Video Notes

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

The formula is simple:

z = (x – mean) / standard deviation

Example

Let’s say you scored 75 out of 100 on a test. At first, that might feel a little disappointing, but let’s calculate its z score to put it in context.

Suppose the other scores in the class were:

scores = c(55, 90, 45, 60, 95, 34)

Let’s calculate the z score:

75 - mean(scores) / sd(scores)

The result is 1.03, meaning your score was 1.03 standard deviations above the mean.

So, even though 75 looked modest, you actually scored above average.

Interpreting Z Scores

• A positive z score means it’s above the mean.
• A negative z score means it’s below the mean.
• If a value has a z score of 0, it’s exactly average.

Comparing Values from Different Distributions

Z scores are also useful for comparing scores across different scales.

For example, suppose you’re comparing two students: one who took the SAT and one who took the ACT.

The SAT ranges from 400–1600, while the ACT ranges from 1–36 — completely different scoring systems.

To compare them fairly, you can use z scores.

The SAT has a mean of 1050 and a standard deviation of 200. The ACT has a mean of 20 and a standard deviation of 5.

• SAT student: 1250 → z = (1250 – 1050) / 200 = 1.0
• ACT student: 26 → z = (26 – 20) / 5 = 1.2

Even though the raw scores are on different scales, the z scores show that the ACT student performed slightly better relative to others who took that test.

That’s the power of z scores — they make scores from different systems comparable on the same standard scale.