Normal Distribution Calculator
Probabilities under a bell curve — below, above or between two values.
Last updated
Your details
Result
- P(X < 115)
- 84.1345%
- As a decimal
- 0.84134475
- Z-score
- 1
- The complement
- 15.8655%
- Density at this point
- 0.01613138
115 is 1 σ above the mean
Height of the curve, not a probability
115 on a curve centred at 100…
z = (x − μ) ÷ σ, then read Φ(z)
z = (115 − 100) ÷ 15 = 1
Φ(1) = 0.841345
P = 84.1345%
Formula — written step by step
An IQ below 115, where μ = 100 and σ = 15…
z = (x − μ) ÷ σ, then read Φ(z)
z = (115 − 100) ÷ 15 = 1
Φ(1) = 0.841345
P(X < 115) = 84.1345%
Φ is the cumulative distribution function — the area under the bell curve to the left of z. Standardising first is what lets one table answer questions about any normal distribution.
A worked example
What proportion falls in a range
- Mean
- 100
- Standard deviation
- 15
- Range
- 85 to 115
- z₁ = (85 − 100) ÷ 15 = −1
- z₂ = (115 − 100) ÷ 15 = +1
- Φ(1) = 0.841345, Φ(−1) = 0.158655
- 0.841345 − 0.158655
68.269% — the familiar "68%" of the empirical rule
This is where the empirical rule comes from. It is not an approximation someone chose: it is the exact area under a normal curve within one standard deviation, 68.269%, rounded for teaching.
How to use this calculator
- Enter the mean and standard deviation of your distribution.
- Choose whether you want the probability below a value, above it, or between two.
- Read the probability as a percentage or a decimal — both are shown.
- The density row is the height of the curve, not a probability. For a continuous distribution the chance of any exact value is zero.
Why the normal distribution is everywhere
The normal distribution describes a great many measurements — heights, measurement errors, test scores — but its real importance is the central limit theorem. That theorem says the mean of a large enough sample is approximately normal almost regardless of the shape of the underlying population.
This is why normal-based methods work on data that is not itself normal. Confidence intervals and hypothesis tests about means rely on the distribution of the sample mean, not of the raw values, and that distribution converges on the bell curve as the sample grows.
Standardising, and what a z-table is
There is not one normal distribution but infinitely many, one for each pair of mean and standard deviation. Standardising — subtracting the mean and dividing by the standard deviation — converts any of them into the single standard normal with mean 0 and standard deviation 1.
That is what makes printed z-tables possible: one table serves every normal distribution. This calculator does the same conversion, then computes the area directly rather than looking it up, so the answer carries more decimal places than any table offers.
Density is not probability
For a continuous distribution, the probability of any single exact value is zero. There are infinitely many possible values, so no individual one can carry positive probability. Probability lives in intervals, which is why every question here is framed as "below", "above" or "between".
The density shown is the height of the curve at that point. It is not a probability and can even exceed 1 for a narrow distribution. What matters is the area under it, which is what the other rows report.
Formula & methodology
What this calculator assumes
- The error function is computed through the incomplete gamma function rather than a short rational approximation. The common Abramowitz–Stegun 7.1.26 fit carries an error near 1e−7, which is visible in the tail probabilities this page exists to report.
- Assumes the underlying variable is normally distributed. By the central limit theorem this is safe for the mean of a large sample even when the raw data is not.
- For a continuous distribution the probability of any exact value is zero; only intervals carry probability.
Sources
- Abramowitz & Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964 — the error and incomplete gamma functions.
- NIST/SEMATECH e-Handbook of Statistical Methods, 2012 — the normal distribution.
Area within k standard deviations
| Range | Area inside | Area outside |
|---|---|---|
| μ ± 0.5σ | 38.29% | 61.71% |
| μ ± 1σ | 68.27% | 31.73% |
| μ ± 1.5σ | 86.64% | 13.36% |
| μ ± 2σ | 95.45% | 4.55% |
| μ ± 2.5σ | 98.76% | 1.24% |
| μ ± 3σ | 99.73% | 0.27% |
Note that ±2σ gives 95.45%, not exactly 95%. The interval that captures precisely 95% is ±1.96σ, which is why that number appears everywhere in statistics.
Frequently asked questions
Why is P(X = exactly 115) zero?
Because the distribution is continuous. Probability is the area under the curve, and the area over a single point has zero width. Ask instead for a narrow interval around it.
What is Φ?
The cumulative distribution function of the standard normal — the area to the left of a given z. Φ(0) is 0.5, because half the curve lies below the mean.
My data is not normal. Can I still use this?
Not for individual values. But if you are working with the mean of a reasonably large sample, the central limit theorem means the sample mean is approximately normal even when the raw data is not.
Why 1.96 rather than 2?
Because ±2σ captures 95.45%, slightly more than 95%. The interval capturing exactly 95% runs to ±1.96σ, and that precision is why the odd-looking number is the standard.