Normal Distribution Calculator

What this calculator does

The normal distribution, sometimes called the Gaussian distribution or the bell curve, describes how many real-world quantities are spread around an average value: test scores, heights, manufacturing measurements, and repeated physical measurements all tend to follow it closely. This calculator finds probabilities, z-scores, percentiles, and raw scores for a normal distribution defined by any mean and standard deviation you choose, including the standard normal distribution (mean 0, standard deviation 1).

Eight calculation modes are supported: the probability below a value, above a value, between two values, outside two values, a two-tailed probability, converting a raw score to a z-score, converting a raw score to a percentile, and the inverse normal calculation that works backward from a probability or percentile to a raw score.

Key concepts

Formulas used

The calculator standardizes every value before finding a probability, then reads the result from the standard normal cumulative distribution function.

• z = (x − mean) / standard deviation
• x = mean + (z × standard deviation)
• P(X ≤ x) = standard normal CDF of z
• P(X ≥ x) = 1 − standard normal CDF of z
• P(a ≤ X ≤ b) = standard normal CDF of upper z − standard normal CDF of lower z

Worked examples

Verifying a result with a Z table

To check a result by hand: first calculate the z-score using z = (x − mean) / standard deviation, then find the cumulative probability for that z in a standard normal (Z) table. For a probability above a value, subtract the table value from 1. For a probability between two values, look up both z-scores and subtract the smaller cumulative probability from the larger one; the same subtraction, applied in reverse, gives the outside-values and two-tailed probabilities.

The empirical rule

The empirical rule, also called the 68-95-99.7 rule, is a quick sanity check for any normal distribution:

• About 68% of values fall within 1 standard deviation of the mean.
• About 95% of values fall within 2 standard deviations of the mean.
• About 99.7% of values fall within 3 standard deviations of the mean.

Common input errors

• Entering the variance instead of the standard deviation. The standard deviation field expects standard deviation, not its square.
• Entering a negative or zero standard deviation. A standard deviation must be strictly greater than zero for the distribution to be defined.
• Confusing a raw score with a z-score. A raw score is on the scale of the original data; a z-score is always in standard-deviation units.
• Entering a percentage instead of a decimal probability in the inverse normal mode when the probability option is selected. Use a value strictly between 0 and 1, or switch to the percentile option and enter a number between 0 and 100.
• Reversing the lower and upper bounds in the between-values or outside-values modes. The calculator sorts the two bounds automatically, but entering them the right way round keeps the inputs easy to read back.

Model assumptions and limitations

This calculator assumes the underlying data genuinely follows a normal distribution. Before treating a calculated probability as a real-world estimate, check that the normal model is a reasonable fit for the data in question — for example, that the distribution is roughly symmetric and bell-shaped rather than skewed, and that the sample is large enough to support the mean and standard deviation used. For counts of successes in a fixed number of trials or events per interval, a binomial or Poisson distribution may be more appropriate than a normal approximation.

This calculator provides a mathematical result based on the normal distribution model and the inputs supplied. It is an informational and educational tool, not statistical, medical, financial, or other professional advice; for decisions where the outcome materially matters, confirm the model fit and consult a qualified professional where appropriate.

Frequently asked questions