Normal Distribution Calculator
Compute cumulative P(X≤x) and right-tail P(X>x) for a normal distribution.
Input Data
Results
At a glance:For a normal distribution, P(X≤x) = Φ((x−μ)/σ), where Φ is the standard normal CDF (area). Standardize the observation to a z-score z=(x−μ)/σ, then look up Φ(z). Right tail P(X>x)=1−Φ(z). Φ has no elementary form; this tool uses an erf approximation.
Formula
z = (x − μ) / σ
P(X ≤ x) = Φ(z)
P(X > x) = 1 − Φ(z)
Interval: P(a < X < b) = Φ(z_b) − Φ(z_a)
$$P(X \le x) = \Phi\!\left(\frac{x-\mu}{\sigma}\right)$$$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$$How to Use
- Enter the observation x, mean μ, and standard deviation σ (σ>0).
- The calculator outputs the z-score z.
- It outputs the cumulative probability P(X ≤ x) and the right-tail probability P(X > x).
- Example: x=85, μ=75, σ=10 → z=1, P(X≤85)≈0.8413.
Standard normal Φ(z) quick reference
| z | Φ(z) | Right tail 1−Φ(z) |
|---|---|---|
| 0 | 0.5000 | 0.5000 |
| 1 | 0.8413 | 0.1587 |
| 1.645 | 0.9500 | 0.0500 |
| 1.96 | 0.9750 | 0.0250 |
| 2 | 0.9772 | 0.0228 |
| 2.576 | 0.9950 | 0.0050 |
Φ(−z) = 1 − Φ(z) by symmetry.
Case Studies
x=85, μ=75, σ=10
z = (85−75)/10 = 1.
P(X≤85) = Φ(1) ≈ 0.8413; right tail P(X>85) ≈ 0.1587.
Grade threshold x=90
μ=75, σ=10 → z=1.5.
P(X≤90)=Φ(1.5)≈0.9332, i.e. ~93.3% score below 90.
Interval μ±σ ≈ 68%
P(μ−σ<X<μ+σ)=Φ(1)−Φ(−1)=0.8413−0.1587=0.6826.
Process yield 70–80
μ=75, σ=2 → z_low=−2.5, z_high=2.5.
P = Φ(2.5)−Φ(−2.5) ≈ 0.9876−0.0062 = 0.9814 (98.14% yield).
Z-table z=1.96
Φ(1.96) ≈ 0.9750; right tail 0.0250.
Corresponds to the two-sided 95% confidence critical value.
68–95–99.7 check
z=2 → Φ(2)≈0.9772, right tail 0.0228.
μ±2σ contains ~95.44%, matching the rule.
FAQ
What is the 68–95–99.7 rule?
A normal distribution has about 68% of values within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ. You can verify this with the cumulative probability from this calculator.
Is the approximation accurate enough?
It uses the standard erf approximation (Abramowitz & Stegun 7.1.26) with a maximum error around 1.5×10⁻⁷, sufficient for general statistics and teaching.
What is a z-score?
The z-score z=(x−μ)/σ standardizes an observation, showing how many standard deviations it is from the mean; positive is right, negative is left.
Why convert to z?
Any normal can be turned into the standard normal N(0,1), so one Φ(z) table (or this calculator) gives probabilities for every normal.
Right tail vs left tail?
P(X>x)=1−Φ(z) is the right tail (probability above x); P(X≤x)=Φ(z) is the left cumulative. They are complementary.
How to get an interval P(a<X<b)?
Standardize a, b to z_a, z_b; interval probability = Φ(z_b) − Φ(z_a). This tool gives the point cumulative; subtract to get the interval.
When to use a normal distribution?
Many natural and social phenomena (height, error, measurement) are roughly normal; the central limit theorem also makes sample means approach normal, so it is widely used in inference.
What if σ is very small or large?
Smaller σ gives a sharper peak (more concentrated); larger σ gives a flatter, more spread-out curve. σ must be greater than 0.
References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.