How this instrument works
The normal distribution is the familiar symmetric, bell-shaped curve describing how many natural and measured quantities are distributed — heights, test scores, and measurement errors all cluster more densely near their mean and thin out symmetrically toward the extremes. It's fully described by just two numbers: the mean (μ), which sets the center of the bell, and the standard deviation (σ), which sets how wide or narrow it is.
This calculator computes three related figures for any value x entered. The z-score, z = (x−μ)/σ, converts x into a distribution-independent number of standard deviations from the mean. The probability density function (PDF) gives the height of the bell curve at x — useful for comparing relative likelihood, though on its own it's not a probability. The cumulative distribution function (CDF), P(X ≤ x), is a true probability: the chance a random draw from this distribution falls at or below x, computed here using the Abramowitz & Stegun erf approximation, which matches the exact mathematical value to within about 7 parts in 100 million.
Because the CDF has no simple closed-form algebraic formula — it's defined as an integral under the bell curve that must be computed numerically or via a well-established approximation — every normal-distribution calculator, spreadsheet function, or statistics table ultimately relies on some numerical method like this one rather than a plain equation evaluated by hand.
- Enter the value being evaluated into x.
- Enter the distribution's mean into μ (mu) — use 0 for the standard normal distribution.
- Enter the distribution's standard deviation into σ (sigma) — use 1 for the standard normal distribution; it must be greater than zero.
- Read z — how many standard deviations x sits from the mean.
- Read pdf, the curve's height at x, and cdf — the probability a random draw falls at or below x.
Worked example — the famous z = 1.96 landmark
You want P(X ≤ 1.96) for the standard normal distribution (μ = 0, σ = 1). Enter 1.96 into x, 0 into μ, and 1 into σ.
z comes out to exactly 1.96, since x already equals the z-score when μ = 0 and σ = 1. The cumulative probability is P(X ≤ 1.96) ≈ 0.9750 — the widely cited standard-normal-table value everyone learns as the boundary of the middle 95% of a normal distribution, since the remaining 5% splits into 2.5% below -1.96 and 2.5% above +1.96. The density at that point is f(1.96) ≈ 0.058441.
Try it again on a real-world scale: an IQ test with μ = 100 and σ = 15, evaluated at x = 115. z = (115−100)/15 = 1.0 exactly, and P(X ≤ 115) ≈ 0.8413 — another widely cited standard-normal-table figure, marking the boundary of the top roughly 16% of scores.
Questions
What's the difference between the PDF and the CDF?
The PDF, probability density function, gives the height of the bell curve at a specific point — useful for comparing how relatively likely different values are, but not itself a probability, since it can exceed 1 for narrow, small-sigma distributions. The CDF, cumulative distribution function, is a genuine probability: it's the area under the curve up to x, representing P(X ≤ x), and it always falls between 0 and 1.
Why can't the CDF be computed with a simple formula?
Because the normal distribution's CDF is defined as the integral of its bell-curve PDF, and that particular integral has no closed-form algebraic solution — it can only be evaluated numerically or approximated. This calculator uses the Abramowitz & Stegun erf approximation, a well-established numerical method accurate to within about 7 parts in 100 million across the range used here, which is why statistics tables and software all rely on some numerical technique rather than a plain equation.
What does it mean if my z-score is negative?
A negative z-score simply means x falls below the mean, not that anything is unusual on its own — z = -1 means x is one standard deviation below average. The CDF at a negative z is correspondingly less than 0.5, since less than half of the distribution's area lies below a point that's below the mean.
What's the empirical rule, and does this calculator confirm it?
The empirical rule says that for a normal distribution, about 68% of values fall within ±1 standard deviation of the mean, about 95% within ±2, and about 99.7% within ±3. This calculator confirms it directly: entering x one sigma above the mean gives a CDF near 0.8413, and one sigma below gives a CDF near 0.1587 — the difference between those two, about 68.3%, matches the empirical rule.
Does this work for any mean and standard deviation, not just the standard normal?
Yes — enter any μ and σ describing your specific distribution, whether test scores, measurements, or anything approximately bell-shaped, and the calculator standardizes internally by computing z = (x−μ)/σ before applying the same CDF approximation. There's no need to convert to the standard normal first.