CDF at z = 1.96
For μ = 0 and σ = 1, P(Z ≤ 1.96) ≈ 0.975002, leaving about 0.024998 in the upper tail.
Normal Distribution Calculator — CDF, Probability & Critical Values
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Normal distribution formulas
For X normally distributed with mean μ and standard deviation σ > 0, first standardize x as z = (x − μ) / σ. The standard normal CDF Φ(z) is the area under the standard normal density to the left of z.
| Quantity | Formula | Meaning |
|---|---|---|
| Density | f(x) = exp(−(x − μ)² / (2σ²)) / (σ√(2π)) | Curve height at x; it is not a point probability. |
| CDF | F(x) = Φ((x − μ) / σ) | P(X ≤ x), the area to the left of x. |
| Upper tail | 1 − F(x) = Φ(−z) | P(X > x), evaluated directly to preserve small tails. |
| Between | F(b) − F(a) | P(a ≤ X ≤ b). |
| Percentile | xp = μ + σΦ−1(p) | The x-value with cumulative probability p. |
Assumptions: The variable is modeled by a continuous normal distribution with finite μ and σ > 0. An exact point has probability zero, so < and ≤ (or > and ≥) give the same result. CDF and inverse-CDF values are numerical approximations. See the NIST normal distribution reference for the density, CDF, survival, and percent-point definitions.
How to use the calculator
- Choose CDF & tails, Range probability, or Critical values.
- Enter the distribution mean μ and standard deviation σ. Use μ = 0 and σ = 1 for the standard normal distribution.
- Enter x or the two range bounds. For a critical value, enter a probability strictly between 0 and 1, or between 0% and 100%.
- Select Calculate. Use Copy or Download to save the text result without sending inputs anywhere.
Normal distribution examples
Within one SD
For any normal distribution, P(μ − σ ≤ X ≤ μ + σ) ≈ 0.682689, or 68.2689%.
Central 95% bounds
The standard normal central 95% interval is approximately −1.959964 to 1.959964, with 2.5% in each tail.
CDF, PDF, probability, and critical value
The PDF is the curve height and may exceed 1 when σ is small; it is not itself a probability. The CDF is accumulated area and always lies from 0 to 1. A range probability is an area between two x-values. A critical value reverses the question: it starts from a specified tail or central probability and finds the matching x-value.
Normal-model results describe the chosen model, not whether a dataset is actually normal. Check the source, measurement process, dependence, skew, and outliers before relying on the model for inference.
Frequently asked questions
What does a normal CDF calculate?
It calculates P(X ≤ x): the area under the normal curve from negative infinity through x. The upper-tail probability is the remaining area, P(X > x).
How do I find the probability between two normal values?
Standardize both values, calculate their CDFs, and subtract: P(a ≤ X ≤ b) = Φ(zb) − Φ(za).
How do I calculate a normal critical value?
For lower-tail probability p, use x = μ + σΦ−1(p). For upper-tail probability α, use cumulative probability 1 − α. For a central probability c, place (1 − c) / 2 in each tail.
What are the z critical values for a central 95% interval?
Approximately −1.959964 and 1.959964. Each tail contains 0.025 and the middle contains 0.95.
Does it matter whether interval endpoints are included?
No. A continuous distribution assigns probability zero to any single exact value, so open and closed endpoints have the same probability.
Can the standard deviation be zero or negative?
No. A normal density needs σ > 0. A zero standard deviation describes a constant rather than a regular normal density.
Why is a very small probability shown in scientific notation?
Scientific notation preserves information that would disappear if a small tail were rounded to 0. The selected decimal setting controls ordinary displayed values.
Are the answers exact?
The probability definitions are exact, but the CDF and inverse CDF have no simple elementary closed form and are evaluated numerically. Values are rounded only for display.
Are my inputs private?
Yes. All calculations happen locally in your browser. This calculator does not upload or store your values.