Z-Score Calculator
How many standard deviations a value sits from the mean, and the percentile that implies.
Last updated
Your details
Result
- Z-score
- 1.5
- Percentile
- 93.3193%
- P(X > your value)
- 6.6807%
- Two-tailed p-value
- 0.133614
- Unusual?
- No — within 2σ
1.5 standard deviations above the mean
93.3193% of the distribution falls below this value
Probability of a value at least this extreme in either direction
About 95% of a normal distribution lies within two standard deviations
85 against a mean of 70…
z = (x − μ) ÷ σ
x − μ = 85 − 70 = 15
15 ÷ 10
z = 1.5
Formula — written step by step
A score of 85 where the mean is 70…
z = (x − μ) ÷ σ
x − μ = 85 − 70 = 15
15 ÷ 10
z = 1.5
x is your value, μ the population mean and σ the standard deviation. A z of 1.5 means the score sits one and a half standard deviations above average — which puts it at roughly the 93rd percentile.
A worked example
Comparing marks from two different exams
- Maths
- 78, class mean 65, SD 8
- History
- 82, class mean 76, SD 12
- Maths: z = (78 − 65) ÷ 8 = 1.625
- History: z = (82 − 76) ÷ 12 = 0.5
- Maths percentile ≈ 94.8%
- History percentile ≈ 69.1%
The maths result is far stronger, despite the lower mark
This is what z-scores are for. The raw marks say history was the better subject; standardising says the opposite. A 78 in a tightly clustered class is a much rarer result than an 82 in a widely spread one.
How to use this calculator
- Enter the value you want to place, then the mean and standard deviation of the population it came from.
- Read the z-score as a count of standard deviations: positive is above the mean, negative below.
- The percentile tells you what share of the distribution falls below your value.
- These readings assume the data is roughly normal. For strongly skewed data the z-score is still defined, but the percentile is not trustworthy.
What a z-score does
A z-score restates a value as a number of standard deviations from the mean. It strips away the original units entirely, which is precisely what makes it useful: a height in centimetres and a test mark out of 100 become directly comparable once both are expressed in standard deviations.
The sign gives the direction and the magnitude gives the distance. A z of 0 is exactly average, ±1 is ordinary, ±2 is uncommon, and ±3 is rare — under a normal distribution, only about one value in 370 falls beyond three standard deviations in either direction.
The empirical rule
For a roughly normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. This is the empirical rule, and it is the fastest sanity check available on any claim involving standard deviations.
It is also where the convention of treating |z| > 2 as "unusual" comes from. That threshold is a convention, not a law of nature, and it is the same reasoning that produces the 5% significance level used throughout hypothesis testing.
When the percentile is not to be trusted
The z-score itself needs no assumption — it is arithmetic, defined for any distribution. The percentile does. Converting z into "what share falls below" requires the data to be approximately normal, and for skewed data that conversion can be badly wrong.
Income is the standard example. Its distribution has a long right tail, so a z of +2 sits at a much higher percentile than the normal table suggests. When data is visibly skewed, report the z-score if it helps, but reach for the actual percentile rank rather than a normal approximation.
Formula & methodology
What this calculator assumes
- The z-score itself needs no assumption — it is arithmetic. The percentile does: converting z into a share of the distribution requires the data to be approximately normal.
- For skewed data such as income, the percentile from a normal table can be badly wrong even though the z-score is correct.
- Uses the population standard deviation. With a small sample estimate, a t-score is the more appropriate measure.
Sources
Z-scores and the area below them
| z | Percentile | Share beyond it (two-tailed) |
|---|---|---|
| −3 | 0.13% | 0.27% |
| −2 | 2.28% | 4.55% |
| −1 | 15.87% | 31.73% |
| 0 | 50.00% | 100.00% |
| 1 | 84.13% | 31.73% |
| 1.645 | 95.00% | 10.00% |
| 1.96 | 97.50% | 5.00% |
| 2.576 | 99.50% | 1.00% |
| 3 | 99.87% | 0.27% |
The 1.96 row is the one worth memorising: it is where the familiar 95% confidence interval comes from.
Frequently asked questions
What is a good z-score?
It depends entirely on context. For a test result, higher is better. For a manufacturing tolerance, a z near zero is the goal. The score measures distance from average, not quality.
Can a z-score be negative?
Yes — it simply means the value is below the mean. A z of −1.5 is exactly as far from average as +1.5, in the other direction.
Should I use the sample or population standard deviation?
Strictly, a z-score uses the population standard deviation. With a sample estimate and a small sample, a t-score is more appropriate, which is what the confidence interval calculator uses.
What z-score counts as an outlier?
A common convention is |z| > 3, but for small samples the interquartile method is more robust — a single extreme value inflates the standard deviation and hides itself.