Normal Distribution Calculator
Calculate normal distribution using your data.
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What this tool does
Calculate normal distribution using your data. Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
How to use the Normal Distribution Calculator
- Enter or select mean (μ).
- Enter or select standard deviation (σ).
- Enter or select lower value (x₁).
- Enter or select upper value (x₂).
- Read the calculated result; change any measurement to compare alternatives.
Formula
z = (x − μ)/σ; areas from the standard normal CDF Φ(z), evaluated with the gammaQ-based tail in statistics.ts (Φ(1) ≈ 0.8413)
- mu
- Mean (μ)
- sigma
- Standard deviation (σ)
- lo
- Lower value (x₁)
- hi
- Upper value (x₂)
Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
Worked example
For normal distribution calculator, the following measurements illustrate the exact method: Mean (μ): 100; Standard deviation (σ): 15; Lower value (x₁): 85; Upper value (x₂): 115.
Inputs
- Mean (μ)100
- Standard deviation (σ)15
- Lower value (x₁)85
- Upper value (x₂)115
Result
- P(lower < X < upper) (%)68.27
- P(X < upper) (%)84.13
- P(X > lower) (%)84.13
- z of lower value-1
- z of upper value1
Results explained
- P(lower < X < upper) (%)
- P(lower < X < upper) (%) from the formula above. Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
- P(X < upper) (%)
- P(X < upper) (%) from the formula above. Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
- P(X > lower) (%)
- P(X > lower) (%) from the formula above. Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
- z of lower value
- z of lower value from the formula above. Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
- z of upper value
- z of upper value from the formula above. Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
Frequently asked questions
z = (x − μ)/σ; areas from the standard normal CDF Φ(z), evaluated with the gammaQ-based tail in statistics.ts (Φ(1) ≈ 0.8413)
Areas assume a normal model with the entered mean and SD. Check the model fits your data before treating these areas as real-world frequencies.
Enter numbers only, in the units and format each label describes. Remove missing values rather than substituting zero, unless zero is a real observation.
No. Results describe the numbers you entered. Statistical inference also depends on sampling design, independence, model fit and interpretation; a p-value is not the probability that a hypothesis is true.
No. Every calculation, including the distribution algorithms, runs entirely in your browser.