Z-Score Calculator with Steps and Percentile

What is a z-score?

A z-score, also called a standard score, shows how far a value lies from the mean, measured in standard deviations. It lets you compare values from different scales, such as an exam mark and a height, because the result has no units. Positive z-scores are above the mean and negative z-scores are below it.

Z-score formulas

To find	Formula
Z-score of a value	z = (x − μ) ÷ σ, or (x − x̄) ÷ s for a sample
Value from a z-score	x = μ + zσ
Percentile (normal data)	the area to the left of z under the standard normal curve

Reading a z-score

Z-score	Percentile (normal data)	Meaning
−2	2.28%	Unusually low
−1	15.87%	Below average
0	50%	Exactly average
1	84.13%	Above average
1.96	97.5%	Edge of the middle 95%
2	97.72%	Unusually high
3	99.87%	Very rare

Worked examples

An exam score

A student scores 85 on a test with a mean of 70 and a standard deviation of 10. Then z = (85 − 70) ÷ 10 = 1.5, which is about the 93.3rd percentile.

Working backwards

For a z-score of −0.5 on a scale with mean 100 and standard deviation 15, x = 100 + (−0.5)(15) = 92.5.

A height

A height of 180 cm where the mean is 170 and the standard deviation is 8 gives z = 1.25, around the 89.4th percentile.

Comparing two scores

A maths mark of 82 (mean 70, standard deviation 8) gives z = 1.5. A science mark of 78 (mean 65, standard deviation 5) gives z = 2.6. The science mark is stronger relative to its class, even though the raw mark is lower.

Z-scores, outliers and the empirical rule

For bell-shaped data, about 68% of values have a z-score between −1 and 1, about 95% between −2 and 2, and about 99.7% between −3 and 3. That is why values with |z| above 3 are often flagged as outliers. The z-score itself works for any data, but the percentile reading needs data that is roughly normal.

Common z-score mistakes

The main mistakes are using the wrong standard deviation, mixing up the sample and population versions, and reading a percentile from skewed data where the normal curve does not fit. Also, a z-score needs a standard deviation above zero. To get the mean and standard deviation from your own list first, use the standard deviation calculator and the mean, median and mode calculator.

Frequently asked questions

What is a z-score?

A z-score says how many standard deviations a value lies above or below the mean. It is found with z = (x − mean) ÷ standard deviation. A z-score of 0 is exactly average, 1 is one standard deviation above, and −1 is one below.

What is a good z-score?

It depends on the goal. A high positive z-score means a value well above average, which is good for an exam mark and bad for a blood-pressure reading. As a guide, values beyond ±2 are unusual, and beyond ±3 are rare.

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

Look up the area under the standard normal curve to the left of z. A z-score of 1.5 gives 0.9332, so the value is at about the 93rd percentile. This works when the data is roughly normally distributed.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean. A z-score of −2 is two standard deviations below average, around the 2.3rd percentile for normal data. The sign shows direction, and the size shows distance.

What is the difference between a z-score and a standard deviation?

Standard deviation measures the spread of a whole data set in its own units. A z-score uses that spread as a ruler to place one value, expressing its distance from the mean in standard deviations. It has no units.

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Related topics on this page: standard score, percentile, standard normal distribution, empirical rule, outliers, raw score, probability