About the Normal Distribution Calculator
This normal distribution calculator finds probabilities under a bell curve. Enter the mean and standard deviation, pick whether you want the area below a value, above it, between two values or outside them, and it returns the probability together with the z-scores and a shaded chart of the region.
It also works in reverse: enter a percentile and it returns the cut-off value, which answers questions like “what score puts you in the top 10%?” Students use it instead of a z-table, teachers use it for grading on a curve, and analysts use it for quality-control limits, test scores, heights and other roughly normal measurements.
For a standard normal distribution set the mean to 0 and the standard deviation to 1, and the values you enter are z-scores. Results assume the data really are normally distributed; for small counts or skewed data, a different distribution may fit better.
With the default inputs, the probability is 97.725%. Change any value above to recalculate instantly.
How to use the normal distribution calculator
- 1Enter the mean and standard deviation of the distribution (0 and 1 for z-scores).
- 2Choose whether you want the area below, above, between or outside your values.
- 3Enter x₁ (and x₂ for between/outside).
- 4Read the probability and z-scores; the chart shades the region.
- 5Optionally enter a percentile to find the matching cut-off value.
Formula and method
Any normal variable X with mean μ and standard deviation σ can be standardised to z = (x − μ)/σ, which follows the standard normal distribution. The probability of falling below x is then Φ(z), the standard normal cumulative distribution function — exactly what a printed z-table lists. Areas above, between or outside values are built from differences of Φ.
The inverse normal runs the other way: it finds the z with Φ(z) = p and converts it back with x = μ + σz. Φ is evaluated with a high-accuracy polynomial approximation (error below 1 in 10 million) and Φ⁻¹ with Acklam’s rational approximation.
- μ
- Mean (centre) of the distribution
- σ
- Standard deviation (spread)
- z
- Number of standard deviations from the mean
- Φ
- Standard normal cumulative distribution function
Worked examples
IQ below 130 (mean 100, SD 15)
An IQ of 130 is (130 − 100)/15 = 2 standard deviations above the mean. Φ(2) = 0.97725, so about 97.7% of people score below 130. The 90th percentile cut-off is 100 + 15 × 1.2816 ≈ 119.2.
Within one SD: between 85 and 115
The area between z = −1 and z = 1 is Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827, the “68” of the 68–95–99.7 rule.
Standard normal: P(Z > 1.96)
The upper tail beyond z = 1.96 holds 1 − 0.9750 = 0.0250 of the area, which is why 1.96 is the critical value for a two-sided 95% confidence interval.
Exam scores outside 50–90 (mean 70, SD 10)
Scores more than 2 SD from the mean on either side make up 2 × 0.02275 = 4.55% of students. The 95th percentile is 70 + 10 × 1.6449 ≈ 86.4.
Frequently asked questions
What is a normal distribution?+
A continuous, symmetric, bell-shaped distribution described by its mean and standard deviation. Many natural measurements — heights, measurement errors, test scores — are approximately normal, and sample means tend toward normal by the central limit theorem.
What is the 68-95-99.7 rule?+
In a normal distribution about 68.27% of values fall within 1 standard deviation of the mean, 95.45% within 2, and 99.73% within 3. It is a quick way to judge how unusual a value is.
How do I find the probability between two values?+
Convert both values to z-scores, look up Φ for each, and subtract: P(a < X < b) = Φ(z_b) − Φ(z_a). Select “between” above and the calculator does this for you.
What is the inverse normal (invNorm)?+
It returns the value x that has a given probability p below it — the opposite of the CDF. For example, invNorm(0.95) on the standard normal is 1.645, so 95% of values fall below z = 1.645.
Is P(X < x) the same as P(X ≤ x)?+
Yes. For a continuous distribution the probability of hitting any exact single value is zero, so “less than” and “less than or equal to” give the same answer.