Normal distribution calculator
Enter the mean and standard deviation, then choose what to find: the area under the bell curve to the left, to the right, between two values, or the value that marks a given percentile.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Enter the mean and standard deviation.
- Pick what to find. For areas, enter x (and b for “between”). For a percentile, enter the probability.
- Read the probability, the percentage and the z-scores.
How it works
z = (x − μ) / σ. The area to the left is Φ(z) = ½ × erfc(−z/√2). The area between a and b is Φ(z_b) − Φ(z_a), computed from the right tails when both bounds are above the mean, so small areas keep their precision. A percentile is found by inverting Φ and then x = μ + zσ.
Worked example
IQ scores with μ = 100 and σ = 15.
- P(X ≤ 115): z = 1, Φ(1) = 0.841345, so about 84.1% score 115 or less.
- P(85 ≤ X ≤ 115): z from −1 to 1, area 0.682689, about 68.3%.
- 95th percentile: z = 1.644854, x = 100 + 1.644854 × 15 ≈ 124.67.
Tips
- For z-score and p-values of a single value, the z-score calculator gives the two-tailed p as well.
- The probability of exactly one value is 0 for a continuous distribution.
- Standardise first (μ = 0, σ = 1) to match a z table.
Frequently asked questions
What is a normal distribution?
A bell-shaped, symmetric distribution fully described by its mean μ and standard deviation σ. About 68% of values lie within one σ of the mean, 95% within two and 99.7% within three.
How is the probability computed?
Each value is converted to a z-score, z = (x − μ) / σ, and the standard normal cumulative distribution Φ(z) is evaluated through the error function. The result is accurate to far better than 1e-7.
How do I find a percentile?
Choose the percentile option and enter the probability, for example 0.95. The tool inverts the cumulative distribution to get z, then x = μ + z × σ. For a mean of 100 and σ of 15 the 95th percentile is about 124.67.
What is the difference between the left and right area?
The left area is P(X ≤ x), the share of values at or below x. The right area is P(X ≥ x) = 1 − left. For a continuous distribution the two always add to 1.
Is my data really normal?
The calculator assumes it is. Heights, measurement errors and many test scores are close to normal, but incomes, waiting times and counts are often skewed, and the result can then be misleading.