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Z-Score Calculator
Use this free z-score calculator to see where a value stands compared to the average. Enter a value, the mean, and the standard deviation to get the z-score, the percentile, and a plain-English reading of the result. You can also work backwards from a z-score to find the value, or paste a list of raw numbers and have the calculator work out the mean and standard deviation for you.
How it works
A z-score answers one simple question: how unusual is this value? It does it by measuring the distance from the mean in units of standard deviation, so a raw score on any scale — test points, heights in inches, prices in dollars — becomes a number you can compare across completely different sets of data. A z-score of 0 means the value is exactly average. A positive z-score means above average, a negative one means below, and the size of the number tells you how far away it is.
To find a z-score, subtract the mean from the value, then divide by the standard deviation. The mean is the average of the group, and the standard deviation measures how spread out the values are — a small standard deviation means most values cluster near the mean, a large one means they are scattered. Dividing by the standard deviation is what makes the score scale-free: a value 15 points above a mean with a standard deviation of 10 is 1.5 standard deviations above average, wherever the numbers come from.
The percentile comes from the standard normal curve. If the data follows a normal distribution, the percentile is the share of all values that fall below your value — the area under the bell curve to the left of your z-score. This calculator computes that area with the Abramowitz–Stegun approximation, which is accurate to about one part in ten million, so the percentile it reports matches a statistics table to two decimal places. Roughly speaking, z = 0 is the 50th percentile, z = 1 is about the 84th, z = 1.5 is about the 93rd, z = 2 is about the 98th, and the same numbers mirror below the mean: z = −1 is about the 16th percentile.
Reverse mode runs the formula backwards. If you know a z-score — say the 90th percentile corresponds to z ≈ 1.28 — and you know the mean and standard deviation, the value is the mean plus the z-score times the standard deviation. That is how test publishers turn a percentile back into a raw score, or how a teacher figures out what score earns the top 10 percent of a class.
The raw-data mode is for when you have the numbers but not the summary. Paste a comma-separated list and the calculator works out the mean, then the standard deviation — both the population version, which divides by N and describes the list itself, and the sample version, which divides by N − 1 and estimates the spread of the larger group the list came from. Pick the one that fits your situation, enter the value you want to score, and it gives the z-score against your own data. A z-score only means what the percentile says if the data is roughly bell-shaped; with a heavily skewed list, treat the percentile as a rough guide rather than a precise fact.
Worked example
A test has a class mean of 70 and a standard deviation of 10. A student scores 85. What is the z-score and percentile?
Step 1: Subtract the mean — 85 − 70 = 15.
Step 2: Divide by the standard deviation — 15 ÷ 10 = 1.50.
Step 3: The z-score is 1.50, so the score is 1.5 standard deviations above the mean.
Step 4: The area under the normal curve below z = 1.50 is 0.9332, so the percentile is 93.32.
Enter 85, 70, and 10 in the calculator and it gives exactly the same result: z = 1.50, 93.32nd percentile. Working backwards, a z-score of 1.50 with the same mean and standard deviation gives X = 70 + 1.50 × 10 = 85. And from the raw list 70, 80, 90, the mean is 80, the population standard deviation is 8.16, and the sample standard deviation is 10.00.
FAQ
What is a z-score?
A z-score measures how many standard deviations a value sits from the mean. It is the value minus the mean, divided by the standard deviation. A z-score of 0 means the value is exactly average, +1 means one standard deviation above average, and -1 means one below.
What does a z-score of 1.5 mean?
A z-score of 1.5 means the value is 1.5 standard deviations above the mean. In a normal distribution, about 93.3 percent of values fall below it, so it sits around the 93rd percentile. Only about 6.7 percent of values are higher.
How do you find the percentile from a z-score?
The percentile is the area under the standard normal curve to the left of the z-score, times 100. For a z-score of 1.5 that area is about 0.933, so the percentile is 93.3. This calculator uses the Abramowitz-Stegun approximation, which is accurate to about one part in ten million.
What is the difference between population and sample standard deviation?
Population standard deviation divides the sum of squared differences by the number of values, N, and is used when you have the whole group. Sample standard deviation divides by N minus 1 instead, which corrects for the fact that a sample tends to understate the spread, and is used when your data is a sample of a larger group.
Can a z-score be negative?
Yes. A negative z-score simply means the value is below the mean. A value one standard deviation below average has a z-score of -1 and sits at about the 16th percentile. The sign tells you the direction, and the size tells you how far from average the value is.