Normal Distribution Probability Calculator
P(X<x), P(X>x) and P(a<X<b) for a normal distribution.
Enter your values
Enter the mean μ you are working with.
Enter the standard deviation σ you are working with.
Enter the value x you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- P(X < x)
- 90.879%
- P(X > x)
- 9.121%
- z-score
- 1.3333
Quick answer
Probabilities under a normal distribution are found by converting to a z-score and applying the cumulative normal distribution function.
z = (x-μ)/σ, P(X<x) = Φ(z)
At a glance
| What it does | P(X<x), P(X>x) and P(a<X<b) for a normal distribution. |
|---|---|
| Category | Statistics |
| Inputs needed | Mean μ, Standard deviation σ, Value x |
| Main output | P(X < x) |
| Formula | z = (x-μ)/σ, P(X<x) = Φ(z) |
| Cost | Free — no sign-up, no download |
How to use the Normal Distribution Probability Calculator
- 1Enter the population mean and standard deviation.
- 2Enter the value x you want a probability for.
- 3The value is converted to a z-score and mapped to a cumulative probability.
Inputs explained
- Mean μ
- Enter the mean μ you are working with.
- Standard deviation σ
- Enter the standard deviation σ you are working with.
- Value x
- Enter the value x you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Mean μ100
- Standard deviation σ15
- Value x120
Results
- P(X < x)90.879%
- P(X > x)9.121%
- z-score1.3333
Frequently asked questions
What is P(X<x)?
The probability that a randomly drawn value from this distribution is less than x.
How do I find P(a<X<b)?
Compute P(X<b) and P(X<a) separately, then subtract.
What do I need to enter into the Normal Distribution Probability Calculator?
Just 3 values: mean μ, standard deviation σ, value x. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Normal Distribution Probability Calculator work out the answer?
Probabilities under a normal distribution are found by converting to a z-score and applying the cumulative normal distribution function. It applies the formula z = (x-μ)/σ, P(X<x) = Φ(z) and shows the working so you can check each step by hand.
Is the Normal Distribution Probability Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.