Statistics
Normal Distribution Calculator
Enter a mean and standard deviation, then find the probability below, above, between, or outside any values, or flip it around and find the value that matches a probability. mathcheck shows the z-scores, percentiles, a shaded bell curve, and the table-style work.
The center of the distribution.
Must be positive. Use σ = 1 with μ = 0 for the standard normal.
Try an example
Result
Result
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- z-score
- —
- Percentile
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- Complement
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Grade planning, algebra checks, and formulas students reach for most.
Study path
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What the normal distribution calculator solves
The normal (Gaussian) distribution is the bell-shaped curve that describes heights, test scores, measurement error, and many averages. This calculator converts a value x into a How many standard deviations a value sits above or below the mean., looks up the standard normal CDF Φ(z), and reports the probability for any region. In inverse mode it starts from a probability and returns the cutoff value, which is what The percent of values in a data set that fall at or below a given value. and 'top 10%' questions need.
Formulas
| Mode | Use it when | Output |
|---|---|---|
| P(X < x) | You want the fraction below a cutoff | Left-tail probability, percentile |
| P(X > x) | You want the fraction above a cutoff | Right-tail probability |
| P(a < X < b) | You want the fraction inside a range | Central probability and both z-scores |
| P(X < a or X > b) | You want the two-tail probability | Outside probability |
| Inverse | You know the percent and need the value | x for a lower tail, upper tail, or symmetric middle |
How to use it
- Enter the mean μ and A number saying how far, on average, the data values sit from the mean. σ. For the standard normal use 0 and 1.
- Pick a probability type and enter the cutoff value(s) on the original scale, not as z-scores.
- For an inverse question, switch to 'Value from a probability', choose which tail is known, and enter p as a decimal.
- Read the probability (or the solved x), the z-scores, and the percentile; open 'Show the work' for the table-style steps.
How to read the answer
A probability of 0.8413 means 84.13% of the Every member of the group you actually care about, not just the ones measured. falls in the shaded region. The z-score tells you how many standard deviations x sits from the mean; z = ±1.96 marks the middle 95%. The percentile is always the left-tail probability, so the 84th percentile is the value with 84% of the data below it.
Common mistakes and edge cases
- Entering a z-score in the x field. If you already have z, set μ = 0 and σ = 1.
- Reading P(X > x) from a left-tail table without subtracting from 1.
- Using a probability above 1 or a percent such as 95 in inverse mode; enter 0.95.
- A single point has probability 0 for a continuous distribution, so P(X = x) is not a meaningful question — use a range.
- Values beyond ±6σ give tail probabilities near 1e-9 or smaller; the calculator keeps full precision where printed tables stop.
Worked examples
Between two values
IQ scores with μ = 100, σ = 15: P(85 < X < 115)
Result
0.6827 (68.27%)
Above a cutoff
P(X > 130) for the same IQ scale
Result
0.02275 (2.28%)
Below the mean
Any value below μ is the 50th percentile: P(X < 100)
Result
0.5 (50.00%)
Negative z-score
Heights with μ = 170 cm, σ = 8 cm: P(X < 160)
Result
0.1056 (10.56%)
Inverse: top 10% cutoff
Find the score with P(X > x) = 0.10 on the IQ scale
Result
119.2233
Inverse: middle 95%
Bounds that hold the central 95% of a standard normal
Result
-1.96 to 1.96
Impossible: zero standard deviation
σ must be positive
Result
Error
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, look up Φ(z) for each, and subtract: P(a < X < b) = Φ(z_b) - Φ(z_a). The calculator does the lookups with full precision and shades the region on the bell curve.
What is the difference between a z-score and a probability?+
A z-score is a position: how many standard deviations a value sits from the mean. A probability is an area under the curve. The standard normal CDF Φ turns a z-score into the left-tail area, and its inverse turns an area back into a z-score.
How do I find x when I know the percentile?+
Use inverse mode. For the 90th percentile enter p = 0.90 with 'P(X < x) = p'. The calculator finds z = Φ⁻¹(0.90) = 1.2816 and returns x = μ + 1.2816σ.
Why is P(X = x) zero?+
The normal distribution is continuous, so probability is measured as area. A single point has no width, so its area is 0. Ask for a range such as P(99.5 < X < 100.5) instead.
Is a normal distribution the same as a standard normal distribution?+
A standard normal has μ = 0 and σ = 1. Every other normal distribution can be converted to it with the z-score formula, which is why one table (or Φ) works for all of them.
When should I use the t distribution instead?+
Use the t distribution when you estimate the standard deviation from a small sample and are building a confidence interval or t test. For probabilities about individual values from a known μ and σ, the normal distribution is the right tool.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026