Normal Distribution Calculator
Enter a mean and standard deviation, then choose the region you want to measure—below, above, between or outside values—or work backward from a percentile.
See the probability on the curve
Enter your values, or try an example.
Your results and calculation will appear here.
Show calculation
How to use this calculator
- Enter the mean and standard deviation in the same units as your values.
- Choose a probability region or the percentile-to-value mode.
- Enter the boundary value(s) and calculate. The chart shows the selected region.
Worked example: test scores
mode: below · mean: 70 · sd: 10 · x: 80
Probability = 0.8413; Percentage = 84.1345%
About 84.1345% of this normal distribution lies below 80.
Example calculation
- z₁ = (80 − 70) / 10 = 1.
- Φ(z₁) = 0.8413.
Formula & assumptions
z = (x − μ) / σ; P(a ≤ X ≤ b) = Φ(zᵦ) − Φ(zₐ); percentile value = μ + σΦ⁻¹(p).
- The variable is modeled by a normal distribution.
- Standard deviation must be positive.
- For a continuous variable, including or excluding a boundary does not change its probability.
Sources
Reviewed: October 2026
Common questions
Does every dataset follow a normal distribution?
No. A mean and standard deviation alone do not establish a normal distribution. This calculator uses the normal model you specify; strongly skewed data or unusually heavy tails may not fit it well.
What is the standard normal distribution?
It is the normal distribution with mean 0 and standard deviation 1.
How is outside a range calculated?
Add the probability below the lower bound to the probability above the upper bound. The boundaries do not matter for a continuous distribution.
Reference tables
Look up standard normal probabilities in the complete Z Table. Browse all statistical tables.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.