Normal Distribution Calculator
Probabilities under the bell curve.
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.
Enter the second value (optional) 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)
- 0.908789
- P(X > x)
- 0.091211
- Z score
- 1.33333
- Percentile
- 90.88%
- P(x < X < x₂)
- 0.068461
Quick answer
The normal distribution converts any value to a z-score, then reads the area under the standard bell curve.
z = (x − μ) / σ
At a glance
| What it does | Probabilities under the bell curve. |
|---|---|
| Category | Statistics |
| Inputs needed | Mean μ, Standard deviation σ, Value x, Second value (optional) |
| Main output | P(X < x) |
| Formula | z = (x − μ) / σ |
| Cost | Free — no sign-up, no download |
How to use the Normal Distribution Calculator
- 1Enter the population mean and standard deviation.
- 2Enter the value you are testing.
- 3Add a second value for a range 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.
- Second value (optional)
- Enter the second value (optional) you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Mean μ100
- Standard deviation σ15
- Value x120
- Second value (optional)130
Results
- P(X < x)0.908789
- P(X > x)0.091211
- Z score1.33333
- Percentile90.88%
- P(x < X < x₂)0.068461
Frequently asked questions
What is the 68-95-99.7 rule?
About 68% of data lies within 1 SD, 95% within 2 and 99.7% within 3.
Is my data normal?
Check a histogram or QQ plot; heavily skewed data needs different tools.
What do I need to enter into the Normal Distribution Calculator?
Just 4 values: mean μ, standard deviation σ, value x, second value (optional). 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 Calculator work out the answer?
The normal distribution converts any value to a z-score, then reads the area under the standard bell curve. It applies the formula z = (x − μ) / σ and shows the working so you can check each step by hand.
Is the Normal Distribution 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.