What Is Normal Probability?
The normal distribution is the symmetric bell-shaped curve defined by a mean μ and a standard deviation σ. To find a probability you standardise each value to a z-score, z = (x − μ) ÷ σ, and read the area under the standard normal curve.
This tool does the standardising and the area lookup for you. It returns the normal probability, the z-score of each value, the left and right tail areas, the density, the percentile rank and the middle 95 % of values. A table of values at μ ± 1, 2 and 3 standard deviations shows the 68-95-99.7 rule for your own distribution.
Below you will find the formulas, a variables table, a worked example and answers to common questions about z-scores, tails and when a normal model is appropriate.
How to Use the Normal Probability Calculator
- Enter the mean and standard deviation of the distribution.
- Choose below, above, between or outside.
- Enter the value or values a and b.
- Read the probability, the z-scores and the shaded area on the curve.
Normal Probability Formula
Standardising
Density
Between two values
Normal Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| μ | Mean of the distribution | centre of the bell curve |
| σ | Standard deviation | must be above 0 |
| x | Value you want a probability for | |
| z | Z-score | (x − μ) ÷ σ |
| Φ(z) | Standard normal cumulative probability | area to the left of z |
Normal Probability Worked Example
IQ scores are scaled to a mean of 100 and a standard deviation of 15. For a score between 85 and 130: z = (85 − 100) ÷ 15 = −1 and z = (130 − 100) ÷ 15 = 2. Then P = Φ(2) − Φ(−1) = 0.97725 − 0.15866 = 0.81859, so about 81.9% of people score between 85 and 130.
With the example values loaded in the calculator above, the steps are:
- Standardise: z = (x − μ) ÷ σ = (85 − 100) ÷ 15 = -1. For the upper value: z = (130 − 100) ÷ 15 = 2.
- Look up the standard normal cumulative probability Φ(z): Φ(-1) = 0.15865525. Φ(2) = 0.97724987.
- P(a < X < b) = Φ(z_b) − Φ(z_a) = 0.97724987 − 0.15865525 = 0.81859461.
Result: P(85 < X < 130) = 0.81859461; z-score of 85 = -1.
The 68-95-99.7 rule
About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. The table below is generated for the distribution you entered.
Where Normal Probability Is Used
Test scores and percentiles
Finding the share of students above or below a score.
Quality control
Estimating the share of parts outside specification limits.
Statistics coursework
Checking z-table and normal probability answers.
Risk and measurement
Estimating how unusual a measurement is.
Common Normal Probability Mistakes to Avoid
- Always sketch the curve and shade the region before computing.
- Probabilities for a single exact value are 0 for continuous distributions; only ranges have probability.
- Use the inverse z-score calculator to go from a probability back to a value.
Normal Probability Terms and Definitions
- Φ(z)
- The standard normal cumulative distribution function: the area to the left of z.
- z-score
- The number of standard deviations a value lies from the mean.
- Percentile
- The percentage of values at or below a given value.
Normal Probability FAQs
How do I find the probability between two values?
Convert both to z-scores, look up Φ for each and subtract: P(a < X < b) = Φ(z_b) − Φ(z_a).
What is the probability of one exact value?
Zero. A continuous distribution gives probability to ranges, not single points.
What is the difference between this and a z-table?
The z-table lists Φ(z) for standard normal values. This calculator standardises for you and handles any mean and standard deviation.
When is the normal distribution appropriate?
For symmetric, bell-shaped data and for sample means with a reasonably large n, by the central limit theorem.