Left-tail probability
Calculates the area below a selected x-value or z-score.
Advanced Math and Statistics
Normal Distribution Calculator
Calculate left-tail, right-tail, interval, outside, z-score, percentile, and inverse normal values with a shaded bell curve and complete summary statistics.
- Probability and percentile calculations
- Z-scores and inverse normal values
- Interactive shaded bell curve
Normal distribution inputs
Enter the mean, standard deviation, and the probability or inverse-normal question.
Distribution parameters
Calculation values
%
%
Quick examples
Advanced Math and Statistics
Analyze probabilities under a normal bell curve
A normal distribution is a symmetric continuous probability distribution centered at its mean. Its standard deviation controls the spread, while z-scores express how far a value lies from the mean in standard-deviation units.
Right-tail probability
Finds the probability of observing a value at or above a selected point.
Interval and outside areas
Measures probability between two values or in both outside tails.
Z-score conversion
Converts raw values into standardized distances from the mean.
Percentile calculation
Finds the percentage of observations expected below a selected x-value.
Inverse normal value
Converts a percentile or tail probability into its corresponding z-score and raw value.
Core formulas
Normal density, standard score, and variance
Normal density
The probability density at a continuous value x.
f(x) = [1 ÷ (σ√2π)]e−(x−μ)²/(2σ²)
Z-score
Standardizes a raw value relative to the distribution.
z = (x − μ) ÷ σ
Variance
The square of the normal distribution's standard deviation.
Var(X) = σ²
How it works
Calculate normal probabilities in four steps
1
Enter mean and spread
Add the distribution mean and a positive standard deviation.
2
Select the calculation
Choose a probability, z-score, percentile, inverse value, or central interval.
3
Enter x or probability
Provide one value, two bounds, or the requested percentile.
4
Review the shaded curve
See the numerical result, z-scores, density, complement, percentile rank, and shaded bell curve.
Frequently asked questions
Normal distribution calculator FAQs
A normal distribution is a continuous, symmetric, bell-shaped probability distribution. Its mean, median, and mode are equal, and the distribution is determined by its mean and standard deviation.
A z-score tells you how many standard deviations a value lies above or below the mean. Positive z-scores are above the mean, negative z-scores are below it, and zero is exactly at the mean.
The x-value is converted to a z-score, and the standard normal cumulative distribution function is used to calculate the area under the curve. Interval probabilities are differences between cumulative areas.
An inverse normal calculation starts with a probability or percentile and returns the z-score and raw x-value whose cumulative probability matches that input.
For a normal distribution, about 68% of observations lie within one standard deviation of the mean, about 95% lie within two, and about 99.7% lie within three.
No. Strongly skewed, multimodal, bounded, discrete, or heavy-tailed data may not follow a normal distribution. Always inspect the data and context before using a normal model.