Calculator
Example data table
Standard normal example (μ=0, σ=1). Values are typical reference points.
| x | z | PDF f(x) | CDF P(X ≤ x) | Right tail P(X ≥ x) |
|---|---|---|---|---|
| -2 | -2 | 0.0539909665 | 0.0227500629 | 0.9772499371 |
| -1 | -1 | 0.2419707245 | 0.1586552638 | 0.8413447362 |
| 0 | 0 | 0.3989422804 | 0.5000000005 | 0.4999999995 |
| 1 | 1 | 0.2419707245 | 0.8413447362 | 0.1586552638 |
| 2 | 2 | 0.0539909665 | 0.9772499371 | 0.0227500629 |
Formula used
This tool uses stable approximations for erf and Φ⁻¹, commonly used in analytics software.
How to use this calculator
- Enter the mean (μ) and standard deviation (σ).
- Set an x value to compute PDF, CDF, and tail probabilities.
- Optionally add an interval [a, b] to get P(a ≤ X ≤ b).
- Provide p to compute the percentile cutoff with InvCDF(p).
- Set α to view two-sided critical z and critical x-range.
- Press Submit; results appear above the form.
- Use CSV or PDF buttons to export your output.
FAQs
1) What does the PDF value mean?
The PDF is a density, not a probability. Probabilities come from areas under the curve, such as CDF values or interval probabilities.
2) Why can a PDF be greater than 1?
Because density is measured per unit on the x-axis. A narrow distribution can have a tall peak while still having total area equal to 1.
3) What is the difference between CDF and right tail?
CDF gives P(X ≤ x). Right tail gives P(X ≥ x) which equals 1 − CDF, useful for exceedance probabilities and thresholds.
4) How is the two-tailed p-value computed here?
It is approximated as 2 × min(P(X ≤ x), P(X ≥ x)). This matches a two-sided extremeness test for symmetric normal distributions.
5) What does InvCDF(p) return?
It returns the cutoff x such that P(X ≤ x) = p. For example, p=0.975 gives the upper 97.5th percentile.
6) What does α control in the results?
α sets the two-sided critical z as z = Φ⁻¹(1 − α/2). The tool also reports the corresponding symmetric critical x-interval around μ.
7) How accurate are the computations?
The erf and inverse CDF use standard approximations with good practical accuracy for most analytics tasks. Extreme tail probabilities may show minor rounding differences.
8) Can I use this for standard normal tables?
Yes. Set μ=0 and σ=1 to replicate standard normal table lookups, and export results to keep a quick reference for your work.