Free calculator
Normal distribution calculator
Work with X ~ N(μ, σ²) in one place: left-tail P(X ≤ x), right-tail P(X ≥ x), between P(a ≤ X ≤ b), or inverse from a left-tail probability p to the cutoff x (percentile / InvNorm style). Toggle standard normal (μ = 0, σ = 1) for table-style Z problems. Use Google Sheets & Excel for NORM.DIST / NORM.INV patterns. Educational normal-model math—not normality-test software.
When to use this calculator
Textbook normal CDF checks before you mirror the same idea in Sheets or Excel—transparent formulas, not a full statistics suite.
- Find tail or interval probabilities for a stated μ and σ (quality specs, simple risk sketches, intro stats).
- Turn a cumulative probability into a cutoff x (percentile) with the same μ and σ.
- Match
NORM.DIST/NORM.INVcells against this page when auditing a template row. - Need z = (x − μ) / σ with tail readouts for one x? Use the z-score calculator; use this page when areas, intervals, or inverse p → x are the goal.
If X is normally distributed with mean μ and standard deviation σ > 0, then Z = (X − μ)/σ is standard normal (mean 0, SD 1). P(X ≤ x) equals P(Z ≤ (x−μ)/σ), computed with the standard normal CDF Φ.
From **x** to a tail probability
Standardize z = (x − μ)/σ, then P(X ≤ x) = Φ(z). The right tail is 1 − Φ(z). For an interval [a, b], P(a ≤ X ≤ b) = Φ((b−μ)/σ) − Φ((a−μ)/σ) (endpoints can be swapped).
From **p** to a cutoff **x**
Given left-tail p in (0,1), solve Φ(z) = p for z, then x = μ + σ z. This matches NORM.INV(p, μ, σ) in Excel when p is a left cumulative probability.
Where do **μ** and **σ** come from?
This page does not estimate parameters from raw rows. If σ is a sample SD, choose STDEV.S vs STDEV.P consistently with your list (see the standard deviation tool).
We do not run Shapiro–Wilk/KS normality tests, t/F distributions, or two-sample probability engines on this v1—use dedicated statistics software when you need those workflows.
For z = (x − μ)/σ with tail readouts centered on one x, open the z-score calculator.
To get σ (and μ) from a pasted list, open the standard deviation calculator.
For two-sided confidence intervals on means or proportions instead of general N(μ, σ²) tail play, open the confidence interval calculator.
FAQs cover left vs right tail, Casio/graphing calculator wording, Excel localized names, and limits of the normal model.
Google Sheets & Excel
In English function names, NORM.DIST(x, mean, standard_dev, TRUE) is P(X ≤ x) for X ~ N(μ, σ) with cumulative TRUE. NORM.INV(p, mean, standard_dev) returns x with P(X ≤ x) = p (left tail). For Z ~ N(0,1), use NORM.S.DIST and NORM.S.INV. Replace cell references with your values. On localized Excel, use Insert function to find the same patterns under your language pack.
=NORM.DIST(A2, B2, C2, TRUE)A2 = x, B2 = μ, C2 = σ (> 0). TRUE = cumulative.
=NORM.INV(D2, B2, C2)D2 = p in (0,1); B2/C2 = μ/σ as above.
=NORM.S.DIST(E2, TRUE)E2 = z on N(0,1). Same as NORM.DIST(E2,0,1,TRUE).
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Frequently asked questions
What does this calculator compute?
Forward modes return a probability for X ~ N(μ, σ²): P(X ≤ x), P(X ≥ x), or P(a ≤ X ≤ b). Inverse returns x with P(X ≤ x) = p for a left-tail p in (0,1)—a percentile cutoff when you multiply p by 100.
What does “left-tail p” mean?
p is always P(X ≤ x) on the same normal curve with your μ and σ. The right tail at x is 1 − p. Some courses label upper-tail areas differently—match your worksheet’s definition before comparing numbers.
What is the standard normal distribution?
The standard normal is N(0,1): mean 0, standard deviation 1. Any N(μ, σ²) variable can be written as X = μ + σ Z with Z ~ N(0,1). The toggle locks μ = 0 and σ = 1 so results match Z-table problems.
How is this different from your z-score calculator?
The z-score page emphasizes standardizing one x with μ and σ and reading N(0,1) tails for that z. This page emphasizes areas, intervals, and inverse p → x for N(μ, σ²)—use whichever matches your question.
Can I enter b smaller than a for “between”?
Yes. The tool uses the ordered interval [min(a,b), max(a,b)], so the probability is the same either way.
Which Google Sheets or English Excel functions match this page?
NORM.DIST(x, mean, standard_dev, TRUE) gives P(X ≤ x). NORM.INV(p, mean, standard_dev) gives x with P(X ≤ x) = p. For Z ~ N(0,1), NORM.S.DIST(z, TRUE) and NORM.S.INV(p) are the standard-normal shortcuts.
What are the German Excel names for NORM.DIST and NORM.INV?
Typical Excel (Deutsch) names are NORM.VERT for NORM.DIST and NORMINV for NORM.INV—confirm in your function library; argument separators may be semicolons.
What are the French Excel names for NORM.DIST and NORM.INV?
Typical Excel (français) names include LOI.NORMALE (cumulative) and LOI.NORMALE.INVERSE—verify on your install.
Is this the same as InvNorm on a Casio or TI calculator?
Conceptually similar for normal inverse and tail setups, but menus differ by model. This page is a transparent web check with explicit μ, σ, and left-tail p—use your course’s device steps for exams.
Does this check whether my data are normally distributed?
No. Histograms, Q–Q plots, and tests like Shapiro–Wilk answer goodness-of-fit questions. This tool only evaluates N(μ, σ²) probabilities and cutoffs once you have already chosen μ and σ.
Is this professional statistics advice?
No. It is a free educational calculator. For regulated reporting, research protocols, or clinical decisions, follow qualified professionals and institutional methods.