Normal Distribution Calculator - PDF & CDF
Calculate the probability density and cumulative probability at a given x-value from a mean and standard deviation.
z-score
1.0000
Probability Density
0.016131
Cumulative Probability
84.13%
How to Calculate Normal Distribution Probability
Enter a mean, standard deviation, and a specific x-value to find that point's probability density (how "tall" the bell curve is there) and its cumulative probability (the chance of landing at or below that value), based on the standard normal distribution.
Example
With a mean of 100, a standard deviation of 15, and x = 115, the z-score is (115 − 100) ÷ 15 = 1.0, meaning 115 is one standard deviation above the mean. The cumulative probability at z = 1.0 is about 84.13%, meaning roughly 84% of values fall at or below 115.
Common Use Cases
- Finding a test score's percentile given the mean and standard deviation of the class.
- Statistics homework involving z-scores and the standard normal distribution.
- Quality control checks estimating how likely a measurement is within an expected range.
FAQs
What's the difference between the PDF and the CDF?
The probability density function (PDF) describes the relative likelihood of the distribution at an exact point, while the cumulative distribution function (CDF) gives the total probability of landing at or below that point — the CDF is generally more useful for real-world questions like percentiles.
What does a z-score of 0 mean?
A z-score of 0 means the x-value equals the mean exactly, putting it right at the center of the distribution with a cumulative probability of 50%.
Is the CDF calculation exact?
This uses a well-known numerical approximation (accurate to about 7 decimal places) for the standard normal CDF, since no exact closed-form expression exists using elementary functions. That level of precision is more than sufficient for virtually all practical and academic uses.