Normal Distribution Calculator

StatisticsLast updated: August 22, 2026

The normal (Gaussian) distribution is the most widely used probability model for continuous variables: height, weight, exam scores, measurement error, and almost any quantity built from the sum of many small effects are approximately normal. The bell-shaped curve is symmetric about the mean, and just two parameters fix its shape — the mean (μ) that sets its location and the standard deviation (σ) that sets its spread.

This calculator works in both directions. Forward, it computes the left tail P(X < x), the right tail P(X > x), and the probability of the interval [a, b] you choose, P(a < X < b), showing the z = (x − μ)/σ transform step by step. Backward, it finds the x value (the percentile) corresponding to a cumulative probability you supply. The results are also drawn as shaded areas on the normal curve.

The center of the distribution. A decimal comma is also accepted.
The measure of spread; it must be greater than 0. If you know the variance, enter its square root.
The left and right tail probabilities are computed relative to this value.
The lower bound for P(a < X < b). Left blank, it is taken as −∞.
The upper bound for P(a < X < b). Left blank, it is taken as +∞.
The x satisfying P(X < x) = q is sought. Enter 0.95 for the 95th percentile.

Normal Distribution Formulas

Density:         f(x) = 1 / (σ·√(2π)) · e^[−(x − μ)² / (2σ²)]
z transform:     z = (x − μ) / σ
Left tail:       P(X < x) = Φ(z)
Right tail:      P(X > x) = 1 − Φ(z)
Interval:        P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)
Inverse:         x = μ + σ · Φ⁻¹(q)
Mean:            E(X) = μ
Variance:        Var(X) = σ²
Empirical rule:  μ ± σ ≈ 68.27%;  μ ± 2σ ≈ 95.45%;  μ ± 3σ ≈ 99.73%
Reporting:       central 95% interval = μ ± 1.96·σ

Φ is the standard normal cumulative distribution function and cannot be written in closed form; this calculator computes Φ with the error function (erf) and Φ⁻¹ with the Acklam rational approximation plus one Halley refinement step. The results match Excel's NORM.DIST / NORM.INV and R's pnorm / qnorm.

How to Calculate

  1. Enter the mean (μ) and the standard deviation (σ) of the distribution; σ must be greater than 0.
  2. Type the x value whose probability you are after; the calculator shows the z = (x − μ)/σ transform step by step.
  3. For an interval probability, enter the lower (a) and upper (b) bounds; leaving one blank treats that side as infinite.
  4. For the inverse direction, enter the cumulative probability (q): the x satisfying P(X < x) = q is computed.
  5. Read the left tail, right tail, and interval probabilities, and confirm those areas visually on the shaded curves.
  6. Take the common z and x equivalents from the percentile table to set your cut-off points.

Worked Examples

IQ score: the probability of falling below 115

In the N(100; 15²) distribution, x = 115 gives z = 1.0000. P(X < 115) = 0.8413 (84.13%) and P(X > 115) = 0.1587. P(85 < X < 115) = 0.6827, which is exactly the 68.27% the empirical rule gives for μ ± σ. In the inverse direction, the boundary of the 95th percentile is x = 124.6728.

z-score — z = (x − μ) / σ: 1.0000 · Left tail — P(X < 115): 0.8413 · Right tail — P(X > 115): 0.1587

Inverse: the exam score needed for the top 10%

On an exam with a mean of 65 and a standard deviation of 12, a score of 80 has a z-score of 1.2500 and P(X < 80) = 0.8944, so 80 points corresponds to roughly the 89.44th percentile. The score required to reach the top 10% is x = 80.3786 for q = 0.90. The probability of scoring between 60 and 70 is 0.3231.

z-score — z = (x − μ) / σ: 1.2500 · Left tail — P(X < 80): 0.8944 · Right tail — P(X > 80): 0.1056

Quality control: the share of production outside tolerance

On a filling line with a target weight of 500 g and a standard deviation of 5 g, 510 g has a z-score of 2.0000 and P(X > 510) = 0.0228. The probability of staying inside the 492-508 g tolerance is P(492 < X < 508) = 0.8904, that is 89.04% of the units; the rest are produced outside tolerance. The 99th percentile boundary is 511.6317 g.

z-score — z = (x − μ) / σ: 2.0000 · Left tail — P(X < 510): 0.9772 · Right tail — P(X > 510): 0.0228

Frequently Asked Questions

What is a z-score and why compute it?
A z-score tells you how many standard deviations a value sits from the mean: z = (x − μ)/σ. The transform reduces distributions with different means and standard deviations to a single scale (the standard normal, N(0,1)), so probabilities can be read from one table. A positive z is above the mean and a negative z is below it.
Is there a difference between P(X < x) and P(X ≤ x)?
Not for continuous distributions. In the normal distribution the probability of any single point is zero (P(X = x) = 0), so including or excluding equality changes nothing. The distinction only matters for discrete distributions such as the binomial and Poisson, where P(X ≤ k) and P(X < k) differ by exactly P(X = k).
How do I compute a normal probability in Excel, SPSS, and R?
In Excel use =NORM.DIST(x, μ, σ, TRUE) for the left tail and =NORM.INV(probability, μ, σ) for the inverse; NORM.S.DIST and NORM.S.INV handle the standard normal. In R, pnorm(x, mu, sigma) and qnorm(q, mu, sigma) give the same results, as do scipy.stats.norm.cdf/ppf in Python. In SPSS the functions are CDF.NORMAL(x, μ, σ) and IDF.NORMAL(q, μ, σ).
What exactly does the empirical rule (68-95-99.7) say?
In a normal distribution about 68.27% of the values lie within μ ± σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ. The widely quoted 95% interval is in fact μ ± 1.96σ rather than μ ± 2σ; the two should not be confused. The rule holds only when the distribution really is close to normal.
Are these calculations valid if my data are not normally distributed?
Not directly. With clear skewness, bimodality, or strong outliers, the tail probabilities the normal model produces become misleading. Check the assumption first with a Q-Q plot and a normality test such as Shapiro-Wilk; apply a transformation if needed, or switch to nonparametric methods.
What should I do for the distribution of a sample mean?
By the central limit theorem, as n grows the sample mean approaches X̄ ≈ N(μ, σ²/n). If it is the distribution of the mean you care about, enter the standard error (σ/√n) instead of σ. For example, with σ = 15 and n = 25 you would type 15/√25 = 3 into the standard deviation field.
Can I model a quantity that cannot be negative with a normal distribution?
Because the support of the normal distribution is (−∞, +∞), it assigns probability to negative values as well. When μ is at least three times σ, that probability is negligible and the model is usable in practice. Otherwise (income, duration, count data, for instance) a distribution with bounded support such as the log-normal, exponential, or Poisson is the better choice.