# Standard z-Scores – How to Explain Them

“Everybody’s different,” I told K8 the other day.

And now I’m prepping standard z-scores for the statistics part of Finite Math.

So I’m thinking about how much everybody’s different. Like really – the actual amount of different each person is.

### Standard z-Scores help us see this!

Let’s say “average” means you’re no different. So being above or below average means you’re different.

And since everybody’s different in different things, how do we measure just how much different each person is – overall?

I propose we can do this with standard z-scores.

Or we can at least teach standard z-scores with this analogy.

### Standard z-scores work in real life.

Let’s look at three qualities: smartness, friendliness and athleticism. Here’s our assumed numbers:

• The average smartness is 100, with a standard deviation of 21.
• The average friendliness is 23, with a standard deviation of 5.
• And the average athleticism is 842 with a standard deviation of 57.

Now let’s take three “different” people.

• Alice is super smart – she’s 132 in the smartness scale.
• Bert is very nice to people – he’s 27 in the friendliness scale.
• Charlie uses his brain more than his body – he’s 753 in the athleticism scale.

Each of these people are different in their own areas. But if we were going to measure how different they were – like who’s the most different – it would be hard.

Is Alice more different than Bert? How about Charlie?

And how can we tell?

### Standardizing their scores levels the playing field!

Using the standard z-score we can see how different each person is.

Here’s the formula:

z = (their score – average) ÷ standard deviation

So for each:

• Alice’s z-score = (132 – 100) ÷ 21 = 1.52
• Bert’s z-score = (27 – 23) ÷ 5 = 0.80
• Charlie’s z-score = (753 – 842) ÷ 57 = – 1.56

Even though Charlie’s score is negative, the absolute value of it is more than either Bert’s standard score or Alice’s.

So Charlie is the most different!

### What is the standard z-score – really?

The Standard z-score is the number of standard deviations above the average something is. But when you say that, it doesn’t make much sense. At least until you have a feel for what that could mean in real life.

You can use this real life example (although less “real” than “life”) to show students what the purpose of a z-score is. And why it’s not just some obscure statistical thing – like it seems!

Will you use this when teaching statistics? Can you improve on it?

Share your thoughts in the comments – and wish me luck in my class today!

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