Statistics & Probability

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Normal Distribution Calculator.

Calculate a normal probability density and cumulative probability.

On-device calculationNo signup
01

Set your values

Results update as you type.

zScore: 1.2

zScore

1.200000
probabilityDensity: 0.194186

probabilityDensity

0.194186
cumulativeProbability: 0.88493

cumulativeProbability

0.884930

Results update automatically as you type.

Use Cases

Find probabilities for a normal distribution

Use the CDF to find the probability that a value falls below a certain threshold. This is useful in quality control, finance, and social sciences.

Example: If test scores are normally distributed with mean 70 and standard deviation 10, find the probability of scoring below 85.

Understand the shape of the distribution

The PDF shows the relative likelihood of different values. Compare PDF values at different x to see which values are more likely.

Example: For mean 0 and standard deviation 1, the PDF at x=0 is about 0.399, while at x=2 it is about 0.054.

Frequently Asked Questions

What does the normal distribution calculator compute?
It computes the probability density function (PDF) and cumulative distribution function (CDF) for a normal distribution. You input a value x, the mean, and the standard deviation, and it returns the height of the bell curve at x (PDF) and the probability that a random variable is less than or equal to x (CDF).
How do I interpret the CDF result?
The CDF gives the probability that a value from the normal distribution is less than or equal to your specified x. For example, if the CDF is 0.84, there is an 84% chance that a randomly selected value will be at or below x.
Can I use this calculator for any normal distribution?
Yes, as long as you know the mean and standard deviation. The calculator works for any normal distribution, regardless of the units. Just enter the appropriate mean and standard deviation for your data.

Tips & Common Mistakes

Tips

  • Ensure the standard deviation is positive; a negative or zero standard deviation is invalid.
  • The mean can be any real number, but it should be representative of your data's center.
  • For a standard normal distribution, set mean to 0 and standard deviation to 1.
  • Use the CDF to find probabilities for ranges by subtracting CDF values: P(a < X < b) = CDF(b) - CDF(a).

Common Mistakes to Avoid

  • Entering the variance instead of the standard deviation. The calculator expects the standard deviation, which is the square root of the variance.
  • Using a negative standard deviation. Standard deviation is always non-negative.
  • Misinterpreting the PDF value as a probability. The PDF value is a density, not a probability; it can be greater than 1.

Last updated: August 13, 2026