Normal Distribution Calculator (CDF / Quantile)
Enter mean & SD to get probabilities (CDF) or critical values (quantiles). Choose one- or two-tailed and see the shaded chart and step-by-step math.
Your inputs
Results
Show steps
z = (x − μ) / σ = (130 − 100) / 15 = 2.0000Φ(z) = 0.9772 (left-area); right-area = 1 − Φ(z) = 0.0228Two-tail = 2 · min(Φ(z), 1 − Φ(z)) = 0.0455Central area = 1 − two-tail = 0.9545
Chart (shaded area)
Shaded region represents the selected tail area. Dashed line marks the mean μ.
Results interpretation
- CDF mode: enter a value x to get left/right/two-tail probabilities.
- Quantile mode: enter an area (e.g., 5%) to get the corresponding critical value(s).
- Two-tail uses symmetry around μ; left/right use a single side.
- Use central area (1 − two-tail) for confidence levels (e.g., 95%).
How this calculator works
Formula & assumptions
For a normal distribution with mean μ and standard deviation σ, standardize with z = (x − μ)/σ . The cumulative probability is Φ(z), where Φ(z) = ½(1 + erf(z/√2)) . Right-tail area is 1 − Φ(z); two-tail area is 2·min(Φ(z), 1 − Φ(z)).
For quantiles, given a cumulative probability p, the critical z* = Φ⁻¹(p). Two-tailed criticals use p = 1 − A/2 for total tail area A, yielding ±z*, then convert to x with x* = μ + z*·σ.
Assumptions: normality, independent observations, and accurate mean/SD. Non-normal data can lead to misleading tail probabilities.
Use cases & examples
Example 1 (CDF): Test scores with μ=100, σ=15. What fraction score ≤130? Standardize z=(130−100)/15≈2. Then Φ(2)≈0.977 → about 97.725%.
Example 2 (Right tail): What fraction ≥115? z≈(115−100)/15≈1. Right area ≈ 1−Φ(1) ≈ 15.866%.
Example 3 (Quantile, two-tail 5%): A=5% → each tail 2.5% → p=0.975 → z≈1.96. Criticals at x≈μ±1.96σ.
Normal Distribution — FAQ
What’s the difference between CDF and quantile?
CDF gives area for a value; quantile gives the value for an area.
When do I use two-tailed?
When deviations on either side of the mean matter equally (e.g., many tests).
How do I get a 95% critical value?
Use two-tail A=5% → z≈1.96 → x=μ±1.96σ.
Does this work for any units?
Yes—just keep x, μ, and σ in the same units.
Is normality required?
Yes; with skewed/heavy tails, probabilities can be off.
What precision should I use?
3–4 decimals is typical for reporting z and probabilities.
Normal Distribution Calculator: CDF, Quantiles, and Clear Decisions
Our normal distribution calculator turns raw numbers into decisions. By converting a measurement to a standardized z-score and applying the cumulative distribution function (CDF), you can answer questions like “What percentage falls below this value?” or “How extreme is this observation?” When you reverse the process with a quantile, you specify the probability and get the corresponding cut score—ideal for thresholds, QA bands, and confidence limits. These two operations—area and critical value—cover most practical needs in statistics, analytics, quality control, and research reporting.
In CDF mode, you enter x along with the mean (μ) and standard deviation (σ). The calculator standardizes to z and returns left, right, and two‐tail probabilities. The left-tail is Φ(z): the fraction at or below x. The right-tail is 1−Φ(z): at or above x. For two-tailedcontexts—where deviations in either direction matter—use 2·min(Φ(z), 1−Φ(z)). The central area (1 − two-tail) is the probability inside the band [μ−|z|σ, μ+|z|σ], which maps neatly to confidence levels (e.g., ~95% for |z|≤1.96).
In quantile mode, you start with an area and get back a critical value. Left‐tail areas use p=A; right‐tail areas use p=1−A; and two‐tail areas split A/2 into each tail and use p=1−A/2 to locate the upper critical. Converting z back to real units via x=μ+zσ makes the result immediately useful: you can draw a line on your process chart or set a policy threshold with a precise false positive rate.
Interpreting results requires context. A z of 2 may be ordinary in a screening pipeline that evaluates thousands of cases (multiple comparisons), yet highly unusual for a single reading. Verify that your data are approximately normal—heavy tails or skew change tail risk. Still, for many aggregate measures and standardized scores, the normal model is an excellent first approximation, and the z/CDF toolkit is the fastest route to trustworthy answers.
Practical tips: keep units consistent; sanity‐check σ (tiny σ inflates z); round to 2–4 decimals; and pair probability with a short explanation (“two‐tail, α=0.05”). Use this calculator to explore scenarios, communicate uncertainty visually with the shaded chart, and set defensible thresholds that stakeholders understand.