Z-Score Calculator
Calculate a z-score (standard score) from a value, mean, and standard deviation, and see the percentile and probability it corresponds to on the normal distribution.
How to Use This Calculator
- Enter your value — the raw data point (x) you want to convert to a z-score
- Enter the mean and standard deviation — of the data set or distribution the value belongs to
- Click Calculate — instantly see the z-score, percentile, and probability breakdown
Enter Your Values
📐 What Is a Z-Score?
- A z-score (or standard score) tells you how many standard deviations a value is from the mean.
- z = 0 means the value equals the mean. Positive z means above the mean. Negative z means below the mean.
- Z-scores let you compare values from different scales or distributions on the same standardized footing.
🧮 Formula Reference
z = (x − μ) ÷ σ
Where x is your raw value, μ (mu) is the mean, and σ (sigma) is the standard deviation. The percentile is found by converting the z-score to a cumulative probability on the standard normal distribution.
📊 Common Z-Score Benchmarks
- z = 0 → 50th percentile (the mean)
- z = +1 → ~84th percentile
- z = +1.645 → 95th percentile
- z = +1.96 → 97.5th percentile (common 95% confidence interval bound)
- z = +2 → ~97.7th percentile
- |z| > 3 → often treated as a statistical outlier
Frequently Asked Questions
What is a z-score in simple terms?
A z-score (or standard score) tells you how many standard deviations a value is above or below the mean of its data set. A z-score of 0 means the value equals the mean; +1 means it's one standard deviation above the mean; -1 means one standard deviation below.
How do you calculate a z-score?
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) ÷ σ. For example, if a test score is 85, the mean is 75, and the standard deviation is 10, the z-score is (85 − 75) ÷ 10 = 1.0.
How do I convert a z-score to a percentile?
Look up the z-score on a standard normal distribution table (z-table), or use a calculator that applies the cumulative distribution function (CDF). A z-score of 0 is the 50th percentile, +1.0 is about the 84th percentile, and +1.96 is about the 97.5th percentile.
What does a negative z-score mean?
A negative z-score means the value is below the mean. For example, a z-score of -1.5 means the value is 1.5 standard deviations below the average, which corresponds to roughly the 7th percentile.
What is considered a high or unusual z-score?
In most contexts, a z-score with an absolute value greater than 2 is considered somewhat unusual, and greater than 3 is often treated as a statistical outlier, since over 99.7% of values in a normal distribution fall within 3 standard deviations of the mean.
What's the difference between a z-score and standard deviation?
Standard deviation measures how spread out an entire data set is. A z-score uses that standard deviation to describe where one specific value falls relative to the mean. You need the standard deviation first to calculate any z-score.