Statistics

Normal Probability Calculator

Compute probability between two bounds in a normal distribution.

Last validated: 2026-02-14

Normal Probability Calculator returns probabilities and quantiles for the normal distribution. It is useful for risk thresholds, scoring models, and statistical coursework. The tool supports interval and tail probability questions quickly. Use it when translating z-scores into intuitive probabilities.

Permalink

Input Pattern

Enter values in the left panel, keep units explicit, run the calculation, then copy or share the result. Invalid fields are highlighted immediately.

Normal Probability Inputs

Result

z1: -1.000000

z2: 1.000000

P(x1 <= X <= x2): 0.68268949

How to use this tool

Enter mean, standard deviation, and value/range of interest.
Run the calculator to obtain tail or interval probability.
Use output to evaluate threshold likelihood or percentile placement.

Worked Example

Auto-generated from the tool's current default or entered inputs.

Example Inputs

Std dev: 15.0
X1: 85.0
X2: 115.0
Z1: -1.0
Z2: 1.0

Expected Outputs

Mean: 100
Std dev: 15
X1: 85
X2: 115

Interpretation

Mean is 100.0. Use this as a decision input, not as a standalone conclusion.
Std dev is 15.0. Use this as a decision input, not as a standalone conclusion.
X1 is 85.0. Use this as a decision input, not as a standalone conclusion.
Interpret this value alongside sample quality and assumptions to avoid overconfidence.

Scenario Compare (A vs B)

Use this to compare two input sets and quantify change in key outputs.

Scenario A

Std dev

Scenario B

Std dev

Confidence and limitations

Outputs are deterministic formula results and depend entirely on input quality and units.
Results are for planning and learning; verify with domain standards before safety-critical use.
Interpretation depends on assumptions (distribution, independence, sample quality).

Formula References

Frequentist estimation and hypothesis-test formulas used in standard textbooks.
Interpretation depends on sample quality and assumptions.

Assumptions

Outputs are deterministic formula results and depend entirely on input quality and units.
Results are for planning and learning; verify with domain standards before safety-critical use.
Interpretation depends on assumptions (distribution, independence, sample quality).

Explore more versions

Tailored guides for specific audiences, regions, and scenarios.

core