How to Use the Normal Distribution Calculator
The normal distribution (bell curve) is the most widely used probability distribution in statistics. It appears naturally in many phenomena such as IQ scores, heights, test results, and measurement errors. This calculator computes the probability and Z-score for any value in a normal distribution defined by its mean and standard deviation.
Enter the mean (μ) and standard deviation (σ), then provide the value of interest x₁. The calculator returns the Z-score, the cumulative probability P(X≤x₁), and the upper tail probability P(X>x₁). To find the probability within a range, enter x₂ as well — the range probability P(x₁≤X≤x₂) will appear automatically.
The empirical rule states that in a normal distribution, about 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. Z-scores standardize values across different distributions, making them useful for comparing scores from different tests or datasets.
Frequently Asked Questions
A Z-score indicates how many standard deviations a value is above (positive) or below (negative) the mean. Z = (x − μ) / σ. A Z-score of +1.5 means the value is 1.5 standard deviations above average.
No. Standard deviation must be a positive number greater than zero. It represents the spread of data, and spread cannot be negative.
This calculator uses the Abramowitz and Stegun polynomial approximation for the normal CDF, accurate to about 7 decimal places — suitable for nearly all practical purposes.