Z Score to Percentile Calculator
Converts a Z-Score (standard deviations) into a simple percentile ranking — like Top 10%.
A percentage that tells you exactly what portion of the population is below or above a specific score.
Preset loaded (95th Perc.)
Percentile rank
95.00%
z = 1.645 — area shaded on the curve below
Above Average: You outperformed the mean.
Probability (0–1)
0.95002
The CDF value
Inverse (right tail)
Top 5.00%
How exclusive your score is
Z-score used
1.645
Direct input
About this calculator
A test score alone does not say much — a score of 75 may rank very high or very low. You need your percentile rank to understand your position. This calculator converts your z-score to a percentile and shows how many people you scored above.
The z-score is the distance: how far a data point is from the average. A z-score of 0 means exactly average; +1 means one standard deviation above. The percentile is the rank: how many values fall below yours — the 80th percentile means you scored higher than 80% of the group.
Formula
Percentile = Φ(z) × 100% · z = (x − μ) ÷ σ
How to use it
- 1
Enter a z-score directly, or switch to Raw Score mode and provide your score, the mean and the standard deviation.
- 2
The calculator evaluates the cumulative distribution function (CDF) — the area under the bell curve from the left up to your z-score.
- 3
Read the percentile, the probability, the right-tail percentage and the bell curve visualization with the shaded area.
A simple example — height
The average height for men is 70 inches with a standard deviation of 3 inches. Your height is 76 inches. Subtract the average: 76 − 70 = 6. Divide by the SD: 6 ÷ 3 = 2. Your z-score is 2.0, which equals the 97.7th percentile — you are taller than about 98% of men.
How this calculator works — the CDF
The calculator uses the cumulative distribution function. The bell curve represents the full population, and the area under the curve shows probability. The curve splits into two equal parts at z = 0, which gives a percentile of 50% — the exact middle.
A positive z-score moves above the average: the calculator measures the area from the left side of the curve to your score. A negative z-score moves below the average, where the area becomes smaller. People once used a large chart called a Z-Table to find percentiles by hand — this calculator replaces it with instant, more exact results.
Percentile = Φ(z) × 100
1. The 68-95-99.7 rule
68% of all people fall between z = −1 and +1 — the average group. 95% fall between −2 and +2; outside this range you are unusual. 99.7% fall between −3 and +3; falling outside is extremely rare. Use this rule to estimate percentiles without a calculator.
2. Percentiles can never reach 100%
The 100th percentile is mathematically impossible — the bell curve stretches forever in both directions without ever touching the bottom. Even a z-score of 5.0 only reaches 99.99997%. Most tests report a highest percentile of 99.9th.
3. Symmetry is key
The normal distribution is perfectly symmetrical. A z-score of +1.0 equals the 84th percentile; a z-score of −1.0 equals the 16th percentile. Notice that 100 − 84 = 16. If you know the percentile for a positive z-score, you already know the negative version — just subtract from 100.
When to use this calculator
Standardized tests
A raw SAT or IQ score does not show your full ranking. An IQ score of 130 has a z-score of +2.0, which falls near the 98th percentile — the top 2% of test takers.
Baby growth charts
Doctors use percentiles to track child growth: a baby in the 15th percentile for weight weighs more than 15 out of 100 babies of the same age. Z-scores help doctors follow growth over time and spot problems early.
Product quality checks
Factories use z-scores to keep products consistent — a soda machine aiming for 500 ml with fills at 498 or 502 ml. Engineers study the z-scores of fill amounts; the Six Sigma method targets a z-score close to 6.
In Excel: In Excel use =NORM.S.DIST(z, TRUE) for the probability, then format as a percentage.
Frequently asked questions
Can a percentile be negative?
No. Percentiles always stay between 0% and 100%. A z-score, however, can be negative — meaning your value falls below the average. A z-score of −2.0 matches the 2.28th percentile: a low rank, but still positive.
What is a good z-score?
It depends on what you measure. Test scores and income favor higher values — a positive z-score above +1.0 is often good. Golf scores and race times favor lower values — a negative z-score below −1.0 is better. In most situations, a z-score between −1 and +1 falls within the normal range.
Why does the graph never touch the bottom line?
The normal distribution follows a special pattern — the curve moves closer to zero as the z-score increases, but never fully reaches it. This means rare events can still happen: the chance becomes very small, but never completely disappears.
How do I calculate my z-score if I don't have it?
Subtract the average from your raw score, then divide by the standard deviation: z = (x − μ) ÷ σ. Or simply switch to Raw Score mode above and the calculator derives it for you.