Z-Score Calculator

StatisticsLast updated: August 17, 2026

A z-score (also called a standard score) is a unitless measure of how many standard deviations an observation lies from the mean of its distribution. A positive z-score means the value is above the mean and a negative one means it is below; z = 0 means the value is exactly at the mean. For example, an exam score with z = 1.50 is 1.5 standard deviations above the mean and higher than roughly 93% of the group.

Standardizing translates scores measured on different scales into a common language: results from a test scored out of 100 and one scored out of 500 become directly comparable once they are converted into z-scores. Alongside the z-score, this calculator returns the percentile (the proportion falling below the score) under a normal distribution, the proportion above it, and the T score (50 + 10z) widely used in educational measurement; it also flags observations with |z| ≥ 3 as candidate outliers.

The observation or test score you want to convert into a standard score.
The population or sample mean.
Must be greater than zero.

Formula

z = (x − μ) / σ
Percentile = Φ(z) × 100   (Φ: standard normal CDF)
Proportion above = 100 − percentile
T score = 50 + 10 · z

Using the variance instead of the standard deviation in the denominator is a common mistake; the denominator must be the square root of the variance (σ). The percentile readings also rest on the normality assumption; if the data is clearly skewed, the percentile should be taken directly from the ranks instead.

How to Calculate

  1. Enter the raw score — the observation you want to convert into a standard score (e.g. 85).
  2. Enter the mean and the standard deviation of the distribution (e.g. μ = 70, σ = 10).
  3. Read the z-score: the sign tells you whether the score is above or below the mean, and the magnitude tells you how far.
  4. Use the percentile, the proportion above, and the T score in your report or comparisons.

Worked Examples

Percentile of an exam score

On an exam with a mean of 70 and a standard deviation of 10, a score of 85 gives z = (85 − 70) / 10 = 1.50. The percentile is 93.32%, the proportion above is 6.68%, and the T score is 65.

z-score (standard score): 1.50 · Percentile (proportion below): 93.32% · Proportion above: 6.68%

A score below the mean (negative z)

On a scale with a mean of 50 and a standard deviation of 5, a score of 45 gives z = −1.00: the score is one standard deviation below the mean, the percentile is 15.87%, and the T score is 40.

z-score (standard score): -1.00 · Percentile (proportion below): 15.87% · Proportion above: 84.13%

A high score on the IQ scale

On the IQ scale (μ = 100, σ = 15) a score of 132 corresponds to a standard score of z = 2.13; the person sits at roughly the 98.36% percentile, with only 1.64% of the scale above that score.

z-score (standard score): 2.13 · Percentile (proportion below): 98.36% · Proportion above: 1.64%

Frequently Asked Questions

What does a negative z-score mean?
A negative z-score means the value falls below the mean of the distribution. For example, z = −1.50 means the score is 1.5 standard deviations below the mean, which corresponds to roughly the 6.7% percentile. The negative sign is not an error; it is simply how the position relative to the mean is expressed.
At what z-score is a value considered an outlier?
In common practice |z| > 2 is noteworthy, |z| > 2.5 is a possible outlier, and |z| > 3 is treated as an outlier. Since about 99.7% of observations in a normal distribution fall within ±3, values beyond that boundary should be examined as either data-entry errors or genuinely unusual observations.
What is a T score, and how does it differ from a z-score?
A T score is the z-score transformed by T = 50 + 10z, giving a scale with a mean of 50 and a standard deviation of 10. Because it avoids negative and fractional values, it is preferred in educational measurement, guidance reports, and personality inventories. It carries exactly the same information as the z-score; only the scale differs.
Does the data have to be normally distributed to read the percentile?
The z-score itself can be applied to any distribution; it always shows how many standard deviations a value lies from the mean. The percentile and probability readings, however, rest on the standard normal distribution, so the data must be approximately normal for those readings to hold; with clearly skewed data the percentile should be taken directly from the ranks.
Can I use the sample mean and standard deviation instead of population values?
Yes. When the population parameters (μ, σ) are unknown, the sample mean and sample standard deviation are used. With small samples these estimates carry more uncertainty, so percentile readings should be treated cautiously; if precise inference is needed, methods based on the t distribution are preferable.
What is the 68-95-99.7 rule?
In a normal distribution about 68% of observations fall within ±1, 95% within ±2, and 99.7% within ±3 standard deviations; this is known as the empirical rule. It gives a quick reading of z-scores: a score beyond ±2 lies in the most extreme 5% or so, and one beyond ±3 lies in an exceptional region of about three in a thousand.