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Z-Score Calculator

Find how many standard deviations a score sits from the mean, the percentile and p-values that go with any z-score, or the z-score for a given percentile, with the area shaded on a normal curve.

Your numbers

What do you want to find?

Result

z-score

1.25

A score of 85 is 1.25 standard deviations above the mean of 70.

Percentile (area below)

89.44%

of a normal population scores below 85

  • Area above: P(Z ≥ 1.25)0.1056
  • Two-tailed p-value: P(|Z| ≥ 1.25)0.2113
  • Within ±1.25 SD of the mean0.7887

How to find a z-score and its percentile

  1. Subtract the mean from the score: 85 − 70 = 15
  2. Divide by the standard deviation: z = 15 ÷ 12 = 1.25
  3. Look up the area to the left in the standard normal (z) table: Φ(1.25) ≈ 0.8944, so the score sits at about 89.44% (its percentile).
  4. Area above is 1 − 0.8944 ≈ 0.1056. The two-tailed p-value doubles the smaller tail: 0.2113.

Area below z = 1.25: 0.8944

Standard normal curve (height)Your shaded area

The highlighted outline traces the area. Its size, not its height, is the probability.

How to use this calculator

Choose one of three modes:

  1. Score → z. Enter a raw score, the mean and the standard deviation. You get the z-score, its percentile, the area above it, the two-tailed p-value and the share of values within that many standard deviations of the mean.
  2. z → probability. Enter any z-score and choose which area to shade: the left tail P(Z ≤ z), the right tail P(Z ≥ z), two-tailed P(|Z| ≥ |z|) or the area between −z and z. All four are listed, with the one you chose in bold.
  3. Percentile → z. Enter a percentile to get the matching z-score. Add a mean and standard deviation to turn it into a score, such as the cutoff for the top 5%.

The chart draws the standard normal curve and outlines the area you’re looking at. The probability is the size of that area, which is the whole curve’s area of 1 split between the shaded and unshaded parts.

The z-score formula

z=x−μσ

Here x is the value, μ the mean and σ the standard deviation. To go the other way, from a z-score back to a value, rearrange it to x = μ + zσ.

The probabilities come from the cumulative distribution function of the standard normal distribution, Φ(z), the area under the curve to the left of z:

Φ(z)=∫−∞z12πe−t2/2dt

The right tail is 1 − Φ(z), the two-tailed p-value is twice the smaller tail, and the area between −z and z is 1 minus the two-tailed value. The calculator evaluates Φ with Hart’s double-precision approximation, which agrees with exact values to at least 13 decimal places, and inverts it for the percentile mode, so it’s more precise than any printed z-table.

Worked example

A class’s test scores have a mean of 70 and a standard deviation of 12. You scored 85.

  1. Distance from the mean: 85 − 70 = 15 points.
  2. In standard deviations: z = 15 ÷ 12 = 1.25.
  3. Percentile: Φ(1.25) = 0.8944, so about 89.44% of a normally distributed class scored below you.
  4. Above you: 1 − 0.8944 = 0.1056, or about 10.56%.
  5. Two-tailed p-value: 2 × 0.1056 = 0.2113, the chance of a score at least as far from the mean in either direction.

Going the other way, the 95th percentile is z = 1.6449, so the score that cuts off the top 5% of that class is 70 + 1.6449 × 12 = 89.7382.

Z-table: common values

Areas under the standard normal curve, computed by the calculator:

z Left tail Φ(z) Right tail Two-tailed Between −z and z
0 0.5000 0.5000 1.0000 0.0000
0.5 0.6915 0.3085 0.6171 0.3829
1 0.8413 0.1587 0.3173 0.6827
1.28 0.8997 0.1003 0.2005 0.7995
1.5 0.9332 0.0668 0.1336 0.8664
1.645 0.9500 0.0500 0.1000 0.9000
1.96 0.9750 0.0250 0.0500 0.9500
2 0.9772 0.0228 0.0455 0.9545
2.576 0.9950 0.0050 0.0100 0.9900
3 0.9987 0.0013 0.0027 0.9973

For a negative z, swap the left and right tails: Φ(−1) = 0.1587. The last column is the source of the familiar 68–95–99.7 rule: about 68.27% of values fall within one standard deviation of the mean, 95.45% within two and 99.73% within three.

Percentile to z-score

Percentile z-score Two-sided cutoff with that % in the middle
50 0 ±0.6745
75 0.6745 ±1.1503
80 0.8416 ±1.2816
90 1.2816 ±1.6449
95 1.6449 ±1.96
97.5 1.96 ±2.2414
99 2.3263 ±2.5758
99.5 2.5758 ±2.807
99.9 3.0902 ±3.2905

The third column gives the critical values for confidence intervals: ±1.6449 for 90%, ±1.96 for 95% and ±2.5758 for 99%.

Interpreting z-scores

  • Between −1 and 1: typical. About two-thirds of values land here.
  • Beyond ±2: unusual. Only about 4.55% of values are this far out, which is why |z| > 1.96 is the usual cutoff for “statistically significant at the 5% level.”
  • Beyond ±3: rare, about 0.27% of values. In quality control, points this far out are often investigated as possible errors or special causes.

Z-scores also let you compare apples and oranges. A z of 1.25 on a chemistry test beats a z of 0.9 on a history test, even if the raw history score was higher, because each is measured against its own class.

Assumptions and limitations

  • Percentiles and p-values assume a normal distribution. The z-score itself is valid for any data, but turning it into a percentile only works if the data are roughly bell-shaped. Skewed data, such as incomes or wait times, can be far from these figures.
  • Known mean and standard deviation. If you estimated them from a small sample (under about 30 values), a t-distribution gives more honest p-values. The standard deviation calculator uses t for its margin of error.
  • Population vs. sample. Use the standard deviation that matches your question; the difference between dividing by n and n − 1 matters most for small samples.

To get the mean of your data first, use the average calculator. If your score is one part of a weighted class grade, the grade calculator works out the overall result.

Sources

Frequently asked questions

What is a z-score?

A z-score tells you how many standard deviations a value is above or below the mean. A z-score of 0 is exactly average, 1.25 is one and a quarter standard deviations above, and −2 is two standard deviations below. Because it's unit-free, it lets you compare scores from different tests or scales.

How do I calculate a z-score?

Subtract the mean from the value and divide by the standard deviation, z = (x − μ) ÷ σ. A test score of 85 in a class with a mean of 70 and a standard deviation of 12 has z = 15 ÷ 12 = 1.25. If you only have raw data, find the mean and standard deviation first.

How do I convert a z-score to a percentile?

Look up the area to the left of z under the standard normal curve, which is what a z-table lists, and multiply by 100. For z = 1.25 that area is 0.8944, so the score is at about the 89th percentile. This assumes the scores follow a normal distribution.

What's the difference between one-tailed and two-tailed p-values?

A one-tailed p-value counts only the area beyond z on one side, while a two-tailed p-value counts both tails, as far from the mean in either direction. For z = 1.96 the right tail is 0.025 and the two-tailed p-value is 0.05. Use two-tailed unless your hypothesis predicted the direction in advance.

What z-score corresponds to 95% confidence?

For a two-sided 95% confidence interval, z = ±1.96, because 2.5% of the area lies beyond each of those values. For a one-sided 95% bound, z = 1.6449, the 95th percentile. The percentile mode shows both for any percentage.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean. A z-score of −1 is at about the 16th percentile, the mirror image of z = 1 at about the 84th.

Updated November 2026. Results are estimates for educational purposes and depend on the inputs you provide.