Many natural measurements, such as heights and test scores, are approximately normal.
How to use the Normal Distribution Calculator
- Mean (μ)
- Standard deviation (σ)
- Lower value
- Upper value
Results update as you type, and every field accepts the units shown in its menu.
How it works
Convert each value to a z-score and use the standard normal cumulative distribution.
Inputs explained
| Field | What to enter |
|---|---|
| Mean (μ) | A number |
| Standard deviation (σ) | A number |
| Lower value | A number |
| Upper value | A number |
What each result means
- P(a < X < b) = (ncdf(([Upper value] – [Mean (μ)])÷[Standard deviation (σ)]) – ncdf(([Lower value] – [Mean (μ)])÷[Standard deviation (σ)]))×100
- P(X < a) = ncdf(([Lower value] – [Mean (μ)])÷[Standard deviation (σ)])×100
- P(X > b) = (1 – ncdf(([Upper value] – [Mean (μ)])÷[Standard deviation (σ)]))×100
Example values
How the main result changes when Mean changes and everything else stays at the example values above:
| Mean (μ) | P(a < X < b) (%) |
|---|---|
| 50 | 0.9808 |
| 75 | 24.8662 |
| 100 | 68.2689 |
| 125 | 24.8662 |
| 150 | 0.9808 |
| 200 | 7.2801e-7 |
Formula
z = (x − μ) ÷ σP(X < x) = Φ(z)Worked example
Inputs — Mean (μ): 100; Standard deviation (σ): 15; Lower value: 85; Upper value: 115.
Results — P(a < X < b) = 68.2689 %; P(X < a) = 15.8655 %; P(X > b) = 15.8655 %.
Frequently asked questions
What is the 68-95-99.7 rule?
About 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.
What is an IQ of 130?
About the 98th percentile, two SDs above the mean.
What formula does the Normal Distribution Calculator use?
z = (x − μ) ÷ σ.