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Normal Distribution Calculator: Probabilities and Inverse

This free normal distribution calculator finds the probability that a value falls below, above or between limits, for any mean and standard deviation. It can also work backwards from a probability to a value.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

The normal distribution is the most used curve in statistics. It turns a value, a mean and a standard deviation into a probability, which tells you how typical or unusual the value is.

Choose what to find, enter the mean and standard deviation and the value or values.

How to Use This Tool

  1. Enter the mean and the standard deviation.
  2. Choose below, above, between or inverse.
  3. Enter x, or a and b, or the probability.
  4. Press Calculate.

How the Calculation Works

The value is converted to a z-score, z = (x − μ) ÷ σ, and the area under the standard normal curve is computed from the error function. P(X > x) is 1 minus the area below, and P(a < X < b) is the difference between two areas. The inverse uses a rational approximation refined by one step of Halley’s method.

Worked Example

IQ scores with mean 100 and standard deviation 15: P(X < 115) is 0.8413, P(85 < X < 115) is 0.6827 and the 97.5th percentile is 129.4.

Common Uses

  • Exam scores and IQ.
  • Quality control limits.
  • Finance and risk.
  • Homework and teaching.

Tips and Common Mistakes

  • The standard deviation must be above zero.
  • Percent signs are accepted for the inverse.
  • z-scores of ±1.96 mark the middle 95%.

Next, you may need the z-score calculator, confidence interval calculator, sample size calculator, binomial calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

The bell-shaped curve that describes many natural measures such as height, test scores and errors. It is set by its mean and standard deviation.
About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.
The probabilities are accurate to about seven decimal places and the inverse to about nine.
Many real data sets are only roughly normal. Treat the results as approximations if your data is skewed.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.