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Standard Scores and Z-Scores

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Source: AP Statistics 2007 Exam

Hi. This tutorial covers standard scores, which are also known as z-scores. Let's just start with an example.

Louise is enrolled in a very large college calculus class. On the first exam, the class mean was 75 and the standard deviation was 8. On the second exam, the class mean was 70 and the standard deviation was 15. Louise scored 85 on both exams.

Assuming the scores on each exam were, approximately normally distributed, on which exam did Louise score better relative to the rest of the class? So that word relative is important there. So even though she scored 85 on each, using this the mean and the standard deviation, we might be able to figure out which score she did or what test she did better on relative to the rest of the class.

All right, so to make relative comparisons between two values on two different normal distributions, standard scores must be calculated. A standard score is abbreviated as a z-score, standard scores are called standardized scores, which is where the z comes from. So we have two formulas, where we can calculate standard scores or z-scores.

I think we'll just call them z-scores from now on. What they are is you always take x. x is just going to be whatever data value you're going to calculate your z-score for-- minus the mean divided by the standard deviation.

So you notice that the difference between these two formulas is these are using sample statistics. These are using population parameters. So this is the formula for sample. This is the formula for population.

In our example, with a college calculus class, we have the entire population there. So we're going to use this version of the z-score formula. But just remember, data value minus mean divided by the standard deviation. So basically, what you're standardized or your standard score or z-score is going to tell you is how many standard deviations above or below a data value-- above or below the mean a data value is.

So let's actually calculate some z-scores for Louise's tests. So let's do exam 1 first. So exam 1, to calculate the z-score, it's x minus the mean divided by the standard deviation. So 85 minus 75 divided by 8, the data value minus the mean divided by the standard deviation.

So if we do that, that's going to give 10/8, which is going to end up being-- that reduces to 5/4. And as a decimal that be 1.25.

So on exam 1, Louise scored 1.25 standard deviations above the mean. Now let's also do it for exam 2. So exam 2, we're still going to do 85, because that's the data value. That's her raw score on the test.

Now we're going to subtract the mean and divide by the standard deviation. So what that ends up giving us is 85 minus 70 is 15 divided by 15. 15/15 it is equal to 1. So what that means is that Louise scored one standard deviation above the mean for exam 2.

So remember the original question was is, on which exam did Louise score better relative to the rest of the class? So even though she got the same score on each, relative to the class she did better on exam 1. Now let's talk a little bit more about what z-scores help indicate.

So a positive z-score indicates that the data value is above the mean. So positive is above the mean. A negative z-score indicates that a data values below the mean.

So if it's negative, it's below the mean. And a z-score of 0 indicates that the data value is the same as the mean. And just to make sure you understand where that's coming from. If we get a negative z-score, something within this value needs to be negative.

Standard deviation can never be negative. So what has to be negative is the numerator. So x minus the mean. So basically, if this value is less than the mean, that's going to give you a negative z-score.

So a negative in z-score indicates that that value was negative. So that means the data value is less than the mean. If the z-score is 0, that means the these two values were the same, because if you subtract them, you get 0.

0 divided by anything is still 0. And then if this z-score is positive, that means this is positive. So that means x is bigger than the mean.

So make sure we remember each of these properties. Well, that has been your tutorial on standard scores, also known as z-scores. Thanks for watching.