Normal distribution calculator for probabilities, z-scores and quantiles
This normal distribution calculator answers the questions that fill statistics homework, quality-control reports and exam prep: what proportion of values fall below a, above b or between a and b for a bell curve with mean μ and standard deviation σ, and the reverse — which value x sits at a given percentile. Each answer is shown as a decimal and a percentage, with the matching z-scores and a shaded curve so you can see the area you are computing.
It also converts between raw values and z-scores, prints the empirical (68–95–99.7) rule for your μ and σ, and includes a helper that checks whether a binomial distribution can be approximated by a normal one and compares the approximation with the exact binomial probability.
How to use the Normal Distribution Calculator
- Enter the mean μ and standard deviation σ of your distribution (leave 0 and 1 for the standard normal Z).
- Pick the question: P(X < a), P(X > b), P(a < X < b), Two tails (outside a…b) or Inverse (quantile).
- Type the bounds a and/or b, or for the inverse mode type the probability (as 0.975 or 97.5%) and choose which area it refers to.
- Read the probability or the x value in the result box; the stats row shows the z-scores and the complement, and the chart shades the area.
- Open z-score converter and empirical rule to convert any x to z (or back) and see μ ± 1σ, 2σ, 3σ ranges, or the Binomial → normal panel to approximate a binomial probability.
- Press Copy result for a text summary or Download SVG for the chart.
Normal Distribution Calculator features
Five calculation modes
Left tail, right tail, between, two tails and inverse — covering percentiles, confidence limits and “what proportion is out of spec” questions in one place.
Precise algorithms
The CDF uses the Hart/West rational approximation (accurate to about 1e-14) and the inverse uses Acklam’s method refined with Newton steps, so z = 1.959964 for p = 0.975 rather than a rounded 1.96.
Empirical rule table
μ ± 1σ, 2σ, 3σ and 1.96σ shown as actual values for your distribution, alongside the exact probabilities (68.27%, 95.45%, 99.73%).
Binomial approximation check
Checks np ≥ 5 and n(1 − p) ≥ 5, applies the continuity correction and shows the exact binomial result for comparison.
Formulas behind the calculator
A normal distribution with mean μ and standard deviation σ has the density f(x) = (1 / (σ√(2π))) · e^(−(x − μ)² / 2σ²). Probabilities are areas under this curve, found by standardising to z = (x − μ) / σ and using the standard normal CDF Φ(z): P(X < a) = Φ(za), P(X > b) = 1 − Φ(zb) and P(a < X < b) = Φ(zb) − Φ(za). The inverse question uses the quantile function: x = μ + σ · Φ⁻¹(p).
Because the curve is symmetric, Φ(−z) = 1 − Φ(z). That is why a two-sided 95% interval uses z = 1.96: Φ(1.96) = 0.9750, leaving 2.5% in each tail.
| z | Φ(z) = P(Z < z) | Two-tailed area outside ±z | Typical use |
|---|---|---|---|
| 1.000 | 0.84134 | 31.73% | μ ± 1σ |
| 1.645 | 0.95000 | 10% | 90% interval, one-sided 5% |
| 1.960 | 0.97500 | 5% | 95% interval |
| 2.326 | 0.99000 | 2% | one-sided 1% |
| 2.576 | 0.99500 | 1% | 99% interval |
| 3.000 | 0.99865 | 0.27% | μ ± 3σ, control limits |
Worked examples
- Exam scores: marks are normal with μ = 65 and σ = 10. P(X > 80) = 1 − Φ((80 − 65)/10) = 1 − Φ(1.5) = 1 − 0.9332 = 6.68% of students score above 80.
- Cut-off for the top 5%: inverse mode with p = 0.05 and “Right tail” gives z = 1.645, so x = 65 + 1.645 × 10 = 81.4.
- Manufacturing tolerance: bolt lengths have μ = 50.0 mm, σ = 0.2 mm and the spec is 49.5–50.5 mm. Between mode gives Φ(2.5) − Φ(−2.5) = 0.9876, so about 1.24% of bolts fall outside the tolerance.
- Binomial approximation: 100 coin flips, P(X ≤ 55). μ = 50, σ = 5; with continuity correction Φ((55.5 − 50)/5) = Φ(1.1) = 0.8643, against the exact binomial 0.8644.
Tips and common mistakes
- Enter the standard deviation, not the variance. If you have the variance, take its square root first — the standard deviation calculator gives both from raw data.
- For “at least” and “at most” with a continuous variable there is no difference between < and ≤; the probability of hitting exactly one value is zero. For a discrete count approximated by a normal curve, apply the continuity correction (±0.5) as the binomial helper does.
- Probabilities for one-sided questions use a single tail; a 95% confidence interval needs 2.5% in each tail (z = 1.96), while a one-sided 95% limit uses z = 1.645.
- If your data are skewed (incomes, waiting times), a normal model will misstate the tails. Fit a line or curve to the data with the linear regression calculator or check the shape with the probability calculator before relying on these figures.
Normal Distribution Calculator FAQ
How do I find P(Z < 1.96) with this normal distribution calculator?
Leave μ = 0 and σ = 1, select P(X < a), and enter a = 1.96. The result is 0.975002, i.e. 97.5% of standard normal values are below 1.96.
What is a z-score?
The number of standard deviations a value sits from the mean: z = (x − μ)/σ. A z of 2 means two standard deviations above the mean; the converter panel does this both ways.
How accurate are the results compared with a z-table?
A printed z-table gives four decimals and needs interpolation. This calculator evaluates the cumulative function to roughly 14 significant digits and the inverse to full double precision after Newton refinement.
How do I get the value for a percentile?
Choose Inverse (quantile), enter the percentile as a probability (0.90 or 90%) and keep “Left tail”. The x shown is the 90th percentile; the stats row gives the matching z.
When can I approximate a binomial with a normal distribution?
A common rule is np ≥ 5 and n(1 − p) ≥ 5 (some texts use 10). The helper checks this, applies the continuity correction and shows the exact binomial probability so you can judge the error yourself.
What does the empirical rule say?
For any normal distribution about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. The table lists the exact figures (68.27%, 95.45%, 99.73%) and the actual ranges for your μ and σ.