# Explaining the concept of $z$-scores in high school statistics

The students have so far studied the uniform probability distribution and have a working familiarity with relative frequency histograms and the 68-95-99.7 empirical rule. They still have trouble with the concept of a random variable, despite numerous examples given in class. The context is a low-income urban school and the students enrolled in this class are considered the best at math. (Statistics is the last course offered in the math program and there are less than 20 students.)

What I've tried doing so far is explaining to them that they could draw a bell-shaped curve over a relative frequency histogram which tells them that the data is normally distributed. I then talked about what would happen if we added 3 to the random variable $X$ and multiplied the random variable $X$ by $\frac{1}{2}$:

My example to them was this: Suppose you had a teacher who is so upset over the performance of his class that he had to bump everyone's grades by 10 points. Let us assume that the data set is normally distributed. What happens to the shape, location and spread of the normal curve? When I told them that everyone from the bottom scorer, the top scorer and the median scorer got a boost and that there was no change in the spread or consistency of the new data (standard deviation), I got blank stares.

I am not allowed to show them these kind of explanations: $Var(cX) = c^2 Var(X)$ and $Var(X+c) = Var(X)$

Three quarters into the lecture, I introduced $z$-scores: $Z = \frac{X - \mu}{\sigma}$ as a way to standardize the normal curve by getting a mean of 0 and variance of 1. (Again, I am not allowed to show them that $E[Z] = 0$ and $Var(Z) = \frac{1}{\sigma^2} Var(X) = \frac{\sigma^2}{\sigma^2} = 1$.) I told them that the area would still be 1 under this $z$-score transformation. A lot of the students gave me confused looks and there was a frustrated kid who yelled "Why do we have to learn this stuff? I don't get this z-score mumbo jumbo."

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The question "Why do I have to learn this stuff?" is almost never an interrogative question. Instead, it is a rephrasing of the statement, "please go away I do not want to learn this stuff." New justifications, no matter how clever, will probably not make it any more interesting. For most students, they actually don't need to know it, anyways. In certain socioeconomic demographics, learning for the sake of learning is often not seen as a path to success among the students' communities. Try learning for the sake of empowerment, instead. – Arkamis Jan 7 '14 at 22:06

When I teach this subject, I work through several examples. One example is the mean height of students at the school, and I give hypothetical numbers like $\mu = 64$, $\sigma = 4$. Then I ask them, based on these values, what is the probability that a randomly selected student will be shorter than $64$? Taller than $68$? And then suppose I go to an elementary school and measure the height of the students there. Is it reasonable to assume the average height there is the same, $64$? Why not? Is it also reasonable to assume the standard deviation is the same? I hypothesize that the distribution of heights for the elementary school has some other mean and standard deviation, say, $\mu = 45$, $\sigma = 2$, and again I go through analogous examples.

So far, your students should be able to understand these concepts and calculate the relevant probabilities using the empirical rule.

Then I ask the question that leads into the need for $z$-scores: Why is it that the chance that a randomly selected student from your school is shorter than $64$ the same as the chance that a randomly selected student from the elementary school is shorter than $45$? Is that the same as saying that $64 = 45$? Why not? Similarly, why is the chance that a randomly selected student at your school is taller than $68$ equal to the chance that a randomly selected student at the elementary school is taller than $47$?

Then I talk about how a $z$-score is a measure of how many standard deviations an observation (i.e. randomly selected student) is away from the average/mean/expected value. In both cases, the observations $68$ and $47$ are one standard deviation greater than their respective means, so both their $z$-scores are 1. As a result, they also represent analogous probabilities. I would talk about how $z$-scores allow us in some sense to compare how likely we are to obtain observations from two different populations. An observation that has a $z$-score of $1$ is not unlikely, but an observation with a $z$-score of $4$ is very, very unlikely--it would be an "outlier," being $4$ standard deviations above the mean.

I could go on, but this should give students a non-formula based understanding of the underlying concepts. From there, I would continue to work out more examples with easy-to-calculate scores, and then finally introduce the $Z = \frac{X-\mu}{\sigma}$ formula, as it formalizes the intuition that you've developed, and finally get $z$-scores that can be any value, like $-1.352$, or $0.579$, etc.

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very nice examples and explanations, thanks heropup! – alo Jan 7 '14 at 22:37

I'm a young mathematics student and remember finding statistics one of the more difficult parts of the curriculum back in high school... I guess this is because teachers are not allowed to make it 'too technical'.

I might have an idea to explain your first example. I think that you make things to abstract when you say that the data set of the points is normally distributed. Just make up a set of scores. Draw a graph on the board. Now add 10 points to each score, and plot this one as well again. Ask if they notice anything. Ask them to calculute the mean and the variance of these two data sets. Ask them if they can draw conclusions themselves. I always thought it was more fun when I found the 'connections' myself.

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