What is a Z-Score?
A z-score, also called a standard score, tells you how many standard deviations a value is from the mean of its distribution. This z score calculator converts any raw value into a z-score using the formula Z = (x - μ) / σ, where x is your value, μ is the mean, and σ is the standard deviation. A positive z-score means the value is above average, while a negative z-score means it is below average.
How to Interpret Z-Scores
A z-score of 0 means the value is exactly at the mean. A z-score of 1 means the value is one standard deviation above the mean, which corresponds to roughly the 84th percentile in a normal distribution. A z-score of -1 falls at about the 16th percentile. Because z-scores are standardized, they let you compare values from completely different scales, such as comparing a student's height to their test score.
Z-Scores and Percentiles
In a normal distribution, z-scores map directly to percentile ranks. A z-score of 1.96 falls at the 97.5th percentile, and a z-score of -1.96 falls at the 2.5th percentile. These values are especially important in statistics because the range between -1.96 and +1.96 covers 95% of the data, which is the foundation of the common 95% confidence interval.
P-Values in Hypothesis Testing
In statistical tests, the z-score is converted into a p-value that represents the probability of observing a result at least as extreme as yours if the null hypothesis were true. This calculator reports both the one-tailed and two-tailed p-value so you can apply the correct interpretation for your test. A smaller p-value provides stronger evidence against the null hypothesis.
For a deeper look at how spread out your data is before computing a z-score, use our standard deviation calculator. Understanding the standard deviation is essential because it is the denominator in the z-score formula.