Free normal distribution calculator. Enter mean, standard deviation and x for the z-score, pdf, and probabilities P(X<=x), P(X>=x), and P(x1<=X<=x2).
Give the calculator a mean μ, a standard deviation σ and a value x, and it returns the z-score, the density f(x), the area to the left P(X ≤ x), the area to the right P(X ≥ x), and x's percentile. Fill in the optional fourth field x₂ and it also gives P(x ≤ X ≤ x₂), the area between the two bounds. Every step is printed with your own numbers substituted in, and the 68–95–99.7 landmarks are translated into your units rather than left abstract.
It runs entirely in your browser — no upload, no account. The tail areas come from an Abramowitz & Stegun error-function approximation (formula 7.1.26) with a maximum absolute error around 1.5×10⁻⁷, which is finer than any printed z-table you would otherwise reach for.
The normal distribution is a whole family of curves, one for every pair of μ and σ. The z-score is what collapses that family down to a single reference curve so a table or a formula can serve all of them:
z = (x − μ) / σ
It answers one question: how many standard deviations is x from the mean, and on which side? Subtracting μ recentres the scale at zero; dividing by σ rescales it so one unit is one standard deviation. A z of 0 sits exactly at the mean. A z of +1.5 is one and a half standard deviations above it. A negative z is below.
Worked through: a test has μ = 72 and σ = 8, and a student scored 85. Then z = (85 − 72) / 8 = 13 / 8 = 1.625. The calculator reports P(X ≤ 85) ≈ 0.947919, so that score is at roughly the 94.79th percentile, and P(X ≥ 85) ≈ 0.052081 — about 5.2% of students did better.
Because the z-score strips out the original units, it is also the honest way to compare across different scales. A 1.625 on this test and a 1.625 on a different test with a different mean and spread describe the same relative standing, even though the raw marks have nothing in common. That is why the calculator shows z prominently rather than burying it.
For a continuous variable, probability is area, not height. This trips people up because the calculator also reports f(x), the probability density — the height of the curve at x — and that number is not a probability. In the example above f(85) ≈ 0.013317, which is not "a 1.3% chance of scoring 85". The chance of scoring exactly 85.000000… is zero. Density only becomes probability once you integrate it across an interval.
So every meaningful question about a normal variable is a question about an interval:
On the same test, P(65 ≤ X ≤ 85) ≈ 0.757132 — about 75.7% of students scored between 65 and 85, and therefore about 24.3% fell outside that band.
One consequence of "probability is area" that saves a lot of confusion: ≤ and < give the same answer for a continuous distribution. The single boundary point contributes zero area. Unlike a discrete count, you never need to worry about whether the endpoint is included.
The empirical rule says roughly 68% of the distribution lies within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. The calculator prints those three intervals converted into your own units, which is the form you can actually use. The exact figures:
| Interval | Area inside | Area in each tail |
|---|---|---|
| μ ± 1σ | 0.682689 | 0.158655 |
| μ ± 1.645σ | 0.900030 | 0.049985 |
| μ ± 1.96σ | 0.950004 | 0.024998 |
| μ ± 2σ | 0.954500 | 0.022750 |
| μ ± 2.576σ | 0.990005 | 0.004998 |
| μ ± 3σ | 0.997300 | 0.001350 |
The 1.96 row is the one behind every 95% confidence interval you have ever seen; the commonly quoted "2 sigma" is a rounding of it, and gives 95.45% rather than 95%. The 1.645 row is the one-sided 5% cutoff. Manufacturing's "six sigma" language reaches much further out than any of these rows — and that is precisely where the assumption of normality is least defensible, because you are asserting something about the extreme tail of a curve that your data almost certainly never sampled.
Use the rule for sanity checks, not for final numbers. If the calculator hands you a tail probability of 0.31 for a value you can see is above 2σ out, you have typed something wrong — a σ where a μ should be, most likely. That is the rule's real job.
Partly because it is genuinely common: quantities produced by many small independent contributions added together — measurement error, manufacturing tolerances, biological variation within a species — do tend towards a bell shape. The deeper reason is the central limit theorem, which says that the average of a sample tends to a normal distribution as the sample grows, largely regardless of the shape of the population it was drawn from. That is what makes the normal the working distribution for sample means, standard errors and confidence intervals even when the underlying data are visibly non-normal.
Two practical conveniences follow. The normal is fully described by just two numbers, μ and σ, so it is cheap to state and cheap to compute with. And it is closed under addition — sums and averages of normals stay normal — which makes the algebra tractable in a way most distributions are not.
The honest version, though, is that a fair share of normality assumptions are made because the software defaulted to it and nobody checked.
The normal curve is symmetric, unbounded in both directions, and has thin tails. Any of those can be wrong for your data, and the calculator has no way to detect it:
A related and quieter error: μ and σ are usually estimated from a sample rather than known. The calculator treats whatever you type as exact. With a small sample your σ is itself uncertain, and the t-distribution is the proper adjustment — not this tool.
When a count of successes has a large enough number of trials, its discrete distribution takes on a bell shape and the normal curve approximates it well. The usual working rule is that both np and n(1 − p) should be at least about 10; set μ = np and σ = √[np(1 − p)] and use this calculator.
Worked: 100 coin flips, what is P(X ≤ 60 heads)? Here μ = 50 and σ = √25 = 5. Entering x = 60 gives 0.977250. The exact binomial answer is 0.982400. Better still, apply the continuity correction — because a discrete count of 60 covers the continuous stretch up to 60.5 — and entering x = 60.5 gives 0.982136, within about 0.0003 of exact. That half-unit adjustment is the difference between a rough approximation and a very good one, and it is the step most often skipped.
For the exact discrete calculation with no approximation involved, use the binomial distribution calculator, which handles up to 1,000 trials directly.
| Field | Meaning | Requirement |
|---|---|---|
| μ | mean — the centre and peak of the curve | any finite number, including negatives |
| σ | standard deviation — the spread | strictly greater than 0 |
| x | the value you are asking about | any finite number |
| x₂ | optional second bound for a range | leave blank for tail probabilities only |
Two field-level cautions. σ must be the standard deviation, not the variance — if you have the variance, take its square root first, or the spread you model will be wrong by that factor. And σ must be positive; zero or negative is rejected, since a zero-spread "distribution" is a single point, not a curve.
To work in standard units instead of your own, set μ = 0 and σ = 1. The calculator then behaves as a standard normal z-table, and x is read directly as z — useful when a textbook question is already phrased in z.
A z-score measures how many standard deviations a value lies from the mean. It is calculated as z = (x − μ) / σ. A z-score of 0 is exactly average, +1 is one standard deviation above the mean, and −2 is two below. Z-scores put any normal distribution onto a common scale so values from different data sets can be compared directly.
Also called the empirical rule, it states that for a normal distribution about 68% of values fall within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three. It is a quick way to judge how unusual a value is without a full calculation.
The probability density function (pdf) gives the height of the bell curve at a point — it shows where values are concentrated but is not itself a probability. The cumulative distribution function (cdf), Φ(x), gives the probability that the variable is less than or equal to x, which equals the area under the curve to the left of x. Probabilities come from areas, not heights.
Compute the cumulative probability at each endpoint and subtract: P(x1 ≤ X ≤ x2) = Φ(x2) − Φ(x1). Enter your lower value as x and your upper value as x₂ in this calculator and it returns the area between them automatically.
The normal distribution is continuous, so probability is spread over a range as area under the curve rather than concentrated at points. The area above a single exact point has zero width and therefore zero probability. That is why meaningful probabilities are always stated over intervals, such as P(X ≤ x) or P(a ≤ X ≤ b).