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Z-scores review

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=data pointmeanstandard deviation
Here's the same formula written with symbols:
z=xμσ
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 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 μ=85 and a standard deviation of σ=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
Michael's z-score is 0.5. 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 μ=82 and a standard deviation of σ=4.
Michael scored 74 on the exam.
Find the z-score for Michael's exam grade.
z=his grademean gradestandard deviationz=74824z=84=2
Michael's z-score is 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 μ=74 and a standard deviation of σ=4.0.
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 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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

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