Case 1
- Raw value x
- 85
- Mean (μ)
- 75
- Standard deviation (σ)
- 5
2.0000
Open with these valuesZ-Score Calculator
2.0000
Result: 2.0000A z-score counts how many standard deviations a value lies above or below the mean. Zero is exactly average, positive is above it, negative below. Because the units cancel out, a z-score compares values from scales that have nothing else in common — a test score against a body weight.
Worked examples
Case 2
- Raw value x
- 90
- Mean (μ)
- 100
- Standard deviation (σ)
- 15
-0.6667
Open with these valuesCase 3
- Raw value x
- 130
- Mean (μ)
- 100
- Standard deviation (σ)
- 15
2.0000
Open with these valuesHow it's calculated
z = (x − μ) ÷ σ
- StepEnter the single value you want to place on the scale.
- StepAdd the mean and the standard deviation of its distribution.
- ResultRead the z-score: the distance from the mean in standard deviations.
What this number means
A z-score puts a single measurement on a scale every distribution shares. The raw number 85 says nothing on its own until you know what is typical and how widely the data spread; the z-score carries both. The formula does two things in turn. Subtracting the mean moves the origin to the centre of the distribution, so the difference reports only how far above or below average the value sits: 85 − 75 = 10. Dividing by the standard deviation then measures that distance in the distribution's own natural unit, turning 10 into (85 − 75) ÷ 5 = 2 — two standard deviations above the mean. Both steps work on the same scale, which is why the answer carries no unit at all. What a z-score does not give you is a probability. Two standard deviations above the mean does not by itself fix a rank: the familiar shares — about 68 % of values within ±1, 95 % within ±2, 99.7 % within ±3 — hold only for a roughly normal distribution. In a skewed one the same z of 2 can sit somewhere quite different in the ordering. And the score inherits everything from its two inputs. A mean and a deviation estimated from a handful of observations make the z-score exactly as uncertain as they are.
σ is the population standard deviation
The formula on this page takes μ and σ from the population, not estimates read off a sample. A sample standard deviation in that slot produces a number that looks the same but does not mean the same thing.
Zero is average, the sign is only direction
A z-score of 0 sits exactly on the mean, positive above it, negative below. Whether below average is good or bad is decided by the context alone.
The units cancel out
Value, mean and deviation all live on the same scale, so their ratio carries no unit. That is what lets a z-score put a test score and a body weight side by side.
The standard deviation must be above zero
At σ = 0 the formula divides by zero and the z-score loses all meaning. That is why the field starts at 0.0001 rather than at 0.
Commonly misread
A z-score of 2 means the value is twice the average.
It means two standard deviations above the mean. With a mean of 100 and a deviation of 15 that is 130, not 200.
A negative z-score is a bad result.
The sign only gives direction. A low z-score is excellent for a golf score and poor for a test result.
Sample or population standard deviation — either one works here.
This formula is defined with μ and σ of the population. A sample estimate in that slot answers a different question, even though the arithmetic looks identical.
Reference table
| Value, mean, SD | Reading | z |
|---|---|---|
| 70, 75, 5 | One standard deviation below the mean | -1 |
| 75, 75, 5 | Exactly average | 0 |
| 85, 75, 5 | Two standard deviations above the mean | 2 |
| 90, 100, 15 | Two thirds of a deviation below | -0.6667 |
| 130, 100, 15 | Two standard deviations above the mean | 2 |
Questions
How do I calculate a z-score?
Subtract the mean from your value, then divide by the standard deviation: z = (x − μ) ÷ σ. For a value of 85 with a mean of 75 and a standard deviation of 5, that is (85 − 75) ÷ 5 = 2 — the value sits two standard deviations above the mean.
What does a z-score actually tell me?
It tells you how many standard deviations a value is from the mean. A z-score of 0 is exactly average, a positive z-score is above the mean, and a negative one is below. Because the units cancel out, z-scores let you compare values from completely different scales on a common footing.
Is a negative z-score bad?
Not inherently — the sign only shows direction. A z-score of −1 means the value is one standard deviation below the mean, while +1 is one above. Whether below-average is good or bad depends on the context: a low z-score is great for a golf score but poor for a test result.
What counts as a high or unusual z-score?
In a roughly normal distribution about 68% of values fall within ±1, 95% within ±2, and 99.7% within ±3. So a z-score beyond about ±3 is unusual — fewer than 1 in 300 values land there. Many fields treat a z above 2 in size as noteworthy and above 3 as an outlier.
What do I need to compute a z-score?
Three numbers on the same scale: the raw value, the mean, and the standard deviation. The standard deviation must be greater than zero, otherwise the formula divides by zero and has no meaning. The value and mean can be any finite number, including negatives.
Sources and last check
Information, not professional advice.