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Normal Distribution Calculator
Exam marks, heights, blood pressure, machine-made parts — a huge amount of real-world data follows the bell-shaped normal distribution, where values bunch around the mean and thin out on both sides. This normal distribution calculator turns any value into its z-score and tells you the probability of landing below it, above it, or between two values — the three questions every statistics student and data analyst asks of a bell curve.
How to use this tool
- Enter the mean (the average) and the standard deviation of your data — for exam marks out of 100, that might be 60 and 10.
- Type the value x you are asking about, and a second value x₂ if you want the "between" probability.
- Pick which probability you want in the list: below x, above x, or between the two values.
- The big number is that probability as a percentage. The line below always shows the z-score and all three probabilities.
Standard normal table (z-table) — main values
| z | P(Z < z) |
|---|---|
| 0.0 | 0.5000 |
| 0.1 | 0.5398 |
| 0.2 | 0.5793 |
| 0.3 | 0.6179 |
| 0.4 | 0.6554 |
| 0.5 | 0.6915 |
| 0.6 | 0.7257 |
| 0.7 | 0.7580 |
| 0.8 | 0.7881 |
| 0.9 | 0.8159 |
| 1.0 | 0.8413 |
| 1.1 | 0.8643 |
| 1.2 | 0.8849 |
| 1.3 | 0.9032 |
| 1.4 | 0.9192 |
| 1.5 | 0.9332 |
| 1.6 | 0.9452 |
| 1.7 | 0.9554 |
| 1.8 | 0.9641 |
| 1.9 | 0.9713 |
| 2.0 | 0.9772 |
| 2.1 | 0.9821 |
| 2.2 | 0.9861 |
| 2.3 | 0.9893 |
| 2.4 | 0.9918 |
| 2.5 | 0.9938 |
| 2.6 | 0.9953 |
| 2.7 | 0.9965 |
| 2.8 | 0.9974 |
| 2.9 | 0.9981 |
| 3.0 | 0.9987 |
The table gives P(Z < z) for the standard normal curve (mean 0, SD 1). Read across the examples on this page: z = 1.5 gives 0.9332, and z = 1.96 gives 0.9750 — the famous 1.96 boundary of the 95% confidence interval. This calculator uses a high-accuracy formula for the same curve, accurate to better than one part in a million, so it also handles negative z-scores and values between the table rows.
How it works
A normal distribution is described by just two numbers: the mean μ, which is the centre of the bell, and the standard deviation σ (SD), which says how spread out the values are. About 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD — the famous 68–95–99.7 rule. Heights of adults, measurement errors, and test scores (when a class is large enough) all follow this pattern closely enough for the maths to be useful.
The z-score is the key that unlocks the standard tables. It asks: how many standard deviations is my value from the mean? A z of 1.5 means one and a half SDs above the mean. Once every value is converted to a z-score, one single curve — the standard normal, with mean 0 and SD 1 — answers every probability question. The area under that curve to the left of z is the probability P(Z < z), written Φ(z). This calculator evaluates Φ with a high-accuracy rational formula for the error function, good to better than 0.000001.
The three probability types cover almost every question. "Below" is a percentile: P(X < 75) = 93.32% means a score of 75 beats about 93% of the class. "Above" is its mirror image, used for cut-offs like "the top 5%". "Between" answers range questions — the chance of scoring between 50 and 75 — and equals Φ at the higher z minus Φ at the lower z.
Worked example
A class test has a mean mark of 60 and a standard deviation of 10. What is the chance that a student scores below 75?
Step 1: Find the z-score — z = (75 − 60) ÷ 10 = 1.5.
Step 2: Look up Φ(1.5) on the standard curve — 0.9332.
Step 3: Read the result — P(X < 75) = 93.32%, so about 93 students in 100 score below 75; only 6.68% score above it.
Step 4: For a range, the chance of scoring between 50 and 75 is Φ(1.5) − Φ(−1) = 0.9332 − 0.1587 = 77.45%.
Those are exactly the numbers this calculator shows for a mean of 60, an SD of 10, and x = 75.
FAQ
What is a normal distribution?
A normal distribution is the bell-shaped pattern that many natural and human measurements follow: most values cluster near the mean, and values get steadily rarer further away. It is fully described by the mean (the centre) and the standard deviation (the spread). About 68% of values lie within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.
What is a z-score?
A z-score says how many standard deviations a value is from the mean: z = (x − mean) ÷ SD. A z of 1.5 means the value is 1.5 SDs above the mean; a z of −1 means 1 SD below it. Converting to z-scores lets one standard table or calculator answer questions about any normal data.
What is the probability of scoring below 75 when the mean is 60 and SD is 10?
The z-score is (75 − 60) ÷ 10 = 1.5, and Φ(1.5) = 0.9332, so the probability is 93.32%. In a class of 100 students, about 93 would score below 75 and about 7 above it.
Why does everyone use 1.96 for 95%?
Because Φ(1.96) = 0.97500. That means 97.5% of values fall below z = 1.96, and by symmetry 97.5% fall above z = −1.96 — leaving exactly 95% of the distribution between −1.96 and +1.96. That is why 1.96 is the boundary of the standard 95% confidence interval.
Can a probability from this calculator be negative or over 100%?
No. Probabilities from a normal distribution always sit between 0% and 100%. A value far below the mean gives a below-probability near 0%, and a value far above gives one near 100%. If you see exactly 0.00% or 100.00%, your x is simply many standard deviations from the mean.
What if my standard deviation is 0?
An SD of 0 would mean every value in your data is identical — there is no spread and no bell curve, so probabilities are meaningless. The calculator asks for an SD above zero. If your data really has no variation, every probability is either 0% or 100% depending on the mean.
Note: probabilities use a high-accuracy formula for the normal curve (error under 0.000001). Real data is only approximately normal — treat close results as estimates.