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.
How to Read a Normal Probability
A normal probability is an area under the bell curve. IQ scores scaled to mean 100 and SD 15 give z = −1 for 85 and z = 2 for 130. Then P(85 < X < 130) = 0.97725 − 0.15866 = 0.8186, so about 82% of scores fall in that range.
Probability for an exact value is zero for a continuous variable, so ask for a range. The percentile rank and the "middle 95%" line help you place a value within the distribution.
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
Between two values
Find the share of a normal population between a and b.
One-sided probabilities
Find the left or right tail beyond a value.
Percentiles
See the percentile rank of a value.
Quality limits
Estimate the share of items outside specification limits.
Common Normal Probability Mistakes to Avoid
- The normal model needs roughly bell-shaped data.
- Standardise with z = (x − μ) ÷ σ if you use a z-table.
- Make sure you use σ, not the variance.
- For a sample mean, use the standard error in place of σ.
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?
Standardise both values to z-scores and subtract: P(a < X < b) = Φ(zb) − Φ(za). For 85 and 130 with μ = 100 and σ = 15 that is 0.97725 − 0.15866 = 0.8186.
What is the probability of one exact value?
Zero, for a continuous distribution. Probabilities are areas, and a single point has no area. Ask for an interval instead.
What is the difference between this and a z-table?
A z-table gives areas for the standard normal only and you must convert first. This calculator takes your mean and SD directly and does the conversion.
When is the normal distribution appropriate?
When the data are roughly symmetric and bell-shaped, or when you work with sample means from large samples by the central limit theorem.