What it means
The NIST Dataplot reference explains that a z-score subtracts the mean and divides by the standard deviation, which scales the data to a standard normal distribution. In symbols, Z = (X - xbar) / s, where xbar and s are the sample mean and standard deviation.
This puts values from different scales on one common footing. The same NIST source describes a related u-score, which subtracts the minimum and divides by the range.
That scales data to between 0 and 1. NIST also documents a variant used in proficiency testing under ISO 13528, where the z-score is (X - assigned value) / sigma.
The assigned value and sigma are set by the testing program, not computed from the data. That standard treats absolute values above 3 as an action signal and above 2 as a warning signal.
Z-scores are often used to flag possible outliers, and the NIST Engineering Statistics Handbook gives a caution. This can mislead, particularly for small samples, because the largest possible z-score is limited by the sample size.
Iglewicz and Hoaglin recommend a modified z-score built on the median and the median absolute deviation, or MAD. The modified score is M = 0.6745 x (x - median) / MAD.
It resists the pull of the very outlier you are trying to find, because the median and MAD hardly move when one value is extreme. In finance, analysts use z-scores to compare returns or ratios across assets with different scales.
How rare a score is depends on the distribution.
In practice
Real-world examples.
Example
A fictional exam has a mean of 70 and a standard deviation of 8. A student scores 86, so her z-score is (86 - 70) / 8 = 2.0. Another scores 64, so his z-score is -0.75, meaning he sits three-quarters of a standard deviation below the average.
Example
A fictional analyst compares a fund's 9.1% return with its peer group, which has a mean of 6.0% and a standard deviation of 1.55%. The z-score is (9.1 - 6.0) / 1.55 = 2.0. The fund is two standard deviations above its peers, which prompts a closer look at how much risk it took to get there.
Example
A fictional lab is graded against an assigned value under a proficiency scheme. Its z-score is 2.4, which is above the warning level of 2 but below the action level of 3. The lab reviews its method and repeats the measurement before the next round of testing.
Formula
Calculation
Z = (x - mean) / standard deviation.
Modified z-score: M = 0.6745 x (x - median) / MAD, where MAD is the median of the absolute deviations from the median.
Worked example with assumed figures: eight values are 40, 42, 43, 43, 44, 45, 46 and 120. The mean is 52.875 and the sample standard deviation is 27.18.
The z-score of 120 is (120 - 52.875) / 27.18 = 2.47.
The largest z-score possible with 8 values is (n - 1) / sqrt(n) = 7 / 2.83 = 2.47, so 120 is close to the cap.
For the modified score, the median is 43.5 and MAD is 1.5. M for 120 is 0.6745 x 76.5 / 1.5 = 34.4, which flags it clearly.
Comparing across scales: a fictional student scores 82 in maths, where the class mean is 70 and the standard deviation is 8, and 74 in science, where the class mean is 60 and the standard deviation is 7. The maths z-score is (82 - 70) / 8 = 1.5 and the science z-score is (74 - 60) / 7 = 2.0. Although the raw science mark is lower, it is the stronger result relative to the class.Case study
Seen in the real world.
This case study is fictional and illustrative. A bank analyst reviews daily processing times in minutes for eight branches. Seven are between 40 and 46, and one reports 120. She computes z-scores first.
The mean is 52.875 and the standard deviation is 27.18, so the 120 has a z-score of 2.47. That is above 2 but below 3, so a rule that flags only values above 3 would miss it. She then notes that with eight values, no z-score can exceed 2.47, so the 120 is already at the ceiling. She switches to the modified z-score based on the median of 43.5 and MAD of 1.5.
The 120 scores 34.4, far above any usual cutoff. She finds a system outage at that branch. The outlier inflated the standard deviation, so the plain z-score understated it.
Watch out
Common mistakes.
- Using a cutoff of 3 on small samples, when the largest possible z-score depends on n and the outlier can be missed.
- Reading a z-score as a probability without checking the distribution, since a score of 2 does not mean a fixed chance.
- Mixing up the sample and the assigned values. A proficiency z-score uses an assigned value and sigma, not the sample mean.
Questions
People also ask.
What is a z-score?
It is the number of standard deviations a value is from the mean, found by subtracting the mean and dividing by the standard deviation.
What does a z-score of 2 mean?
The value is two standard deviations above the mean. In proficiency testing under ISO 13528, an absolute value above 2 is a warning signal and above 3 an action signal.
Is a z-score good for finding outliers?
It can help, but NIST warns it can mislead for small samples. Iglewicz and Hoaglin suggest a modified z-score based on the median and MAD.
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