Z-Score Calculator
Convert a raw value to a z-score and see the exact normal probability and percentile it corresponds to.
Details
Must be greater than zero.
Z-score
2
2 standard deviations above the average
Chance of scoring below
97.725%
Chance of scoring above
2.275%
This converts a value into a z-score: how many standard deviations it sits above or below the average.
It works in both directions, so you can enter a raw value to get its z-score, or a z-score to find the raw value it corresponds to.
What a z-score is
A z-score says how far a value is from the average, measured in standard deviations. A z-score of 2 means two standard deviations above the mean; −1 means one below.
That is more useful than it sounds, because it strips out the units. A test mark, a height in centimetres and a cholesterol reading all become plain numbers on the same scale, so they can be compared directly.
An IQ of 130, on a scale with a mean of 100 and a standard deviation of 15, is exactly two standard deviations up. Its z-score is 2, which places it in roughly the top 2% of people.
Subtract the mean, then divide by the standard deviation. A negative result simply means the value sits below average.
What to enter
- Your value
- The raw measurement you want to convert: a test score, a height, a lab result.
- Average of the group
- The mean of the population or sample you are comparing against.
- Standard deviation
- How spread out that group is. Work it out from raw data with the standard deviation calculator if you do not have it.
What a z-score tells you about rank
- z = 0
- Exactly average. Half the group is above and half below.
- z = ±1
- About 68% of values fall within this band, so roughly 84th percentile at +1.
- z = ±2
- About 95% fall within. A z of +2 is around the 97.7th percentile.
- z = ±3
- About 99.7% fall within. Beyond ±3 is genuinely rare.
- Negative z
- Simply below average. It is not a bad score in itself; it depends entirely on what is being measured.
What this assumes
The percentile figures assume a normal, bell-shaped distribution. On skewed data a z-score is still calculable but the percentiles no longer hold.
The mean and standard deviation describe the group you are comparing against, not just your own sample.
How to calculate a z-score
Subtract, then divide. The whole method is one line.
- value − mean
- The gap, which can be negative
- ÷ standard deviation
- Converts that gap into a count of standard deviations
Find the gap. Subtract the mean from your value. 130 − 100 = 30.
Divide by the standard deviation. 30 ÷ 15 = 2. So the value is two standard deviations above average.
Read the sign. Positive is above the mean, negative is below. Zero is exactly average.
To reverse it, multiply. value = mean + (z × standard deviation). A z of −1 gives 100 − 15 = 85.
See a worked example: comparing a maths mark against an English mark
- Maths
- 78, class mean 70, SD 8
- English
- 84, class mean 80, SD 2
Maths: (78 − 70) ÷ 8 = 1.0.
English: (84 − 80) ÷ 2 = 2.0.
The English mark is lower relative to the raw maths mark difference, yet its z-score is twice as high.
Because English marks are tightly clustered, being 4 above the mean there is more impressive than being 8 above in maths.
English z = 2.0 beats maths z = 1.0
Frequently asked questions
It only means below average, and whether that is bad depends entirely on what you are measuring.
A z of −1.5 on an exam is poor. The same z for blood pressure or cholesterol is usually good news. The sign is direction, not judgement.
Look it up in a standard normal table, or use the rough landmarks: z = 0 is the 50th percentile, z = 1 is about the 84th, and z = 2 is about the 97.7th.
This only works if the data is roughly bell-shaped. For skewed data the z-score is still valid but the percentile conversion is not.
A common rule of thumb is a z beyond ±3, which covers only about 0.3% of a normal distribution. Some fields use ±2 as a flag for closer inspection.
An outlier is not automatically an error. It may be the most interesting data point you have, so investigate before deleting anything.
Because raw numbers cannot be compared across different scales. Scoring 78 in one subject and 84 in another tells you nothing until you know how each class performed.
The z-score puts everything on one ruler. That is also why it underpins confidence intervals and hypothesis testing.
A z-score is used when you know the population standard deviation, or your sample is large. A t-score is used for small samples where the standard deviation is estimated.
The t-distribution is slightly wider to account for that extra uncertainty. Above roughly 30 observations the two converge and the distinction stops mattering much.
Problems people actually run into
Comparing z-scores from different groups
A z-score is always relative to the group it came from. Scoring z = 2 in a weak class is not the same achievement as z = 2 in a strong one, even though the number is identical.
Standardising makes values comparable within a shared distribution. It does not make two separate populations equivalent, which is a subtle but important limit.
Reading percentiles off skewed data
The 68-95-99.7 landmarks only hold for a bell curve. Income, house prices and waiting times are all heavily skewed, so a z of 2 there does not mean the 97.7th percentile.
Plot the data or check the median against the mean first. If those two diverge sharply, treat any percentile from a z-score with suspicion.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: September 4, 2026