Teacher Guide: 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.

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Worksheet Answer key

Class quiz

10 questions on Probability Distributions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Calculate a z-score with $z = \frac{x - \mu}{\sigma}$
Interpret the sign and size of a z-score in context
Find a value from a given z-score with $x = \mu + z\sigma$
Compare values from different distributions using z-scores
Identify unusual values using the rule $|z| > 2$

Prerequisites

• Introduction to Normal Distribution (mean, standard deviation, 68-95-99.7 rule)
• Calculating the mean and standard deviation of data
• Solving simple linear equations
• Working with negative numbers and decimals

Discussion Starters

1. You got 70 points on a test. Without knowing anything else, can you say if that is good? What else would you want to know?
2. Two students both scored 8 points above their class mean. How can one of them still have done relatively better?
3. In a race, is a z-score of $-2$ good or bad? What about on a test?
4. Why does a z-score have no unit, and why is that useful?

Common Misconceptions

A higher raw score is always the better result

Remediation: Use the Mia and Leo example: Leo has more points but a lower z-score. Ask students to explain in their own words why spread matters.

A negative z-score is a bad result

Remediation: Show a context where lower is better (sprint times, prices, errors) and let students decide when negative z is good.

A z-score is a percentage or a probability

Remediation: Stress that z counts standard deviations. Connect $z = 1$ to the 68-95-99.7 rule from the previous lesson to show that the percentage is a separate step.

Differentiation Ideas

For Struggling Students:

• Start with whole-number z-scores (values exactly 1 or 2 standard deviations away)
• Draw a number line with the mean and steps of one standard deviation marked
• Do the two steps separately: first the distance from the mean, then divide

For On-Level Students:

• Calculate and interpret decimal z-scores in different contexts
• Back-solve values from z-scores with $x = \mu + z\sigma$
• Compare results from two or three different distributions

For Advanced Students:

• Find $\sigma$ or $\mu$ when a value and its z-score are given
• Find both $\mu$ and $\sigma$ from two values and their z-scores
• Explain why adding the same number to every value does not change the z-scores

Standards Alignment

HSS-ID.A.4 (CCSS.MATH.CONTENT.HSS.ID.A.4)
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages
HSS-ID.A.2 (CCSS.MATH.CONTENT.HSS.ID.A.2)
Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets

Lesson Resources

visualZ-Score Explorer
Drag a test score along the bell curve and watch its z-score change on the aligned z axis
activityWho Did Better?
Students compare results from two classes with different means and spreads, first by points, then by z-scores
worksheetZ-Scores in Context
Calculate, interpret and back-solve z-scores for tests, heights, sprint times and factory data

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A z-score (also called a standard score) tells you how many standard deviations a value lies above or below the mean.

z = \frac{x - \mu}{\sigma}

$x$ is the value you look at
$\mu$ (mu) is the mean of the distribution
$\sigma$ (sigma) is the standard deviation

How to read it:

$z > 0$ : the value is above the mean
$z < 0$ : the value is below the mean
$z = 0$ : the value is exactly the mean
$|z|$ says how far away it is, measured in standard deviations

Example: on a math test with

\mu = 74

and

\sigma = 6

, a score of

82

has

z = \frac{82 - 74}{6} = \frac{8}{6} \approx 1.33

. The score is

1.33

standard deviations above the mean.

Going back from z to x: solve the formula for

x

x = \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

is often called unusual, because it lies more than 2 standard deviations from the mean.

Worked Examples

On a math test, the scores have mean

\mu = 74

points and standard deviation

\sigma = 6

points. Mia scored

82

. What is her z-score?

Write the formula

z = \frac{x - \mu}{\sigma}

→

x = 82

\mu = 74

\sigma = 6

Subtract the mean (distance from the mean)

82 - 74 = 8

→ Mia is 8 points above the mean

Divide by the standard deviation

z = \frac{8}{6} = \frac{4}{3} \approx 1.33

→

z \approx 1.33

Answer: $z \approx 1.33$ : Mia scored about 1.33 standard deviations above the class mean.

Common Mistakes

Subtracting in the wrong order: $\frac{\mu - x}{\sigma}$

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: $z = \frac{x - \mu}{\sigma}$ . For $x = 82$ , $\mu = 74$ , $\sigma = 6$ : $z = \frac{8}{6} \approx 1.33$ , not $-1.33$ .

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

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

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

Comparing raw scores from different tests

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

Correct: Convert each value to a z-score first, then compare: $1$ versus $1.33$ , 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 $z = -2$ can be an excellent result. For test scores, higher is better.

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 $z = -2.5$ 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 $|z| > 2$ or $|z| > 3$
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 $\mu = 11.0$ s and $\sigma = 0.2$ s. A time of $11.3$ s has $z = \frac{0.3}{0.2} = 1.5$ : slower than average.

1Try It Yourself

In the same race ( $\mu = 11.0$ s, $\sigma = 0.2$ s), Sara runs $10.6$ s.

What is Sara's z-score?

Step 1: Write the mathematical expression

Calculate $\frac{10.6 - 11.0}{0.2}$ :

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 $z = 0$ is exactly average for the age, $z = 1$ is one standard deviation taller than average.

2Try It Yourself

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

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

Step 1: Write the mathematical expression

Use $x = \mu + z\sigma$ with $z = -2$ :

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 $\mu = 500$ g and $\sigma = 4$ g. A box of $504$ g has $z = 1$ : perfectly normal.

3Try It Yourself

A box on the line weighs $489$ g ( $\mu = 500$ g, $\sigma = 4$ g).

What is the z-score of this box?

Step 1: Write the mathematical expression

Calculate $\frac{489 - 500}{4}$ :

Key Takeaways

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

Frequently Asked Questions

Can a z-score be a decimal or bigger than 3?

Yes. Most z-scores are decimals like

1.33

-0.75

. Values beyond

3

-3

are possible, just very rare in a normal distribution.

Does a z-score need the data to be normal?

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.

Why is the z-score of the mean always 0?

Because

x - \mu = 0

when

x = \mu

, and

0

divided by

\sigma

0

. The mean is zero standard deviations away from itself.

What is the difference between a z-score and a percentile?

A z-score measures distance in standard deviations, a percentile says what percent of values lie below. They are linked (for normal data,

z = 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 $x$ into its z-score with $z = \frac{x - \mu}{\sigma}$
Unusual value: A value more than 2 standard deviations from the mean, so $|z| > 2$

Formula Card

Z-score

z = \frac{x - \mu}{\sigma}

How many standard deviations x is from the mean

Value from a z-score

x = \mu + z\sigma

Turn a z-score back into the original units

Unusual values

|z| > 2

More than 2 standard deviations from the mean

Teacher Guide: Z-Scores

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Sports: Comparing Sprint Times

Medicine: Growth Charts

Quality Control: Cereal Boxes

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Z-score

Value from a z-score

Unusual values

More in This Topic

Introduction to Probability Distributions

Binomial Probability

Discrete vs Continuous Distributions

Expected Value