Enter a value, the mean and the standard deviation to get the z-score — and the percentile it sits at.
What Is a Z-Score Calculator?
A z-score calculator standardizes a value: (85 − 70) ÷ 10 = 1.50 — the value sits 1.5 standard deviations above the mean, at the 93.32nd percentile. Z-scores let you compare apples and oranges (a test score vs a height) on one scale. The percentile comes from the standard normal distribution: z = 1.50 → 93.32% of values fall below.
How to Use
- Enter the value x.
- Enter the mean μ.
- Enter the standard deviation σ.
- Click “Calculate Z-Score”.
Formula
z = (x − μ) ÷ σ. Percentile = Φ(z) × 100 (standard normal CDF).
Worked Example
x = 85, μ = 70, σ = 10: z = 1.50; percentile 93.32%.
How It Works, In Plain Words
A z-score strips the units away from a value so positions can be compared. It takes three inputs: the value x you care about, the mean μ of the set it belongs to, and the standard deviation σ, which measures how spread out that set is. The first step is the gap: x − μ. In the worked example that is 85 − 70 = 15 points above average. The second step divides the gap by the spread: 15 ÷ 10 = 1.50. The answer is in standard deviations, not points — this value sits one and a half spreads above the mean.
The tool then converts z into a percentile using the standard normal curve (the Φ function in the formula). The percentile is the share of values falling below yours: 93.32% here, so the value beats all but a small slice at the top. A value exactly at the mean has z = 0 and sits at the 50th percentile — half below, half above. Negative z-scores mirror the positive side: below the mean, with percentiles under 50.
Real-World Uses
A student compares results across two subjects with different averages and spreads: raw marks cannot be compared fairly, but z-scores can — the higher z is the stronger relative performance. A quality-control checker measures parts coming off a line and flags anything beyond the page’s outlier guide of |z| > 3 for a second look before the batch ships. A researcher standardises measurements recorded in different units before combining them into one analysis. A parent reading a child’s growth chart is, in effect, reading z-scores: how far from the average for that age, in standard deviations. In each case the raw number stays the same story; the z-score tells you where it stands.
Benefits
- Z-score and percentile together
- The standardization idea explained
- Useful for test scores, QC and research
- Pairs with the P-Value calculator
Features of Our Tool
- Instant results in your browser
- No sign-up, no download, 100% free
- Works on mobile, tablet and desktop
- Clear step-by-step explanation below the tool
- Private — your entries never leave your device
Common Mistakes
- Using sample SD vs population SD inconsistently
- Assuming normality — percentiles assume a normal distribution
- Reading z = 1.5 as “1.5%” — it is 1.5 standard deviations
Common Mistakes to Avoid
Three traps recur. Mixing standard deviations: a sample SD and a population SD differ, and the mistakes list warns against swapping them halfway through an analysis — use the one your μ belongs with. Assuming normality: the percentile half of the answer leans on the normal curve, so for heavily skewed data the z-score is still a fair rescaling but the percentile can mislead — the FAQ makes exactly this distinction. Reading z as a percentage: z = 1.5 is not “1.5%” of anything; it is 1.5 standard deviations, as the page’s own example stresses. And a practical one: σ must be above zero — with no spread, there is nothing to standardise against.
FAQ
What is a z-score?
How many standard deviations a value is from the mean — here 1.50.
What percentile is z = 1.5?
93.32nd percentile.
What is a good z-score?
Depends on context — |z| > 2 is often flagged as unusual, |z| > 3 as an outlier.
Can z-scores be negative?
Yes — negative means below the mean.
Do I need normal data?
For the percentile, yes — the z-score itself is just a rescaling.
How do I turn a z-score back into a raw value?
Reverse the formula: x = μ + z × σ. For this page’s example, 70 + 1.50 × 10 = 85 — the value you started with. It is handy when a report gives you a z-score and you want the measurement in its original units.
What is the difference between a z-score and a percentile?
The z-score counts distance from the mean in standard deviations; the percentile counts values below you. They move together but are not proportional: near the mean, a small z change moves the percentile a lot, while out at |z| > 3 the percentile barely shifts.