How accurate is the inverse CDF (quantile) computation?

We implement Peter Acklam’s rational approximation with one Newton refinement, yielding errors around 10^{-6} for representative percentiles—suitable for coursework and exam prep.

Normal Distribution Calculator — PDF/CDF/Quantile & Z-Score

Q: When should I use left-tail, right-tail, or two-tailed probabilities?

Left-tail probability reports P(X ≤ x), right-tail reports P(X ≥ x), and two-tailed sums both tails where |X-μ| is at least |x-μ|. Choose the tail that matches your hypothesis test or classroom example.

Results

Value: —

Steps

For educational purposes only. Always exercise professional judgement when interpreting statistical outputs.

Normal curve preview

We plot the range from −4σ to +4σ, highlighting the selected tail or interval in blue to visualise probabilities at a glance.

Teacher mode

Standardise with z = (x − μ) / σ and exploit symmetry Φ(−z) = 1 − Φ(z) to explain one- and two-tailed probabilities.
Interval probability is Φ((b−μ)/σ) − Φ((a−μ)/σ); splitting the calculation helps learners follow each bound.
Quantiles use x = μ + σΦ⁻¹(p), ideal for referencing in lesson slides without scanning printed Z tables.
Approximation error is around 10⁻⁶ for typical percentiles after one Newton refinement—sufficient for coursework and exams.

FAQ

When should I use left-tail, right-tail, or two-tailed probabilities?

Left-tail covers P(X ≤ x), right-tail covers P(X ≥ x), and two-tailed combines both ends where |X−μ| is at least |x−μ|. Match the choice to your hypothesis test or instructional example.

How accurate is the inverse CDF (quantile) result?

We rely on Peter Acklam’s rational approximation with a single Newton step. The typical absolute error stays near 10⁻⁶, which is more than adequate for teaching, homework, and exam preparation.

Normal Distribution & Z-Score Calculator

Results

Normal curve preview

Teacher mode

FAQ

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