How to Find a Z-Score

Lesson Plan - Get It!

Audio:

What Is a Z-Score, Really?

A z-score tells you how far a value is from the average (mean), measured in standard deviations.

In plain terms, a z-score answers this question: How unusual is this value compared to everything else?

A z-score of 0 means the value is exactly average.

A positive z-score means the value is above average.

A negative z-score means the value is below average.

The bigger the number (positive or negative), the farther the value is from the mean.

Why Z-Scores Matter

Z-scores allow you to:

Compare values from completely different data sets.

Understand how extreme or typical a value is.

Analyze data that follows a normal distribution (the classic bell curve).

This is why z-scores show up everywhere—test scores, sports stats, psychology studies, and even business predictions.

The Formula You Need

To calculate a z-score, use this formula:

Here’s what each part means:

x = the value you are analyzing

? (mu) = the mean (average)

? (sigma) = the standard deviation

This formula measures how far your value is from the mean, then scales it by the spread of the data.

Step-by-Step: How to Find a Z-Score

Follow this process every time:

1. Identify the value (x).

2. Find the mean (?).

3. Find the standard deviation (?).

4. Subtract the mean from the value.

5. Divide by the standard deviation.

That’s it. No shortcuts needed.

Let’s Apply It

Go back to the test score example.

SAT math scores:

• Mean = 527
  Standard deviation = 107
  Student score = 740

ACT math scores:

• Mean = 20.4
  Standard deviation = 5.6
  Student score = 33

Now calculate.

SAT:

• (740 ? 527) ÷ 107 ? 1.99
• This score is about 2 standard deviations above the mean.

ACT:

• (33 ? 20.4) ÷ 5.6 ? 2.25
• This score is more than 2 standard deviations above the mean.

Even though 740 looks bigger than 33, the ACT score is actually more impressive relative to its distribution.

That’s the power of z-scores—they level the playing field.

Real-Life Thinking

Z-scores are not just about tests. They help answer questions like:

• Is this athlete performing above average?
• Is this company financially stable compared to others?
• Is this data point typical or unusual?

Once you understand z-scores, raw numbers stop being confusing—and start telling a clear story.

Wrapping It Up

Now you know how to:

Define a z-score.

Use the formula correctly.

Interpret what the result means.

Compare values across different scales.

Next, it’s time to put this into action. You’ll calculate z-scores, interpret results, and test your understanding with guided practice.