Normal Distribution Calculator
Find the probability below, above, between or outside any values for a normal distribution with your own mean and standard deviation, or go the other way from a probability to the value, with the area shaded on the bell curve.
How to Use
- Enter the mean μ and the standard deviation σ of the distribution (not the variance).
- Choose what to find: the area below a value, above it, between two values, outside them, or the inverse.
- Type the value or values. For the inverse, type a probability (0.05 or 5%) and say where the area is: left, right, middle or both tails.
- Read the probability, the z-scores and the percentage; the bell curve shades the area.
- The table underneath gives the other areas and the 68–95–99.7 ranges for your mean and SD.
Worked Example
Above a value. Birth weights are roughly normal with μ = 3,400 g and σ = 500 g. For 4,000 g, z = (4,000 − 3,400) ÷ 500 = 1.2 and Φ(1.2) = 0.8849, so the share of babies over 4,000 g is 1 − 0.8849 = 0.1151, about 11.5%.
Outside two values. Bolts must be 9.9 to 10.1 mm, and the machine makes them with μ = 10.02 mm and σ = 0.05 mm. The limits are z = (9.9 − 10.02) ÷ 0.05 = −2.4 and z = (10.1 − 10.02) ÷ 0.05 = 1.6. Below the lower limit: Φ(−2.4) = 0.0082; above the upper: 1 − Φ(1.6) = 0.0548. Out of spec: 0.0630, or 6.3%. Centring the machine on 10.00 mm would make both limits z = ±2 and cut the rejects to 4.55%.
The common mistake: using the variance instead of the SD. IQ has σ = 15, so its variance is 225. Dividing by 225 gives z = (115 − 100) ÷ 225 = 0.067 and an area of 0.5266, as if 115 were barely above average. The right z is 15 ÷ 15 = 1, and the area below 115 is 0.8413.
Show Work
Formulas
How the Curve Became Normal
Abraham de Moivre found the curve in 1733 while approximating the probabilities of long runs of coin tosses. Carl Friedrich Gauss derived it again in 1809 as the law of errors in astronomical measurements, which is why it is also called the Gaussian, and Pierre-Simon Laplace’s central limit theorem of 1810 explained why it appears whenever many small independent effects add up.
In the 1840s Adolphe Quetelet fitted it to the chest measurements of Scottish soldiers, and Francis Galton and Karl Pearson later made “normal curve” the usual name. Because every normal distribution is the standard one stretched by σ and shifted by μ, a single printed table of Φ(z) served every problem until calculators replaced it.
About This Calculator
This calculator finds areas under a normal curve with any mean and standard deviation: below a value, above it, between two values or outside them, and the inverse, from an area on the left, on the right, in the middle or split between both tails back to the values. It computes the normal CDF to full double precision, and works out upper tails directly rather than as 1 minus a number close to 1, so areas far out in the tails keep their accuracy. For a single value’s z-score and percentile, use the Z-Score Calculator; for test statistics, the P-Value Calculator.
Everything runs in your browser; nothing is sent anywhere.
Related tools: Z-Score Calculator, P-Value Calculator, and Probability & Statistics Lab.
Frequently Asked Questions
How do I find the probability below a value?
Standardise it, z = (x − μ) ÷ σ, then find the area to the left of z under the standard normal curve, Φ(z). For IQ scores with μ = 100 and σ = 15, an IQ of 115 has z = 1, and Φ(1) = 0.8413, so 84.13% of people score below 115. The area above is 1 − 0.8413 = 0.1587.
How do I find the probability between two values?
Subtract the two areas to the left: P(a < X < b) = Φ(z_b) − Φ(z_a). With μ = 500 and σ = 100, the values 400 and 650 have z = −1 and 1.5, so the probability is 0.9332 − 0.1587 = 0.7745, or 77.45%. The probability outside them is 1 − 0.7745 = 0.2255.
What is the 68–95–99.7 rule?
In any normal distribution, 68.27% of values lie within 1 standard deviation of the mean, 95.45% within 2 and 99.73% within 3. With μ = 500 and σ = 100, that is 400 to 600, 300 to 700 and 200 to 800. It is a quick check on any answer from the calculator.
How do I find the value for a given probability?
Use the inverse normal: find the z with that area, then x = μ + zσ. The top 5% has 0.95 to its left, so z = 1.6449, and for IQ the cut-off is 100 + 1.6449 × 15 = 124.67. The middle 95% has 2.5% in each tail, so z = ±1.96 and the values are μ ± 1.96σ.
Does it matter whether I use < or ≤?
Not for a normal distribution: a single exact value has probability 0, so P(X < 115) and P(X ≤ 115) are both 0.8413. It matters when the normal curve approximates whole-number counts or scores: then P(X ≤ 115) is usually worked out at 115.5 (a continuity correction), giving z = 1.0333 and 0.8493.
How do I use the Normal Distribution Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Quality control
Bolts specified at 9.9 to 10.1 mm from a process with μ = 10.02 mm and σ = 0.05 mm: 6.30% are out of spec.
Health data
Birth weights with μ = 3,400 g and σ = 500 g: 11.51% are over 4,000 g.
Exam scores
Scores with μ = 500 and σ = 100: 77.45% fall between 400 and 650.
Cut-offs
The top 5% of IQ scores (μ 100, σ 15) starts at 124.67.
Tolerance limits
The middle 95% of any normal distribution is μ ± 1.959964σ.
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