A z-score tells how many standard deviations a value is above or below the mean. Find it with z = x − μσ: subtract the mean, then divide by the standard deviation. A positive z-score is above the mean, a negative one is below, and z = 0 is exactly at the mean.
In Spanish
puntuación z
Also called
standard score, z-value, standardized score
How to say it
ZEE-score
Grade level
Statistics
Algebra 2StatisticsCCSS HSS-ID.A.4SATAPACT
Z-Score in a picture
The z-score re-labels the axis in standard deviations. With mean 70 and standard deviation 10, the score 85 sits at z = 1.5.
Formal definition and notation
For a value x from a distribution with mean μ and standard deviation σ > 0, the z-score is z = x − μσ. For sampledata, use x̄ and s in place of μ and σ. The z-score has no units, so it lets you compare values measured on different scales. If X is normal, Z = X − μσ follows the standard normal distribution, with mean 0 and standard deviation 1.
the population standard deviation (s for a sample)
Key facts
z > 0: above the mean. z < 0: below the mean. z = 0: equal to the mean.
A z-score of 2 means 2 standard deviations above the mean, not 2 points above.
For bell-shaped data, about 95% of z-scores fall between −2 and 2 (the empirical rule).
Values with z beyond ± 3 are unusual and are often checked as possible outliers.
To compare scores from different tests, compare their z-scores: the larger z is relatively better.
Converting every value to a z-score gives a data set with mean 0 and standard deviation 1.
Z-score formula
Z-score formula
z = x − μσ and, solved for the value, x = μ + zσ
x
the data value
μ
the mean
σ
the standard deviation
The units cancel: inches over inches, points over points. That is why z-scores from different tests can be compared.
Try it: how far from the mean?
Interactive
The band shows one standard deviation on each side of the mean. Drag a dot outside the band and estimate its z-score: (value minus mean) divided by the standard deviation.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Z-Score examples, step by step
Example 1: A score below the mean
Warm-up
A test has mean 64 and standard deviation 8. Find the z-score of a score of 52.
Subtract the mean: 52 − 64 = −12.
Divide by the standard deviation: −128 = −1.5.
Answer:z = −1.5: the score is 1.5 standard deviations below the mean.
Example 2: Who did better?
Core
Ana scored 82 on a history test with mean 74 and standard deviation 4. Ben scored 90 on a math test with mean 80 and standard deviation 8. Relative to their classes, who did better?
Ana: z = 82 − 744 = 84 = 2.
Ben: z = 90 − 808 = 108 = 1.25.
Ana’s score is 2 standard deviations above her mean; Ben’s is 1.25 above his.
Answer: Ana did better relative to her class: z = 2 compared with z = 1.25.
Example 3: From a z-score back to a value
Test-style
Heights of adult women in a study have mean 165 cm and standard deviation 7 cm. What height has a z-score of −2?
Use x = μ + zσ.
x = 165 + (−2)(7) = 165 − 14.
x = 151.
Answer: A height of 151 cm has a z-score of -2.
Common mistakes
✗ MistakeForgetting to divide: z = 85 − 70 = 15.
✓ Insteadz = 85 − 7010 = 1.5.
The difference is in points. Dividing by the standard deviation turns it into a number of standard deviations.
✗ MistakeSubtracting in the wrong order: z = 70 − 8510 = −1.5 for a score above the mean.
✓ InsteadAlways value minus mean: 85 − 7010 = 1.5.
The sign carries meaning: positive is above the mean, negative is below.
✗ MistakeDividing by the variance: with variance 100, z = 15100 = 0.15.
✓ InsteadUse the standard deviation √100 = 10: z = 1.5.
The variance is in squared units; the z-score needs the standard deviation.
The standard deviation measures the spread of the whole data set, like 10 points. A z-score describes one value, like 1.5 standard deviations above the mean.
A percentile tells what percent of values are at or below a value. A z-score tells how many standard deviations away it is. For normal data, z = 0 is the 50th percentile and z = 1 is about the 84th.
Quick check: 3 questions
Question 1 of 3
The mean is 100 and the standard deviation is 15. What is the z-score of 130?
Show answer and explanation
Answer: C.z = 130 − 10015 = 3015 = 2. 30 forgets to divide, −2 subtracts in the wrong order, and 0.5 divides the wrong way.
Question 2 of 3
What does z = −0.8 mean?
Show answer and explanation
Answer: B. A z-score counts standard deviations, and the minus sign means below the mean. It is not a number of points and not a percent.
Question 3 of 3
The mean is 50 and the standard deviation is 6. Which value has z = 1.5?
Show answer and explanation
Answer: D.x = 50 + 1.5 × 6 = 59. 51.5 adds 1.5 points, 41 goes below the mean, and 56 is only 1 standard deviation above.
Z-Score: frequently asked questions
What is a good z-score?
It depends on what you measure. For test scores, higher is better, so z = 1 means you beat the mean by one standard deviation, which in bell-shaped data is about the 84th percentile. For times in a race, a negative z-score is better. A z-score near 0 is typical; beyond ± 2 is unusual.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. For example, with mean 64 and standard deviation 8, a score of 52 has z = 52 − 648 = −1.5. Negative is not bad in itself; it only describes the direction from the mean.
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