Normal Distribution Calculator
Probability under the bell curve, without a z-table.
Normal Distribution Calculator
Probability between two values, z-scores, and the 68% and 95% ranges for a normal distribution.
Uses the normal CDF with a maximum error below 1.5 × 10⁻⁷. Probabilities are two-sided within the bounds you enter.
P(x₁ < X < x₂)
68.27%
- z-scores
- -1 → 1
- P(X < x₁)
- 15.87%
- P(X < x₂)
- 84.13%
- Middle 68%
- 85.08 – 114.92
- Middle 95%
- 70.6 – 129.4
Density at the lower bound: 0.01613. The normal model assumes a symmetric, bell-shaped distribution — check a histogram before relying on it.
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What This Calculator Does
The Normal Distribution Calculator returns the probability that a normally distributed value falls between two bounds. It also reports the matching z-scores, the cumulative probabilities, and the middle 68% and 95% ranges for your mean and standard deviation.
How It Works
A normal distribution is defined by its mean and standard deviation. Converting a value to a z-score — (x − μ) / σ — places it on the standard normal curve. The area under the curve between two z-scores is the probability of landing in that range. This calculator evaluates the normal CDF numerically, with an error below 1.5 × 10⁻⁷.
Example
For a mean of 100 and a standard deviation of 15, the probability of a value between 85 and 115 is about 68.27%, because those bounds sit one standard deviation either side of the mean. The middle 95% range runs from about 70.6 to 129.4.
Formula
z = (x − μ) / σP(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)Φ is the standard normal cumulative distribution function.
≈68% within 1σ, ≈95% within 2σ, ≈99.7% within 3σReading the Results
Probability between bounds
The area under the normal curve between your two values, expressed as a percentage of all observations.
Middle 95% range
The central interval that contains 95% of values for this mean and standard deviation — the same 1.96σ logic used in confidence intervals.
Sources and References
NIST/SEMATECH e-Handbook of Statistical Methods
Normal distribution and standard normal table
Section 1.3.6.6 documents the normal PDF, CDF, and the z transformation.
Abramowitz & Stegun, Handbook of Mathematical Functions
Error-function approximation 7.1.26
The approximation used here has a maximum absolute error of 1.5 × 10⁻⁷.
References identify the method or data behind this calculator. Where a standard or tariff changes, the current edition applies, and the assumptions stated above take precedence over a remembered figure.
Frequently Asked Questions
What is the 68–95–99.7 rule?
In a normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. It is a quick sanity check before using the exact calculation.
Can I use this if my data are not normal?
The arithmetic still runs, but the probabilities will be wrong. Check a histogram or a normality test first. Income, house prices, and wait times are typically skewed and need a different model or a transformation.
What is the difference between this and a z-score calculator?
A z-score calculator standardises one value. This calculator integrates the curve between two values to give a probability, and also reports the central 68% and 95% ranges.
Why is the middle 95% range not exactly mean ± 2σ?
Two standard deviations capture about 95.4%. The exact central 95% interval uses z = ±1.96, which is slightly narrower — the same critical value used in a 95% confidence interval.