Z-Scores

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.

Calculating a Z-Score

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.

A Negative Z-Score

Heights of 15-year-olds in a school have $\mu = 170$ cm and $\sigma = 8$ cm. Tom is $158$ cm tall. Find his z-score and explain what it means.

Subtract the mean: $158 - 170 = -12$ = Tom is 12 cm below the mean

Divide by the standard deviation: $z = \frac{-12}{8} = -1.5$ = $z = -1.5$

Interpret the sign and the size: Negative sign: below the mean. Size $1.5$: one and a half standard deviations = 1.5 SD below the mean

Answer: $z = -1.5$: Tom is 1.5 standard deviations shorter than the average. A negative z-score is not bad, it only means below the mean.

From a Z-Score Back to a Value

Exam scores have $\mu = 64$ and $\sigma = 8$. The top grade goes to everyone with $z \ge 1.5$. What is the lowest score that gets the top grade?

Solve the formula for x: $z = \frac{x - \mu}{\sigma} \Rightarrow x = \mu + z\sigma$ = $x = \mu + z\sigma$

Find how far above the mean that is: $z\sigma = 1.5 \times 8 = 12$ = 12 points above the mean

Add it to the mean: $x = 64 + 12 = 76$ = $x = 76$

Answer: A score of $76$ points or more gets the top grade.

Who Did Relatively Better?

Mia scored $82$ on a math test ($\mu = 74$, $\sigma = 6$). Leo scored $88$ on a physics test in another class ($\mu = 80$, $\sigma = 8$). Leo has more points. Who did better compared with their own class?

Find Mia's z-score: $z_{Mia} = \frac{82 - 74}{6} = \frac{8}{6} \approx 1.33$ = $z_{Mia} \approx 1.33$

Find Leo's z-score: $z_{Leo} = \frac{88 - 80}{8} = \frac{8}{8} = 1$ = $z_{Leo} = 1$

Compare the z-scores: $1.33 > 1$ = Mia stands out more in her class

Answer: Mia did relatively better. Both were 8 points above their mean, but in Mia's class the scores were less spread out, so her 8 points count for more standard deviations.

Is This Value Unusual?

Birth weights of newborns are about normal with $\mu = 3.4$ kg and $\sigma = 0.5$ kg. A baby weighs $2.2$ kg. Is this weight unusual? Also find the range of usual weights (within 2 standard deviations).

Calculate the z-score: $z = \frac{2.2 - 3.4}{0.5} = \frac{-1.2}{0.5} = -2.4$ = $z = -2.4$

Compare with the rule of thumb $|z| > 2$: $|-2.4| = 2.4 > 2$ = Unusual (unusually light)

Find the lower end of the usual range: $x = 3.4 + (-2)(0.5) = 2.4$ = $2.4$ kg

Find the upper end of the usual range: $x = 3.4 + 2(0.5) = 4.4$ = $4.4$ kg

Answer: $z = -2.4$, so $2.2$ kg is unusually light. Usual weights lie between $2.4$ kg and $4.4$ kg.

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.

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

Why: 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 $-1.33$.

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

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

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

Mistake: Comparing raw scores from different tests

Why: 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.

Mistake: Thinking a negative z-score means a bad result

Why: 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.

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.

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.

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.

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

Quality Control: Cereal Boxes

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

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

A z-score tells how many standard deviations a value is from the mean: $z = \frac{x - \mu}{\sigma}$

Positive z: above the mean, negative z: below the mean, $z = 0$: exactly the mean

To go back from z to a value, use $x = \mu + z\sigma$

A z-score has no unit, so values from different distributions can be compared: the larger z is relatively higher

A value with $|z| > 2$ is usually called unusual, it lies more than 2 standard deviations from the mean

Always subtract first, then divide: $(x - \mu) / \sigma$

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

A: Yes. Most z-scores are decimals like $1.33$ or $-0.75$. Values beyond $3$ or $-3$ are possible, just very rare in a normal distribution.

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

A: 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.

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

A: Because $x - \mu = 0$ when $x = \mu$, and $0$ divided by $\sigma$ is $0$. The mean is zero standard deviations away from itself.

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

A: 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.

Exit Presentation

Z-Scores

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Probability / Probability Distributions

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 min

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