Z-Scores

Learn what a z-score tells you, how to calculate it, how to turn it back into a value, and how to compare results from different tests.

Advanced25 min18 problems
Practice Problems

Definition

A z-score (also called a standard score) tells you how many standard deviations a value lies above or below the mean.
z=x−μσz = \frac{x - \mu}{\sigma}
  • xx is the value you look at
  • μ\mu (mu) is the mean of the distribution
  • σ\sigma (sigma) is the standard deviation
How to read it:
  • z>0z > 0: the value is above the mean
  • z<0z < 0: the value is below the mean
  • z=0z = 0: the value is exactly the mean
  • ∣z∣|z| says how far away it is, measured in standard deviations
Example: on a math test with μ=74\mu = 74 and σ=6\sigma = 6, a score of 8282 has z=82−746=86≈1.33z = \frac{82 - 74}{6} = \frac{8}{6} \approx 1.33. The score is 1.331.33 standard deviations above the mean.
Going back from z to x: solve the formula for xx:
x=μ+zσx = \mu + z\sigma
Because the units cancel (points divided by points), a z-score has no unit. That is what makes it so useful: you can compare a test score, a height and a running time on one common scale. A value with ∣z∣>2|z| > 2 is often called unusual, because it lies more than 2 standard deviations from the mean.

Try it now

Test scores have mean μ=50\mu = 50 and standard deviation σ=5\sigma = 5. What is the z-score of a score of 6060?

Worked Examples

On a math test, the scores have mean μ=74\mu = 74 points and standard deviation σ=6\sigma = 6 points. Mia scored 8282. What is her z-score?
1

Write the formula

z=x−μσz = \frac{x - \mu}{\sigma} → x=82x = 82, μ=74\mu = 74, σ=6\sigma = 6
2

Subtract the mean (distance from the mean)

82−74=882 - 74 = 8 → Mia is 8 points above the mean
3

Divide by the standard deviation

z=86=43≈1.33z = \frac{8}{6} = \frac{4}{3} \approx 1.33 → z≈1.33z \approx 1.33

Common Mistakes

Subtracting in the wrong order: μ−xσ\frac{\mu - x}{\sigma}

Why it's wrong: This flips the sign. A value above the mean would get a negative z-score, which says the opposite of the truth.

Correct: Always value minus mean: z=x−μσz = \frac{x - \mu}{\sigma}. For x=82x = 82, μ=74\mu = 74, σ=6\sigma = 6: z=86≈1.33z = \frac{8}{6} \approx 1.33, not −1.33-1.33.

Forgetting the brackets: typing 82−74/682 - 74 / 6 into a calculator

Why it's wrong: The calculator divides first and gets 82−12.33=69.6782 - 12.33 = 69.67, which is not a z-score at all.

Correct: Subtract first, then divide: (82−74)/6≈1.33(82 - 74) / 6 \approx 1.33.

Comparing raw scores from different tests

Why it's wrong: 88 points on a test with μ=80\mu = 80, σ=8\sigma = 8 is not automatically better than 82 points on a test with μ=74\mu = 74, σ=6\sigma = 6. The tests have different centers and spreads.

Correct: Convert each value to a z-score first, then compare: 11 versus 1.331.33, so the 82 is relatively better.

Thinking a negative z-score means a bad result

Why it's wrong: The sign only tells you below or above the mean. Whether that is good depends on the context.

Correct: For running times or prices, lower is better, so z=−2z = -2 can be an excellent result. For test scores, higher is better.

Interactive Visual

Math test: μ = 74 points, σ = 6 points. Drag the score and watch z

82 · z = 1.33xz56-362-268-1740801862923Math test score (points)

μ = 74 points, σ = 6 points · Drag a handle or type x or z.

z = (x − μ)/σ = (82 − 74)/6 = 1.33

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Practice Problems

18 problems
Problem 1 of 18
Easy
Test scores have mean μ=50\mu = 50 and standard deviation σ=5\sigma = 5. What is the z-score of a score of 6060?

Why It Matters

Raw numbers alone often do not tell you much. Is 82 points a good score? It depends on how everyone else did. A z-score answers that question in one number.
  • School and exams: comparing results from different classes, subjects or years, even when the tests had different difficulty
  • Medicine: growth charts for children and bone density reports are given as z-scores, a child with z=−2.5z = -2.5 for height is checked by a doctor
  • Sports: comparing athletes across different events, like a sprinter and a long jumper
  • Quality control: factories flag products whose measurements have ∣z∣>2|z| > 2 or ∣z∣>3|z| > 3
  • Finance and science: spotting unusual values (outliers) in data
In the next lessons you will use z-scores to find areas and probabilities under the normal curve, so this is the key step.

Real World Applications

Sports: Comparing Sprint Times

Coaches compare athletes with z-scores. In races a lower time is better, so a negative z-score means faster than average.

Example:

In a regional 100 m race the times have μ=11.0\mu = 11.0 s and σ=0.2\sigma = 0.2 s. A time of 11.311.3 s has z=0.30.2=1.5z = \frac{0.3}{0.2} = 1.5: slower than average.

1Try It Yourself

In the same race (μ=11.0\mu = 11.0 s, σ=0.2\sigma = 0.2 s), Sara runs 10.610.6 s.

What is Sara's z-score?

Step 1: Write the mathematical expression

Calculate 10.6−11.00.2\frac{10.6 - 11.0}{0.2}:

Medicine: Growth Charts

Pediatricians describe a child's height or weight as a z-score compared with children of the same age. Values below $z = -2$ are checked more closely.

Example:

A child with height z=0z = 0 is exactly average for the age, z=1z = 1 is one standard deviation taller than average.

2Try It Yourself

For 8-year-olds, heights have μ=128\mu = 128 cm and σ=5.5\sigma = 5.5 cm. A doctor checks every child with z<−2z < -2.

Below which height (in cm) is a child checked?

Step 1: Write the mathematical expression

Use x=μ+zσx = \mu + z\sigma with z=−2z = -2:

Quality Control: Cereal Boxes

A factory fills cereal boxes. Boxes whose weight has $|z| > 2$ are taken off the line and checked.

Example:

The filling machine gives μ=500\mu = 500 g and σ=4\sigma = 4 g. A box of 504504 g has z=1z = 1: perfectly normal.

3Try It Yourself

A box on the line weighs 489489 g (μ=500\mu = 500 g, σ=4\sigma = 4 g).

What is the z-score of this box?

Step 1: Write the mathematical expression

Calculate 489−5004\frac{489 - 500}{4}:

Key Takeaways

  • 1A z-score tells how many standard deviations a value is from the mean: z=x−μσz = \frac{x - \mu}{\sigma}
  • 2Positive z: above the mean, negative z: below the mean, z=0z = 0: exactly the mean
  • 3To go back from z to a value, use x=μ+zσx = \mu + z\sigma
  • 4A z-score has no unit, so values from different distributions can be compared: the larger z is relatively higher
  • 5A value with ∣z∣>2|z| > 2 is usually called unusual, it lies more than 2 standard deviations from the mean
  • 6Always subtract first, then divide: (x−μ)/σ(x - \mu) / \sigma

Frequently Asked Questions

Yes. Most z-scores are decimals like 1.331.33 or −0.75-0.75. Values beyond 33 or −3-3 are possible, just very rare in a normal distribution.
Yes. Most z-scores are decimals like 1.331.33 or −0.75-0.75. Values beyond 33 or −3-3 are possible, just very rare in a normal distribution.
You can calculate a z-score for any data with a mean and a standard deviation. Turning it into a percentage (like the 68-95-99.7 rule) only works well when the data are about normal.
Because x−μ=0x - \mu = 0 when x=μx = \mu, and 00 divided by σ\sigma is 00. The mean is zero standard deviations away from itself.
A z-score measures distance in standard deviations, a percentile says what percent of values lie below. They are linked (for normal data, z=1z = 1 is about the 84th percentile), and you will learn to convert between them in the next lessons.

Glossary

Z-score (standard score)
The number of standard deviations a value lies above (positive) or below (negative) the mean
Mean (μ)
The center of the distribution, the average value
Standard deviation (σ)
A measure of how spread out the values are around the mean
Standardizing
Converting a value xx into its z-score with z=x−μσz = \frac{x - \mu}{\sigma}
Unusual value
A value more than 2 standard deviations from the mean, so ∣z∣>2|z| > 2

Formula Card

Z-score

z=x−μσz = \frac{x - \mu}{\sigma}

How many standard deviations x is from the mean

Value from a z-score

x=μ+zσx = \mu + z\sigma

Turn a z-score back into the original units

Unusual values

∣z∣>2|z| > 2

More than 2 standard deviations from the mean

Today's challengeOne problem, three tries

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