A z-score reports how many standard deviations a value sits from its mean.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The formula
z = (value − mean) ÷ standard deviation.
The sign
A positive z is above the mean and a negative one is below. A z of zero is exactly at the mean.
Comparing across distributions
Because z has no units, scores from different tests can be compared directly. That is its main purpose.
How unusual
By the empirical rule, a z beyond 2 puts a value in the outer 5%, which is usually called unusual.
A caution
A z-score can be computed for any distribution, but the percentage interpretation needs a roughly normal shape.
How unusual is one value?
z = (value − mean)/standard deviation counts how many standard deviations from the mean a value sits. It puts any measurement on a common scale, so unlike quantities become comparable.
Comparing across different distributions
A test score and a height cannot be compared directly, but their z-scores can. A z of 2 is equally unusual in both, which is exactly what standardising is for.
The sign says which side
A negative z is below the mean, positive above. The magnitude says how far. Both parts carry information, and dropping the sign loses half the statement.
z-scores work for any distribution
Standardising requires only a mean and a standard deviation. Converting a z to a percentile is what requires normality, and confusing the two steps is a frequent slip.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Rosa Compare Two Tests
Rosa scored 80 on a test with mean 70 and standard deviation 5, and 90 on one with mean 85 and standard deviation 10.
- Step 1
For the first test, z = (80 − 70) ÷ 5 = 2.
Step 4: Your Turn
Practice makes it stick.
The Score
Problem 1 of 2
Value 80, mean 70, standard deviation 5. What is the z-score?
The Below
Problem 2 of 2
Value 60, mean 70, standard deviation 5. What is the z-score?
Standardise It
1 of 8
Value 85, mean 75, sd 5. z-score?
2 of 8
Value 70, mean 70, sd 4. z-score?
3 of 8
Value 55, mean 60, sd 10. z-score?
4 of 8
z = 1.5, mean 100, sd 10. What is the value?
5 of 8
z = −1, mean 50, sd 8. What is the value?
6 of 8
Which z is more unusual? 1 for z = 0.4, 2 for z = −2.6.
7 of 8
Put the z-score comparison in order.
- 1Divide by the standard deviation.
- 2Repeat for the other distribution.
- 3Compare the two z-scores.
- 4Subtract the mean from the value.
8 of 8
Value 92, mean 80, sd 6. z-score?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Value 90, mean 78, sd 6. What is the z-score?
Question 2 of 2
Why can z-scores compare results from different tests?
What You Learned
- z = (value − mean) ÷ standard deviation.
- The sign says above or below the mean; the size says how far.
- Because z has no units, values from different distributions become comparable.