Convert any value to a z-score, find its percentile rank, and calculate tail probabilities.
A z-score (or standard score) tells you how many standard deviations a value is above or below the mean of a distribution. The formula is: z = (x − μ) / σ
A z-score of +1.5 means the value is 1.5 standard deviations above the mean. A z-score of −2 means it is 2 standard deviations below the mean. Z-scores allow you to compare values from different distributions on a common scale.
The percentile tells you what percentage of the population scores at or below a given value. A z-score of 0 is the 50th percentile — exactly half the distribution is below it. A z-score of +1 is approximately the 84th percentile.
Example: if a student scores at the 92nd percentile on a standardised test, 92% of students scored at or below them.
Known Mean & SD: Use this when you know the population parameters — for example, IQ tests have a defined mean of 100 and SD of 15, or when your textbook provides the distribution parameters.
From raw data: Use this when you have a dataset and want to evaluate where a specific value sits relative to that data. The calculator will estimate the mean and SD from your data first.
Left tail P(X ≤ x): The probability that a randomly chosen value falls at or below your value. This is the same as the percentile expressed as a proportion.
Right tail P(X ≥ x): The probability that a randomly chosen value falls at or above your value.
Two-tailed P(|X| ≥ |z|): The probability of being this extreme in either direction. Used in hypothesis testing.
In a standard normal distribution:
Values with |z| > 2.5–3 are commonly flagged as potential outliers.