Z-Score Calculator: z, Percentile & Probability Above

What does z = 1.50 mean?

It means 85 sits one and a half standard deviations above the mean of 70 — and 93.32% of values in the distribution fall below it, while 6.68% sit above. A z-score turns a raw value into a position: 0 is exactly average, 1 is one standard deviation above, -1 is one below, and the percentile is the share of values below, read off the normal curve. The same z-score means the same position whatever the units — 85 on a test with mean 70 and sd 10 is the same standing as 115 on a test with mean 100 and sd 10.

Three things to check before you trust the number

The mean and the spread both matter. A z-score of 1.50 needs both: 85 minus 70 is 15, and 15 divided by 10 is 1.50. The same value against a mean of 80 is z = 0.50, and against a spread of 5 it is z = 3.00 — the same number, three different standings. If the mean or the spread is wrong, the z-score is wrong with it.

The percentile assumes the normal curve. 93.32% is the area under the bell curve up to z = 1.50, so it is only as good as the distribution being roughly normal. On a skewed distribution — incomes, house prices — the percentile is a rough guide, not a count, and the z-score itself still tells you how far from the mean the value sits in spread units.

A z-score of zero is average, not nothing. z = 0 means the value is the mean, and the percentile is 50.00% — half the values below, half above. A negative z-score is below the mean, not a mistake: x = 60 against a mean of 70 is z = -1.00, 15.87% below, the mirror of x = 80.

Worked example

On the defaults: x = 85, mean 70, standard deviation 10.

1. Subtract the mean: 85 − 70 = 15.
2. Divide by the standard deviation: 15 ÷ 10 = 1.50 — the z-score.
3. Read the percentile: the area under the normal curve up to 1.50 is 93.32%.
4. Take the share above: 100 − 93.32 = 6.68%.
5. Smallest value for a z-score of at least 2: 90 — at 90 z is 2.00 (unrounded 2.000000, percentile 97.72%), and at 89.5 it is 1.95 (unrounded 1.950000, percentile 97.44%).
6. Smallest value for a percentile of at least 95%: 86.5 — at 86.5 z is 1.65 (unrounded 1.650000, percentile 95.05%), and at 86 it is 1.60 (unrounded 1.600000, percentile 94.52%).

Every branch, checked

• x = 80: x = 80: z = 1.00; 84.13% below, 15.87% above. x = 80 — z is 1.00, 84.13% below, one standard deviation above the mean.
• x = 60: x = 60: z = −1.00; 15.87% below, 84.13% above. x = 60 — z is -1.00, 15.87% below, one standard deviation below the mean, the mirror of x = 80.
• x = 100: x = 100: z = 3.00; 99.87% below, 0.13% above. x = 100 — z is 3.00, 99.87% below, three standard deviations out, almost nothing above.
• x = 70: x = 70: z = 0.00; 50.00% below, 50.00% above. x = 70 — z is 0.00, 50.00% below, the value is the mean.
• sd 5: sd 5: z = 3.00; 99.87% below, 0.13% above. sd 5 — z is 3.00, the same value is three standard deviations out when the spread is half as wide.
• x = 90, mean 80, sd 20: x = 90, mean 80, sd 20: z = 0.50; 69.15% below, 30.85% above. x = 90, mean 80, sd 20 — z is 0.50, 69.15% below.
• x = 55, sd 15: x = 55, sd 15: z = −1.00; 15.87% below, 84.13% above. x = 55, sd 15 — z is -1.00, 15.87% below.
• x = 72.5: x = 72.5: z = 0.25; 59.87% below, 40.13% above. x = 72.5 — z is 0.25, 59.87% below.
• x = 100, mean 50, sd 25: x = 100, mean 50, sd 25: z = 2.00; 97.72% below, 2.28% above. x = 100, mean 50, sd 25 — z is 2.00, 97.72% below.
• x = 65: x = 65: z = −0.50; 30.85% below, 69.15% above. x = 65 — z is -0.50, 30.85% below.

The reverses, worked backwards

• Smallest value for the z-score to REACH at least 2: 90 — at 90 the figures are z = 2.00; 97.72% below, 2.28% above (unrounded z 2.000000, percentile 97.724994%, above 2.275006%); at the lower neighbor the figures are z = 1.95; 97.44% below, 2.56% above (unrounded z 1.950000, percentile 97.441201%, above 2.558799%); at the upper neighbor the figures are z = 2.05; 97.98% below, 2.02% above (unrounded z 2.050000, percentile 97.981785%, above 2.018215%) — verdict: STEP.
• Smallest value for the percentile to REACH at least 95%: 86.5 — at 86.5 the figures are z = 1.65; 95.05% below, 4.95% above (unrounded z 1.650000, percentile 95.052855%, above 4.947145%); at the lower neighbor the figures are z = 1.60; 94.52% below, 5.48% above (unrounded z 1.600000, percentile 94.520071%, above 5.479929%); at the upper neighbor the figures are z = 1.70; 95.54% below, 4.46% above (unrounded z 1.700000, percentile 95.543457%, above 4.456543%) — verdict: STEP.
• Largest standard deviation for the z-score to REACH at least 2: 7.5 — at 7.5 the figures are z = 2.00; 97.72% below, 2.28% above (unrounded z 2.000000, percentile 97.724994%, above 2.275006%); at the upper neighbor the figures are z = 1.88; 96.96% below, 3.04% above (unrounded z 1.875000, percentile 96.960370%, above 3.039630%); at the lower neighbor the figures are z = 2.14; 98.39% below, 1.61% above (unrounded z 2.142857, percentile 98.393777%, above 1.606223%) — verdict: STEP.
• Largest value for the z-score to STAY at or below 1.5: 85 — at 85 the figures are z = 1.50; 93.32% below, 6.68% above (unrounded z 1.500000, percentile 93.319277%, above 6.680723%); at the upper neighbor the figures are z = 1.55; 93.94% below, 6.06% above (unrounded z 1.550000, percentile 93.942923%, above 6.057077%); at the lower neighbor the figures are z = 1.45; 92.65% below, 7.35% above (unrounded z 1.450000, percentile 92.647070%, above 7.352930%) — verdict: STEP.
• Largest mean for the z-score to REACH at least 2: 65 — at 65 the figures are z = 2.00; 97.72% below, 2.28% above (unrounded z 2.000000, percentile 97.724994%, above 2.275006%); at the upper neighbor the figures are z = 1.90; 97.13% below, 2.87% above (unrounded z 1.900000, percentile 97.128351%, above 2.871649%); at the lower neighbor the figures are z = 2.10; 98.21% below, 1.79% above (unrounded z 2.100000, percentile 98.213564%, above 1.786436%) — verdict: STEP.
• Smallest value for the share above to be at most 5%: 86.5 — at 86.5 the figures are z = 1.65; 95.05% below, 4.95% above (unrounded z 1.650000, percentile 95.052855%, above 4.947145%); at the lower neighbor the figures are z = 1.60; 94.52% below, 5.48% above (unrounded z 1.600000, percentile 94.520071%, above 5.479929%); at the upper neighbor the figures are z = 1.70; 95.54% below, 4.46% above (unrounded z 1.700000, percentile 95.543457%, above 4.456543%) — verdict: STEP.

What the defaults assume

The defaults are x = 85, mean 70, standard deviation 10 — a test score 15 points above the mean with a 10-point spread — and the z-score is 1.50, the percentile 93.32% and 6.68% above. They are a starting point, not a benchmark: your value may be a height, a salary or a lap time, and the right answer for you is the one your numbers describe. Change the value, the mean and the spread to your own and the calculator recomputes instantly. Subtract the mean, divide by the spread, and read the percentile off the curve.

These notes repeat in every explainer section because this page is built to be read start to finish: the defaults are one value, not a standard; the mean and the spread both matter; the percentile assumes the normal curve; and a z-score of zero is average, not nothing.

Output · formula · meaning

Who uses this

• A student checking a test score — 85 against a mean of 70 and a spread of 10 is z = 1.50, 93.32% below, in the top 7% of the class.
• A teacher grading on a curve — a z-score of 2.00 is the 97.72nd percentile, and a z-score of -1.00 is the 15.87th, whatever the raw marks were.
• A researcher standardizing measurements — a z-score turns different units into one scale, so a height and a weight can be compared on the same footing.
• A quality-control engineer watching a process — a reading six standard deviations out, z = 6.00, is 100.00% below, and the process needs looking at.

Accuracy & limitations

The calculator does the arithmetic exactly — the z-score is the value minus the mean over the standard deviation, and the percentile is the normal CDF at that z-score, good to 1.5e-7 — and every figure on this page is checked against an independent model. What it cannot do is check that your distribution is normal: the percentile is the area under the bell curve, so on a skewed distribution it is a rough guide, not a count, and the z-score itself still tells you how far from the mean the value sits in spread units. It also takes one value at a time — a whole dataset needs the mean-median-mode-calculator page or the standard-deviation-calculator page first. And the page prints the unrounded figures so the rounding does not mislead you — z = 1.50, unrounded 1.500000, percentile 93.32%, unrounded 93.319277%.

Cite this calculator

A z-score is the value minus the mean, divided by the standard deviation. On the defaults — x = 85, mean 70, standard deviation 10 — the z-score is 1.50, the percentile is 93.32% and 6.68% of values sit above. Smallest value for the z-score to reach at least 2: 90. Smallest value for the percentile to reach at least 95%: 86.5. Largest standard deviation for the z-score to reach at least 2: 7.5. Largest value for the z-score to stay at or below 1.5: 85. Largest mean for the z-score to reach at least 2: 65. Smallest value for the share above to be at most 5%: 86.5. Source: Bright Side Kit, brightoolbox.com.

Frequently asked questions