The Z-Score Calculator helps you standardise values, find raw values from z-scores, and determine probabilities from the standard normal distribution. A z-score tells you how many standard deviations a data point is from the mean.
Enter your values and the calculator instantly computes the result with a step-by-step breakdown. The interactive bell curve visualisation highlights the region of interest, making it easy to understand where your value falls within the distribution.
Your calculations will appear here
A z-score (also called a standard score) measures how many standard deviations a data point is above or below the mean of a distribution. The formula is z = (x - mu) / sigma, where x is the raw value, mu is the population mean, and sigma is the population standard deviation.
Z-scores allow you to compare values from different distributions on a common scale. A z-score of 0 means the value equals the mean. Positive z-scores indicate values above the mean, and negative z-scores indicate values below it. Roughly 68% of values fall within one standard deviation of the mean (z between -1 and 1), 95% within two standard deviations, and 99.7% within three.
The standard normal distribution has mean 0 and standard deviation 1. Any normal distribution can be converted to the standard normal by computing z-scores. The cumulative distribution function (CDF) gives P(Z < z), the probability that a standard normal variable is less than z.
Problema: A student scored 85 on a test where the class mean is 70 and the standard deviation is 10. What is their z-score?
Solucion: z = (85 - 70) / 10 = 15 / 10 = 1.5
Respuesta: z = 1.5 (the student scored 1.5 standard deviations above the mean)
Problema: In a distribution with mean 100 and standard deviation 15, what value corresponds to z = -2?
Solucion: x = mu + z * sigma = 100 + (-2) * 15 = 100 - 30 = 70
Respuesta: x = 70
Problema: What proportion of values fall below z = 1.96 in a standard normal distribution?
Solucion: Look up z = 1.96 in the standard normal table (or use the CDF): P(Z < 1.96) = 0.9750.
Respuesta: Approximately 97.5% of values fall below z = 1.96
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