects, like measurement errors, heights or test scores. It is a poor model for strongly skewed data such as incomes or waiting times.

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