How to Find a Z-Score

Contributor: Lynn Ellis. Lesson ID: 13773

Z-scores are not what you get when zebras play video games, but they are an important concept in statistics! Learn what they are and how to calculate them.


Measurement and Data, Statistics and Probability

learning style
personality style
Grade Level
High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

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Andrew and Jorge were very competitive in math class.

sat test

In 2019, Andrew took the SAT and got 740 on the math portion. Jorge, on the other hand, took the ACT that same year and got a score of 33 on the math portion. Of course, they want to know who did better.

  • How can they answer that question?

Andrew and Jorge can settle this question by looking at z-scores.

To learn what a z-score is and how to calculate it, watch Z Scores from Parabola Magic:

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Now that you know about z-scores, let's apply that to Andrew and Jorge.

SAT scores on the math portion followed a normal distribution with a mean of 527 and a standard deviation of 107. ACT scores on the math portion followed a normal distribution with a mean of 20.4 and a standard deviation 0f 5.6.

We will calculate Andrew's z-score first. Remember that:

  z = (x - mu)


In this case:

  z = (740 - 527) or z = 1.99


That tells us that Andrew's score is 1.99 standard deviations above the mean score.

Now for Jorge's z-score:

  z = (33 - 20.4) or z = 2.25


That tells us that Jorge's score is 2.25 standard deviations above the mean.

Without calculating z-scores, we couldn't compare the performance of Andrew and Jorge because their scores were on different scales. By calculating their z-scores, we have put the scores onto the same scale, and we can compare them.

Jorge did better on the ACT than Andrew did on the SAT in this example.

Now, you practice finding z-scores.

For this exercise, let's work with a distribution that is normal with a standard deviation of 4 and a mean of 30.

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Let's do a little practice comparing z-scores.

Assume that you sleep 8 hours one night, and your father sleeps 7.5 hours that same night. Your father claims that you got more sleep than he did, but you want to use z-scores to argue against that.


You know that teens' sleep follows a normal distribution with a mean of 7.6 hours and a standard deviation of 1.8 hours. You also know that adults' sleep follows a normal distribution with a mean of 6.43 hours and a standard deviation of 1.3 hours.

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If you feel like you understand the concept of a z-score well, move on to the Got It? section.

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