What are z-scores?

A z-score measures exactly how many standard deviations above or below the mean a data point is.
Here's the formula for calculating a z-score:
z, equals, start fraction, d, a, t, a, space, p, o, i, n, t, minus, m, e, a, n, divided by, s, t, a, n, d, a, r, d, space, d, e, v, i, a, t, i, o, n, end fraction
Here's the same formula written with symbols:
z, equals, start fraction, x, minus, mu, divided by, sigma, end fraction
Here are some important facts about z-scores:
  • A positive z-score says the data point is above average.
  • A negative z-score says the data point is below average.
  • A z-score close to 0 says the data point is close to average.
  • A data point can be considered unusual if its z-score is above 3 or below minus, 3.
Want to learn more about z-scores? Check out this video.

Example 1

The grades on a history midterm at Almond have a mean of mu, equals, 85 and a standard deviation of sigma, equals, 2.
Michael scored 86 on the exam.
Find the z-score for Michael's exam grade.
z=his grademean gradestandard deviationz=86852z=12=0.5\begin{aligned}z&=\dfrac{\text{his grade}-\text{mean grade}}{\text{standard deviation}}\\ \\ z&=\dfrac{86-85}{2}\\ \\ z&=\dfrac{1}{2}=0.5\end{aligned}
Michael's z-score is 0, point, 5. His grade was half of a standard deviation above the mean.

Example 2

The grades on a geometry midterm at Almond have a mean of mu, equals, 82 and a standard deviation of sigma, equals, 4.
Michael scored 74 on the exam.
Find the z-score for Michael's exam grade.
z=his grademean gradestandard deviationz=74824z=84=2\begin{aligned}z&=\dfrac{\text{his grade}-\text{mean grade}}{\text{standard deviation}}\\ \\ z&=\dfrac{74-82}{4}\\ \\ z&=\dfrac{-8}{4}=-2\end{aligned}
Michael's z-score is minus, 2. His grade was two standard deviations below the mean.

Practice problems

Problem A
The grades on a geometry midterm at Oak have a mean of mu, equals, 74 and a standard deviation of sigma, equals, 4, point, 0.
Nadia scored 70 on the exam.
Find the z-score for Nadia's exam grade. Round to two decimal places.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Want to practice more problems like these? Check out this exercise.