Normal Distribution Calculator: Probabilities Below, Above, Between and Percentiles
Enter the mean and standard deviation, then choose below, above, between or percentile. The shaded curve shows the area you asked for.
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P(x₁ < X < x₂)
0.682689
- As a percentage
- 68.2689%
- z-score
- −1
Rule of thumb: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD.
Show working
- z₁ = (x₁ − μ) / σ = (85 − 100) / 15 = −1
- z₂ = (115 − 100) / 15 = 1
- P(85 < X < 115) = Φ(1) − Φ(−1) = 0.841345 − 0.158655 = 0.682689
How to use
- Enter the mean μ and standard deviation σ.
- Choose below, above, between, or percentile → value.
- Enter the value(s) or the percentile and read the probability.
Good to know
Each value is converted to z = (x − μ)/σ and the area is read from the standard normal CDF Φ(z), computed to near double precision. Between two values: Φ(z₂) − Φ(z₁).
Percentile → value uses the inverse normal: x = μ + σ·Φ⁻¹(p). For IQ-style scores (mean 100, SD 15) the 95th percentile is about 124.7.
The 68–95–99.7 rule: about 68.27% of values lie within 1 SD of the mean, 95.45% within 2 SD and 99.73% within 3 SD. Real data is only approximately normal, so treat results as estimates.
Questions
What share of values lies within one SD of the mean?
About 68.27%. With mean 100 and SD 15 that is between 85 and 115.
How do I find the top 5% cut-off?
Choose “percentile → value” and enter 95%. The answer is μ + 1.645σ.
What is the difference from the z-score calculator?
Here you work directly in your own units (marks, cm, ₹) with any mean and SD; the z-score page focuses on z itself.
Does it handle very small probabilities?
Yes. Tail areas are accurate far into the tails, e.g. P(Z > 5) ≈ 2.87 × 10⁻⁷.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.