Calculate probabilities, z-scores, and percentiles for normal distributions. Visualize shaded areas under the bell curve.
Translate a raw value, probability, or percentile into its corresponding position on a normal distribution. Enter the distribution mean and standard deviation, then choose whether the desired area lies below a cutoff, above it, or between two values. The calculator relates raw values to z-scores and shows the probability as a shaded region under the bell curve. The most common mistake is selecting the wrong tail: a cumulative probability below a value and the probability above that value are complements, not the same result.
A concrete input and expected output from the current implementation.
Input
Mean: 100 Standard deviation: 15 Find P(X <= 115)
Expected output
z-score: 1 P(X <= 115) approximately 0.8413447, or 84.13447%
The cutoff is one standard deviation above the mean because (115 - 100) / 15 = 1. The standard normal cumulative probability at z = 1 is approximately 0.8413447.
For a normal variable with mean m and positive standard deviation s, the corresponding standard score is z = (x - m) / s. A positive z-score places x above the mean, a negative score places it below the mean, and z = 0 places it at the center. This conversion allows probabilities from different normal distributions to be evaluated on the same standard normal scale.
A below-cutoff result is the cumulative area to the left of that value. An above-cutoff result is one minus the left-tail area. For an interval, the left-tail probability at the lower bound is subtracted from the left-tail probability at the upper bound. A percentile calculation reverses this relationship, locating the value whose left-tail area matches the requested percentage.
No, not for an ideal continuous normal distribution. The probability of observing any one exact value is zero, so P(X < x) equals P(X <= x). This does not mean an observed measurement is impossible; it means probability is assigned to ranges of values rather than to an infinitely precise single point.
Not automatically. The calculator returns probabilities implied by a normal distribution with the mean and standard deviation you enter. Strong skewness, heavy tails, multiple peaks, truncation, or many extreme observations can make that model unsuitable. Check the distributional assumption or use a model that matches the data before interpreting the calculated area as an empirical frequency.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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