Enter a value x, a mean, and a standard deviation to get the z-score and the probability that X is less than or greater than x.
Estimates only.
The normal distribution is the familiar bell curve, symmetric around the mean. Given any value x, this calculator converts it to a z-score and looks up the cumulative probability using the error function (erf), which is equivalent to the area under the curve up to x.
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%. This means that in a standard normal distribution, about 93.3% of values fall below 1.5.
The normal distribution models many real-world phenomena where values cluster around an average, such as heights, test scores, measurement errors, and financial returns. It is the foundation of z-tests, t-tests, confidence intervals, and many other statistical methods. When a sample size is large enough, the Central Limit Theorem guarantees the sampling distribution of the mean is approximately normal regardless of the original distribution.
In a normal distribution, approximately 68% of values fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. This is also called the empirical rule. For example, if IQ scores have mean 100 and SD 15, about 95% of people have IQs between 70 and 130 (two SDs on each side).
Convert the value to a z-score using z = (x - mean) / SD, then look up the cumulative probability from a standard normal table or use the CDF formula. P(X less than x) is the area to the left of x under the bell curve. P(X greater than x) is 1 minus that area. This calculator does both steps automatically.
A z-score of 1.5 means the value is 1.5 standard deviations above the mean. In a standard normal distribution, about 93.32% of values fall below this point and about 6.68% fall above it. A z-score of 1.5 corresponds roughly to the 93rd percentile.