Normal distribution calculator
Compute the normal CDF, survival probability, and PDF for any x, mean, and standard deviation.
What this calculator covers
Use this calculator when you want the left-tail probability, upper-tail probability, and density from the same normal-curve input.
It is useful as a bridge tool because many inference tasks become easier once a raw value has been standardized and read back through the normal curve.
Frequently asked questions
- What does the CDF result tell me?
- The cumulative distribution function (CDF) gives the probability that a randomly drawn value from the specified normal distribution falls at or below your input x. For example, a CDF of 0.84 means about 84 percent of values in that distribution are expected to be at or below x.
- What is the survival probability?
- The survival probability, also called the upper-tail probability, is simply 1 minus the CDF. It tells you what fraction of the distribution lies above your input value. The two together always sum to 1.
- What does the PDF value represent?
- The probability density function (PDF) gives the height of the normal curve at exactly x. Unlike the CDF, it is not a probability by itself — it becomes a probability only when integrated over a range. A higher PDF value means x sits near the peak of the distribution.
- How is the z-score used in the calculation?
- The z-score standardizes x by subtracting the mean and dividing by the standard deviation. Once expressed in standard-deviation units from the mean, the value can be looked up on the standard normal curve, which is how the CDF and PDF are computed.
Tool
Run the calculation
Result
RESULT · CDF
â„–166
Primary result
50%
For a normal distribution with mean 0 and standard deviation 1, the probability of observing a value at or below 0 is 50%.
- Z-score
- 0
- CDF P(X ≤ x)
- 50%
- Survival P(X > x)
- 50%
- PDF at x
- 0.398942
Step-by-step solution
- 1.Standardize the input: (0 - 0) / 1 = 0.
- 2.Evaluate the normal CDF at that standardized location to get P(X ≤ x) = 50%.
- 3.The remaining upper-tail probability is P(X > x) = 50%, and the point density at x is 0.398942.
Walkthrough
Visual walkthrough
The normal calculator translates a raw x-value into a standardized position, then reads both the cumulative probability and the density at that point.
01
Center and scale the value
z = (0 - 0) / 1
Normal-curve lookups begin by converting the raw value into standard-deviation units from the mean.
02
Read the left-tail probability
P(X ≤ x) = 50%
The cumulative distribution function reports the area under the curve to the left of the chosen value.
03
Pair probability with density
The survival tail is whatever probability remains above x, while the PDF shows how tall the curve is right at x.
50% upper tail · PDF 0.398942