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4 Tutorials that teach Standard Scores and Z-Scores
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Standard Scores and Z-Scores
Common Core: S.ID.4

Standard Scores and Z-Scores

Author: Katherine Williams

Calculate a z-score using a given value and the corresponding mean and standard deviation.

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Video Transcription

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This tutorial covers standard scores. Standard scores are a way of making a comparison between two values that are from different data sets and on different normal distribution curves. Now, when you calculate your standard score, you can get a positive number or a negative number. If it's a positive number, that means that your score is above the mean. If you get a negative number, that means your score is less than the mean. If you get a 0, that means your score is the same as the mean.

When you're calculating a standard score, which is also called a z-score, you're doing a particular calculation. You're saying whatever value you're looking at minus the mean divided by the standard deviation. Here is an example. I'm going to start by reading it. And then we'll go back and pick out the important information.

Sasha earns an 84% on her chemistry test where the mean was an 80 and the standard deviation was 5. She earns an 82 on her physics test where the mean was a 76 and the standard deviation was 6. Which class did she perform better in?

So first, I like to start recording my important information so that we can get rid of all this text. So she has a chemistry class and a physics class. And she took tests in both.

So in chemistry, her score was an 84. The mean was an 80. And the standard deviation is 5. Because these are my notes and I'm trying to write them shorthand, I'm going to be using the symbols for mean and standard deviation.

Now in physics, her score was an 82. The mean was 76. And the standard deviation was 6.

So now that I have all the important information from the text, I don't need to look at that anymore. So now that we have the information out, we have some room to calculators our z-scores. So here we have a score of 84, a mean of 80, and a deviation of 5. So we're going to do the value that we're looking at, the 84 minus 80 divided by 5, divided by the deviation. 84 minus 80 is 4, divided by 5, 4/5.

Now, it's usually easier to compare decimals than fractions, so I'm going to convert that into decimal. 4 divided by 5 is going to give me 0.8. So Sasha's z-score for chemistry is a 0.8.

Now we're going to repeat for physics. So for physics, her score was an 82. She had a mean of 76 and a deviation of 6. So we're going to have-- and I'm going to [INAUDIBLE] 82 minus 76 divided by 6. 82 minus 76 is going to be 6 divided by 6. And then I have a fraction. 6 divided by 6 converts to a decimal of 1.

So we can see her score in chemistry gave her a z-score of 0.8. Her score in physics gave her a z-score of 1. In both cases, they're positive numbers. So in both cases, she scored above the mean, which we could tell right away from our data.

Now, which class did she do better in? Sasha does better in physics than in chemistry, because her z-score ended up being higher. Even though she had a higher score in chemistry, because of where the mean and the standard deviation fell, her physics score was better overall. So this has been your tutorial on standard scores, which are also called z-scores.

Terms to Know
Standard Scores/z-scores

A value that explains how many standard deviations away from the mean an observation is. It can be positive (if the value is above the mean) or negative (if the value is below the mean).

Formulas to Know

z space minus s c o r e space f o r space s a m p l e equals fraction numerator x minus top enclose x over denominator s end fraction
z space minus s c o r e space f o r space p o p u l a t i o n equals fraction numerator x minus mu over denominator sigma end fraction