Normal Distribution Calculator
For a normally distributed variable X ~ N(μ, σ), this calculator finds the probability of landing below or above a value x. It converts x to a z-score and evaluates the cumulative distribution function Φ(z).
Standardising, then reading the area
The normal distribution is the bell-shaped curve that measurement errors, biological variation and averages of almost anything tend to follow. Its usefulness rests on a single practical fact: every normal distribution is the same shape, differing only in where it is centred and how wide it is.
That is what makes one table, or one function, enough for all of them. This calculator converts a value to its position on the standard curve and reports the probability of falling below it or above it.
The two steps are worth keeping separate. Standardising turns a value into a z-score — how many standard deviations it sits from the mean — and that number is comparable across completely different quantities. Only then is the probability read.
Reporting both tail probabilities rather than one saves a subtraction and, more usefully, makes the complementary relationship visible. They always sum to 1, which is a check available on every result at no cost.
How to use this calculator
- Enter the value of interest The point the probability is measured up to. It carries the units of whatever is being measured, and may be above or below the mean.
- Enter the mean The centre of the distribution, where the curve peaks. Because the shape is symmetric, this is also the median.
- Enter the standard deviation It must be positive. It sets the width: a larger value spreads the same total probability over a wider range and flattens the peak.
- Read the z-score before the probabilities It says how unusual the value is in a way the raw number cannot. Roughly, beyond two is uncommon and beyond three is rare.
The formula, and where it comes from
z = (x − μ) / σ P(X ≤ x) = Φ(z) = ½·(1 + erf(z/√2)) P(X ≥ x) = 1 − Φ(z)
Standardising subtracts the mean to move the centre to zero, then divides by the standard deviation to make the width one unit. Any normal distribution becomes the same standard curve under that transformation, which is the whole reason a single table ever sufficed.
The cumulative function gives the area under the curve to the left of a point — the probability of landing at or below it. It has no expression in elementary functions, which is why it is written in terms of the error function rather than as a formula you could evaluate by hand.
The error function itself is computed from a published rational approximation rather than by integration. That is the standard approach, and it delivers several correct decimal places across the whole range, which is well beyond what any table offered.
The upper tail is obtained by subtracting from 1 rather than by a second evaluation. Total probability is 1, so the two tails are complementary by definition and cannot disagree.
What each input means
- x Value — form field “Value x”
- The point of interest, in the units of the measured quantity.
- μ Mean — form field “Mean μ”
- The centre of the distribution. Shifting it moves the whole curve without changing its shape.
- σ Standard deviation — form field “Standard deviation σ”
- The width, which must be strictly positive. A value of zero would describe a quantity with no variation and is rejected.
- z Standard score
- How many standard deviations the value lies from the mean. Negative below the mean, positive above it, and unitless either way.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A point on the standard curve
A value of 1.2 on a distribution already centred at zero with unit width. Standardising changes nothing, so the value is its own z-score.
Inputs Value x = 1.2, Mean μ = 0, Standard deviation σ = 1
- Distribution X ~ N(μ = 0, σ = 1)
- Z-score z = (x − μ) / σ = 1.2
- P(X ≤ x) Φ(z) = 0.88493
- P(X ≥ x) 1 − Φ(z) = 0.11507
Result P(X ≤ 1.2) = 0.88493, P(X ≥ 1.2) = 0.11507
The lower probability is a little under 0.9, so roughly a ninth of the distribution lies above this point. A z-score of 1.2 is mildly high rather than remarkable.
The two reported probabilities sum to 1 exactly. That is guaranteed by how the upper tail is computed, which makes the check a confirmation that the right value was read rather than of the arithmetic.
A score on a scaled distribution
A value of 115 where the mean is 100 and the standard deviation 15 — the conventional shape of many standardised test scales.
Inputs Value x = 115, Mean μ = 100, Standard deviation σ = 15
- Distribution X ~ N(μ = 100, σ = 15)
- Z-score z = (x − μ) / σ = 1
- P(X ≤ x) Φ(z) = 0.841345
- P(X ≥ x) 1 − Φ(z) = 0.158655
Result P(X ≤ 115) = 0.841345, P(X ≥ 115) = 0.158655
The z-score is exactly 1, since the value sits one standard deviation above the mean. The probability below is about 0.84, so roughly one in six results exceeds it.
That probability is identical to the one for a raw score of 1 on the standard curve. Standardising erases the units entirely, which is what allows results from different scales to be compared at all.
Reading the result
What a z-score conveys
Distance from the mean measured in standard deviations, with no units attached. About 68% of a normal distribution lies within one, 95% within two and 99.7% within three — the rule of thumb that turns a z-score into an instinct.
The probability of an exact value is zero
For a continuous distribution any single point has no probability at all; only intervals do. That is why the strict and inclusive versions of these tails are identical, and it is not a rounding artefact.
Probabilities between two points
Run the calculation twice and subtract the smaller lower-tail probability from the larger. There is no dedicated interval mode, but two evaluations answer the question exactly.
When you would use this
Judging how unusual an observation is
Given a mean and standard deviation from historical data, the upper-tail probability says how often a value that extreme should occur — the basic move behind quality control limits and reference ranges.
Converting between different scales
Two measurements from distributions with different means and widths become comparable once both are standardised, which is what a z-score is for.
Assumptions and limitations
What this calculator assumes
- The quantity is normally distributed with the mean and standard deviation supplied.
- The standard deviation is strictly positive.
- The cumulative probability is computed from an approximation to the error function, accurate to several decimal places.
- The upper tail is derived as one minus the lower tail, so the two always sum to 1.
Where it stops being the right tool
- One point at a time: a probability between two values requires two runs and a subtraction.
- The inverse direction is not offered, so finding the value corresponding to a given probability needs a different tool.
- No probability density is reported, only the cumulative probabilities.
- Normality is assumed rather than tested; the calculation says nothing about whether the data warrants it.
Common mistakes
Entering the variance instead of the standard deviation
Why it happens. Both describe spread and both are routinely quoted, but the variance is the square. Using it makes the distribution far wider than intended and every probability closer to a half.
How to avoid it. Take the square root first if what you have is a variance. The z-score line will look implausibly small when the width has been overstated.
Reading the wrong tail
Why it happens. Both are reported adjacently and the question is often phrased as at least or more than, which corresponds to the second rather than the first.
How to avoid it. Match the wording: below or at most is the lower tail, above or at least is the upper. Whichever you take, the two must sum to 1.
Treating a probability near zero as impossible
Why it happens. A z-score beyond four produces a tail so small it displays as effectively nothing, which reads as a hard bound on the distribution.
How to avoid it. Remember the normal curve extends without limit in both directions. Very small is not zero, and for a large enough sample such values do occur.
Key terms
Frequently asked questions
What is the cumulative probability?
The area under the curve to the left of the value — the chance of observing something at or below it. Because the total area is 1, subtracting it from 1 gives the chance of exceeding the value instead.
How accurate is the result?
The cumulative probability comes from a published rational approximation to the error function, correct to several decimal places across the range. That is more precision than a printed table offers and ample for any ordinary use.
What is the standard normal distribution?
The normal distribution with mean 0 and standard deviation 1. Any normal variable becomes standard once the mean is subtracted and the result divided by the standard deviation, which is why one curve serves for all of them.
How do I find the probability between two values?
Run the calculation at each value and subtract the smaller lower-tail probability from the larger. The difference is the area between the two points.