Enter a mean and standard deviation with the range you care about. Leave either bound empty for a one-sided probability.
The shaded region is 68.27% of the area under the curve. Ticks mark the mean and each standard deviation either side of it.
Show the working, step by step
How it works
Every normal distribution is the same shape stretched and shifted, so any probability question can be answered by converting to the standard normal — mean 0, standard deviation 1 — and reading the area from there.
z = (x − μ) / σ P(a < X < b) = Φ(z_b) − Φ(z_a)
Φ is the cumulative distribution function: the area under the curve to the left of a given z. This is what a printed z-table contains, and what the calculator evaluates directly.
A worked example
IQ scores are conventionally modelled as normal with μ = 100 and σ = 15. What fraction of people score between 85 and 115?
- z for 85 is (85 − 100) / 15 = −1.
- z for 115 is (115 − 100) / 15 = +1.
- Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827.
About 68.3% — which is exactly the first line of the empirical rule, since 85 and 115 are one standard deviation either side of the mean.
The shape, and what σ does to it
- The curve is symmetric about the mean, so the mean, median and mode all coincide.
- Its inflection points — where the curvature flips — sit at exactly μ ± σ. That is the standard deviation made visible.
- Doubling σ halves the peak height and doubles the width. The area stays 1.
- The tails never touch the axis, so every value has a non-zero probability, however remote.
When the normal model is the wrong one
The normal distribution is symmetric and unbounded, which rules it out for a good deal of real data. Incomes, waiting times and file sizes are right-skewed with a hard floor at zero. Counts of rare events follow a Poisson distribution. Proportions from a fixed number of trials follow a binomial.
Using a normal model on such data produces confident answers that are simply wrong — most visibly, it will assign real probability to impossible negative values.
Related calculators
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Bell curve generator
Draw and label the curve for any μ and σ.
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Z-score calculator
Standardise a single value and get its percentile.
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Empirical rule
The 68–95–99.7 shortcut for the same curve.
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Standard deviation
Get the σ this calculator needs from raw data.
Common questions
How do I find the probability between two values?
Enter the mean, the standard deviation and both bounds. The calculator converts each bound to a z-score, reads the area below each from the standard normal distribution, and subtracts.
How do I get a one-sided probability?
Leave one bound empty. An empty lower bound means −∞, an empty upper bound means +∞, so clearing the lower box gives you P(X ≤ upper).
What is the area under a normal curve?
Always exactly 1, for any mean and any standard deviation — it is a probability distribution. Changing σ makes the curve wider and flatter, or narrower and taller, but never changes the total area.
What if my data is not normally distributed?
Then these probabilities do not apply, however precisely they are computed. Check the shape first — a box plot showing a badly off-centre median is a strong sign that the normal model is the wrong one.