What the normal curve gives you
The normal distribution assigns probability to a continuous quantity through the area under a bell-shaped curve. The mean fixes where the curve is centred and the standard deviation fixes how wide it is; between them those two numbers determine the entire distribution, which is unusual and is a large part of why the normal is so heavily used.
Probability is area. The chance of landing between 85 and 115 is the area under the curve between those two points, and it comes out at 0.6827 when the mean is 100 and the standard deviation is 15. The chance of landing below 85 is the area to the left of 85. The chance of landing on exactly 100 is zero, because a single point has no width - which sounds pedantic until you notice it is the reason continuous and discrete distributions need different handling.
The curve owes its ubiquity to the central limit theorem: sums and averages of many independent contributions tend towards a normal shape whatever the individual contributions look like. Measurement error, sample means, and quantities built from many small additive effects are therefore often close to normal. Quantities built from multiplicative effects, such as incomes and asset prices, are not - they are closer to lognormal, and treating them as normal understates the extremes badly.
The density, the cumulative function, and standardising
The density is f(x) = exp(-(x - mu)2 / (2 sigma2)) divided by sigma times the square root of 2 pi. Read it in pieces. The squared term in the exponent makes the curve symmetric about mu and makes the height fall off rapidly - at three standard deviations the exponential factor is e-4.5, about 1.1% of its peak. Dividing by sigma in the exponent sets the scale, and the constant out front is exactly what is needed to make the total area equal 1.
Note that f(x) is a density, not a probability. Its units are probability per unit of x, so on a distribution measured in millimetres the density is per millimetre and can exceed 1 when sigma is small. Only areas are probabilities.
To get an area you need the cumulative function Phi, which is the integral of the density from minus infinity up to a point. That integral has no closed form in elementary functions, which is why printed z tables exist and why software evaluates it with an error-function approximation instead.
Standardising is what makes a single table sufficient for every normal distribution. Convert your bound to a z-score with z = (x - mu) / sigma, and the probability depends only on z. So P(X < 85) with mu = 100 and sigma = 15 equals P(Z < -1), which any table gives as 0.1587. The area between two bounds is then Phi(zb) - Phi(za), and the right tail is 1 - Phi(zb) by symmetry of the total area.
Symmetry gives you two more shortcuts worth remembering: Phi(-z) = 1 - Phi(z), and the area between -z and +z is 2Phi(z) - 1. Older z tables list only positive z for exactly that reason.
Worked example: IQ scores between 85 and 115
IQ is scaled to a mean of 100 and a standard deviation of 15. What share of people score between 85 and 115?
- Standardise the lower bound. za = (85 - 100) / 15 = -1.00.
- Standardise the upper bound. zb = (115 - 100) / 15 = +1.00.
- Look up the left areas. Phi(-1.00) = 0.158655 and Phi(1.00) = 0.841345.
- Subtract. 0.841345 - 0.158655 = 0.682689, so about 68.27% of people fall in that band.
- Read the tails. Below 85 is 0.158655 and above 115 is 1 - 0.841345 = 0.158655. The two tails are equal because the bounds are symmetric about the mean, and 0.682689 + 2 x 0.158655 = 1 exactly.
Now change the question to an asymmetric one: what share score above 130? z = (130 - 100) / 15 = 2.00, and the right tail is 1 - Phi(2.00) = 1 - 0.977250 = 0.022750, about 1 person in 44.
Finally the density at 85, which is what the curve's height is there. Compute exp(-0.5 x (-1)2) = exp(-0.5) = 0.606531, then divide by sigma x sqrt(2 pi) = 15 x 2.506628 = 37.599424. That gives 0.016131 per IQ point. Multiply by a narrow width to approximate a probability: the chance of scoring between 84.5 and 85.5 is roughly 0.016131 x 1 = 1.61%, and the exact interval calculation agrees to four decimal places at 0.01613 - as it should over a width that small.
Reading the areas, and checking normality first
Every probability here is a proportion of the population under the assumed model. Read them as long-run frequencies: a right tail of 0.0228 means about 23 in 1,000 exceed the bound, if the normal model holds.
The empirical rule is the fast mental version. About 68% of the distribution lies within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The exact figures are 68.27%, 95.45% and 99.73%, and the table on this page prints them against your own bounds. The commonly quoted 95% interval uses 1.96 standard deviations rather than 2, which is where the difference between 95.00% and 95.45% comes from.
Before trusting any of this, check that a normal model is defensible. Three quick tests: is the quantity roughly symmetric? Can it plausibly take values several standard deviations below the mean without becoming impossible - a time, a count or a price cannot go negative? And are the extremes you have already observed consistent with the thin tails the normal implies? A distribution that produces a five-sigma event every few months is not normal, whatever the histogram looks like near the centre.
The tail behaviour is where the model is most often wrong and most often consequential. The normal density falls off as exp(-z2/2), which is extraordinarily fast: a six-sigma event has probability about 2 in a billion under the model. Financial returns, network latencies and insurance losses all produce such events far more often, which means normal-based risk estimates for those quantities are systematically too optimistic. For empirical data, reading a percentile straight off the observations makes no distributional assumption at all.
Standard normal areas at the values that keep appearing
| z | Phi(z), left area | Right tail | Area between -z and +z |
|---|---|---|---|
| 0.00 | 0.500000 | 0.500000 | 0.000000 |
| 0.50 | 0.691462 | 0.308538 | 0.382925 |
| 1.00 | 0.841345 | 0.158655 | 0.682689 |
| 1.28 | 0.899727 | 0.100273 | 0.799455 |
| 1.645 | 0.950015 | 0.049985 | 0.900030 |
| 1.96 | 0.975002 | 0.024998 | 0.950004 |
| 2.00 | 0.977250 | 0.022750 | 0.954500 |
| 2.576 | 0.995005 | 0.004995 | 0.990010 |
| 3.00 | 0.998650 | 0.001350 | 0.997300 |
| 4.00 | 0.999968 | 0.000032 | 0.999937 |
Every entry is the standard normal cumulative function evaluated at that z. The 1.645, 1.96 and 2.576 rows are the one- and two-tailed critical values behind the 90%, 95% and 99% confidence levels.
Where normal calculations go wrong
- Assuming normality because the sample size is large. The central limit theorem describes the distribution of sample means, not of individual observations. A large sample of a skewed quantity is still skewed.
- Applying the model to a quantity with a hard boundary. Times, counts, prices and concentrations cannot go below zero. If the mean is less than about three standard deviations above zero, the normal model assigns real probability to impossible values.
- Reading the density as a probability. f(x) is a height per unit of x and can exceed 1 when sigma is small. Only the area beneath it is a probability.
- Trusting the far tail. The normal has exponentially thin tails, so it is the least reliable exactly where the stakes are highest. Extreme value theory, not the normal curve, is the right tool for rare maxima.
- Using a discrete count without a continuity correction. Approximating a binomial count of 12 or more with a normal curve should integrate from 11.5, not 12. Skipping the half-unit shift biases the tail.
- Mixing one-tailed and two-tailed areas. The area above 1.96 is 2.5%; the area outside plus or minus 1.96 is 5%. Both are correct answers to different questions, and confusing them halves or doubles a p-value.
Related distributions and tools
The standard normal is this distribution with mu = 0 and sigma = 1. Standardising any normal quantity with the z-score calculator maps it onto that single curve, which is why one table suffices for all of them. Going the other way, from a probability back to a cut-off, is the inverse problem the z critical value calculator solves.
Several distributions are close relatives. The t distribution has the same bell shape with heavier tails and is what you use when sigma has been estimated from a small sample. The chi-square distribution describes sums of squared standard normals and underlies variance tests. The lognormal describes a quantity whose logarithm is normal, which fits incomes, particle sizes and asset prices far better than the normal does.
For discrete counts, the binomial distribution calculator gives exact answers where a normal approximation would only be close, and it is worth using directly rather than approximating now that the exact computation is cheap. For empirical data with no assumed shape, the percentile calculator and the interquartile range calculator describe the distribution you actually have rather than one you have assumed.
Two applied uses deserve mention. In quality engineering, process capability indices are ratios of specification width to six standard deviations, and every parts-per-million figure quoted from them is a normal tail area of the kind this page computes. In finance, value at risk under a normal assumption is a quantile of exactly this curve - and the persistent underestimation of extreme losses is the clearest practical demonstration that real data have fatter tails than the model.