Education
Normal Distribution Calculator
Last updated: June 19, 2026
A normal distribution calculator is a statistical tool that computes probabilities and percentiles for a normally distributed dataset. Using the population mean and standard deviation, the calculator determines the probability of a random variable falling within a specific range, above a threshold, or below a threshold by calculating the corresponding z-scores. It can also perform the inverse operation, finding the raw data value that matches a target percentile. Students, statisticians, and data analysts use this utility to evaluate standard distributions and test hypothesis models.
Pick raw-score or z-score mode and a probability direction (left, right, or between two values). Enter the inputs and the calculator returns the probability under the standard normal curve as a decimal and a percent.
Quick Answer
Calculate normal distribution probabilities and percentiles. Enter your mean, standard deviation, and range bounds to find areas under the bell curve.
Probability
e.g. 110
e.g. 100
Must be greater than 0. · e.g. 15
Uses the Abramowitz & Stegun 7.1.26 approximation of the standard normal CDF. Accurate to about 7 decimal places across the practical z range.
P(X ≤ x) · z = 0.667
0.747508
74.7508% area under the curve
Examples
Left tail at x = 110, μ = 100, σ = 15
z = 0.667, P ≈ 0.747508 (74.75%)
Right tail at z = 1.96
P(Z ≥ 1.96) ≈ 0.025002 (2.50%)
Between z = −1 and z = +1
P ≈ 0.682689 (68.27%)
How it works
Convert any raw score to a z-score using z = (x − μ) ÷ σ. The standard normal CDF Φ(z) is the cumulative area under the standard normal curve to the left of z. The calculator uses a closed-form polynomial approximation, then derives the requested probability:
Left tail · Φ(z)
Right tail · 1 − Φ(z)
Between · Φ(z₂) − Φ(z₁)
68.27% within ±1σ · 95.45% within ±2σ · 99.73% within ±3σ. Common critical values: ±1.645 (90%) · ±1.96 (95%) · ±2.576 (99%).
Related probability and stats calculators
- Z-score calculator for converting a raw value to a z-score without computing a probability.
- P-value calculator for hypothesis-test framing of the same tail probabilities.
- Confidence interval calculator uses z-critical values from this distribution to build the interval bounds.
- Margin of error calculator uses the same z-criticals for sample-proportion margins.
- Standard deviation calculator for computing σ from a list of values.
- Binomial probability calculator for the discrete counterpart on a fixed number of trials.
- All education calculators.
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Frequently asked questions
The normal distribution is a symmetric, bell-shaped distribution defined by two parameters: mean μ and standard deviation σ. It is the most-used distribution in statistics because many natural processes are approximately normal, and because the Central Limit Theorem makes sums and averages of many independent random variables tend toward a normal distribution. The standard normal has μ = 0 and σ = 1.
The calculator first converts the raw score x to a z-score using z = (x − μ) ÷ σ. Then it computes Φ(z), the standard normal CDF, using the Abramowitz and Stegun 7.1.26 polynomial approximation, which is accurate to about 7 decimal places across the practical z range. Left-tail probability is Φ(z); right-tail is 1 − Φ(z); between two z-scores is Φ(z₂) − Φ(z₁).
Use raw score mode when you have the actual values from your data, plus the mean and standard deviation of the underlying distribution. Use z-score mode when you already have the z-score (a standardized value) and just want the corresponding probability. The math is the same; the modes just save you the conversion step in one direction or the other.
Between mode computes P(a ≤ X ≤ b), the area under the curve strictly between two values. It is the difference between two CDFs: Φ(z₂) − Φ(z₁). For example, the probability that a value falls within 1 standard deviation of the mean (between z = −1 and z = +1) is about 0.6827, the well-known 68 percent rule for the normal distribution.
Φ(0) = 0.5 (the mean). About 68.27 percent of the area lies within z = ±1, 95.45 percent within z = ±2, and 99.73 percent within z = ±3. Common critical values: z = ±1.645 (90 percent two-sided), ±1.96 (95 percent), ±2.576 (99 percent). The calculator returns these directly when those z-scores are entered.
The result is exact to about 7 significant figures. The Abramowitz and Stegun approximation is the same one used by the p-value calculator on this site and by most introductory statistics software for closed-form CDF approximations. For research-grade precision (10+ decimals), use a statistical package or library that integrates the density directly.
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