Z-Score
In short: A z-score counts the number of spread-sized steps between a data value and its group’s average, positive above and negative below, so unlike scales can be compared fairly.
Algebra 2StatisticsS-ID.A.4
What is Z-Score?
A z-score turns a raw measurement into a number of steps away from typical. A result of 84 means little alone, but being 1.75 steps above typical shows exactly how unusual it is.
Given a value \(x\), a center \(\mu\) and a spread \(\sigma\), the formula is \(z = \frac{x – \mu}{\sigma}\). Above-center values produce positive answers, below-center values produce negative ones, and the center itself gives zero. Units cancel, so a quiz grade and a race time can be placed side by side.
How to spot it
- Two results come from different tests or scales, and you must decide which one is better relative to its group.
- A question asks for the z-score of a value, meaning how many standard deviations it lies from the mean.
- A value must be called unusual or typical, and a common cutoff is a z-score beyond 2 or below -2.
- A table of normal areas is needed, and the entry point is a standardized value.
The method
- Find the mean \(\mu\) and standard deviation \(\sigma\) of the group the value belongs to.
- Subtract the mean from the value: \(x – \mu\).
- Divide by the standard deviation to get \(z\).
- Interpret the sign and size: how far, and in which direction, from the mean.
Z-Score: solved example
Problem
Maya earned 84 on a unit exam whose class average was 70 with a spread of 8. Her result on a later quiz was 61, where the class average was 50 with a spread of 5. Which result stands out more compared with classmates?
Solution
First test: \(z = \frac{84 – 70}{8} = 1.75\). Second test: \(z = \frac{61 – 50}{5} = 2.2\).
Because \(2.2 > 1.75\), the quiz result sits farther above its own class average. Relative to classmates she did better on the quiz, even though the raw number is smaller.
Try it
Question 1
Standardize \(x = 45\) for a group with center 60 and spread 12.
Show the answer
\(z = \frac{45 – 60}{12} = -1.25\): the value is 1.25 steps under the center.
Question 2
A standardized value is 2.5 for a group centered at 18 with spread 1.6. Recover \(x\).
Show the answer
\(x = 18 + 2.5(1.6) = 22\).
Z-Score vs. Standard Deviation
Standard deviation measures the typical spread of a whole data set and is expressed in the data’s own units. A standardized score is about one value: it uses that spread as its yardstick and reports how many yardsticks the value sits from the mean. One describes the group, the other places an individual inside it.
Keep going
Lessons
Worksheets and quizzes
- The Normal Distribution – Worksheet
- The Normal Distribution – Quiz
- z-Scores and the Normal Distribution – Worksheet
- z-Scores and the Normal Distribution – Quiz
Related words
- Normal Distribution
- Empirical Rule (68-95-99.7)
- Standard Deviation
- Mean, Median and Mode
- Margin of Error
Whole dictionary: A–Z · Algebra 2 · Statistics