Normal Distribution Calculator
Find the probability below, above, between or outside values for any mean and SD, or the inverse normal cut-off for a percentile, on a shaded bell curve.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Normal Distribution Calculator?
Enter the mean and standard deviation of a normal distribution and ask a question in your own units: what share of values fall below 85, above 130, between 85 and 115, or outside a tolerance band. The Normal Distribution Calculator returns the probability, the z-scores it used and the complementary areas, and shades the region on a bell curve labelled at the mean and at 1, 2 and 3 standard deviations.
Switch to "Values from probability" to work backwards. Give a probability and the calculator finds the cut-off: the value with 90% of the distribution below it, the score needed to be in the top 5%, or the symmetric limits that hold the middle 95%. This is the inverse normal, the same calculation as Excel's NORM.INV or a TI-84's invNorm, and it is what you need for percentile targets, grading cut-offs and specification limits.
- Probability below, above, between or outside any values
- Inverse normal for left, right, central and two-tailed areas
- Works in your own units with any mean and standard deviation
- Shaded bell curve labelled at μ and ±1, 2, 3σ
- Accurate tiny tail probabilities down to about 1e-300
- Runs in your browser; shareable links
How to use the Normal Distribution Calculator
- Enter the mean (μ) and the standard deviation (σ) of the distribution.
- For a probability, choose P(X < a), P(X > a), between a and b, or outside a and b, and enter the value or values.
- For a cut-off, switch to "Values from probability", enter the probability as a decimal (0.9 for 90%) and choose whether it is the area to the left, to the right, in the centre or in the two tails.
- Read the main result, the z-scores and the complementary areas, and check the shaded curve.
- Copy the result or share a link with the same settings.
Worked examples
IQ between 85 and 115 (the default)
Input
μ = 100, σ = 15, between 85 and 115
Output
P = 0.68268949 (68.27%); 15.87% below 85 and 15.87% above 115
Both bounds are one standard deviation from the mean (z = ±1), so the answer is the 68% of the empirical rule.
Share above a threshold
Input
μ = 100, σ = 15, P(X > 130)
Output
P = 0.0227501 (about 1 in 44)
130 is z = 2. The right tail beyond 2 standard deviations is 2.275%, half of the 4.55% outside ±2.
A percentile cut-off
Input
μ = 500, σ = 100, left area 0.9
Output
x = 628.155 (z = 1.28155)
A score of about 628 marks the 90th percentile. The same answer comes from a right-tail area of 0.1.
Middle 95% of a distribution
Input
μ = 100, σ = 15, central 0.95
Output
70.6005 to 129.3995 (z = ±1.959964)
This is μ ± 1.96σ, the band that holds 95% of values and the basis of the 95% confidence multiplier.
Background and insight
Abraham de Moivre found the bell curve in 1733 as an approximation to coin-toss probabilities, and Gauss and Laplace developed it as the law of measurement errors around 1809 and 1810. The central limit theorem explains why it appears so often: sums and averages of many small independent effects tend towards a normal shape whatever the shape of each effect. Heights, measurement errors and averages of samples are close to normal; incomes, waiting times and house prices are not.
Every normal distribution is the standard normal stretched and shifted. A value x becomes z = (x - μ) / σ, and the probability below x is Φ(z), the area under the standard curve to the left of z. Φ has no closed form, so tables list it to four decimals. This calculator computes it with Marsaglia's series and a continued fraction in the tails, which keeps tiny areas accurate: the chance of exceeding 8 standard deviations is about 6.2e-16 and is reported as such rather than as 0.
The empirical rule summarises the shape: about 68.27% of values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three. On the IQ scale (mean 100, SD 15), that puts 68% of people between 85 and 115 and about 2.3% above 130. The inverse runs the other way using Acklam's rational approximation refined by a Halley step, so the familiar 1.959964 for the central 95% comes out to full precision.
Common use cases
Exam and test scores
Turn a mean and SD into the share of students above a pass mark, or find the score for the top 10%.
Manufacturing tolerances
Estimate the fraction of parts outside specification limits from the process mean and standard deviation.
Health reference ranges
Find the share of a population above or below a cut-off when a measurement is close to normal.
Statistics coursework
Replace z-table lookups and check each step of a normal probability problem.
Tips and limitations
- Check the data really are close to normal before trusting tail probabilities. Skewed data such as incomes give badly wrong answers in the tails.
- For a continuous normal distribution, P(X < a) and P(X ≤ a) are the same. When approximating whole-number counts, a continuity correction of ±0.5 improves accuracy.
- Use the population standard deviation of the distribution, not the standard error of a mean, unless you are asking about sample means.
- Enter probabilities as decimals: 0.95, not 95.
- The curve is drawn from -4 to +4 standard deviations; values further out are still calculated exactly but sit at the edge of the plot.
Frequently asked questions
How do I find the probability between two values in a normal distribution?
Convert both values to z-scores with z = (x - μ) / σ, then subtract: P = Φ(z_upper) - Φ(z_lower). Choose "Between a and b" and the calculator does both steps and shades the area.
What is the inverse normal distribution?
It answers the reverse question: given a probability, which value cuts it off? For example, with μ = 500 and σ = 100, the value with 90% below it is 628.16. It is the same as Excel's NORM.INV and the TI-84's invNorm.
What is the 68-95-99.7 rule?
In a normal distribution about 68.27% of values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three. It is a quick check, and the calculator gives the exact areas for any bounds.
How is this different from a z-score calculator?
A z-score calculator works in standard units. This one takes the mean and standard deviation of your own data and answers in those units, handles between and outside regions directly, and finds cut-off values from probabilities.
Can I use this for data that are not normal?
Only as a rough guide near the centre. Tail probabilities in particular can be far off for skewed or heavy-tailed data. For sample means, the central limit theorem often makes the normal model reasonable even when the raw data are not.
What does P(X > a) mean on the curve?
It is the area under the curve to the right of a, which equals the share of the population with values greater than a. The total area under the curve is 1.
Is there an in-depth guide to the Normal Distribution Calculator?
Yes. “Normal Distribution Probabilities and the Inverse Normal” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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