Enter a value, a mean, and a standard deviation to get the exact area under the bell curve on either side.
The panel recalculates the moment you change a field; no data leaves your browser.
The normal distribution is the familiar bell curve, symmetric around the mean. This calculator converts your value to a z-score, then finds the area under the curve to its left using the error function, which is mathematically equivalent to the cumulative probability.
For the defaults (x = 1.5, mean = 0, SD = 1): z = 1.5000; P(X < 1.5) = 93.32%; P(X > 1.5) = 6.68%. A z-score of 1.5 lands close to the 93rd percentile of the distribution.
The familiar "68-95-99.7" shorthand rounds off; here is what this calculator's own erf-based method returns at exactly 1, 2, and 3 standard deviations from the mean:
Notice these are slightly different from the rounded "68-95-99.7" most textbooks quote; the difference shows up past the second decimal. See the z-score calculator for the percentile framing of the same math, or the probability calculator for binomial and Poisson alternatives when your data isn't continuous.
It's a rounded shorthand for how much of a normal distribution sits within 1, 2, and 3 standard deviations of the mean. This calculator doesn't rely on the rounded version; it computes the exact cumulative probability with the error function for whatever x, mean, and SD you enter, so results for values in between the whole-SD marks are just as precise.
Because every outcome falls on one side of x or the other; together they cover the entire distribution. The calculator computes P(X < x) directly and simply subtracts it from 1 to get P(X > x), so the two will always sum to exactly 100% by construction.
No, not reliably. Every formula here assumes a symmetric, bell-shaped distribution. Skewed data, like income or wait times, will give you a technically computed but practically misleading answer if you force it through a normal-distribution calculator instead of checking the shape first.
The calculator stops and shows placeholder dashes instead of a number. A standard deviation of zero means every value in the distribution is identical, which breaks the z-score formula's division step, so the tool declines to guess rather than return a meaningless result.