# The Normal Curve and Z-Scores

Psychology I · Statistics · https://tryals.app/learn/psychology-i/the-normal-curve-and-z-scores

## The Bell Curve and Empirical Rule

The **normal distribution** is the familiar symmetric bell curve: unimodal with its mean, median, and mode all equal at the exact center. Naturally occurring variables like height and IQ approximate this shape.

The **empirical rule** describes how data spreads around the mean in a normal distribution:

| Range | Share of data |
| --- | --- |
| Mean ± 1 SD | About 68% |
| Mean ± 2 SD | About 95% |
| Mean ± 3 SD | About 99.7% |

Use this rule to quickly estimate where individual values fall.

## Computing and Using Z-Scores

To compare a specific score to a distribution, compute a **z-score**: how many standard deviations a raw score sits above or below the mean.

$$z = \frac{x - \mu}{\sigma}$$

Where $x$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation. A z-score of 0 is at the mean; positive is above, negative is below. Standardization allows comparing completely different scales.

> **Common pitfall:** treating a z-score as a percentile. A z-score measures distance in SD units; converting it to a percentile requires the empirical rule or a standard normal table.

## Practice questions

3 of this lesson's 8 practice questions, with answers. The full set is in the app.

### 1. Order the steps to standardize a raw score and turn it into a percentile.

**Answer:**

1. Subtract the mean from the raw score.
2. Divide that difference by the standard deviation to get the z-score.
3. Note the sign: positive is above the mean, negative below.
4. Convert the z-score to a percentile using the empirical rule or a z-table.

**Why:** You compute the deviation, divide by the SD to get z, read its sign, and only THEN convert to a percentile. Skipping the last step - treating the z-number itself as a percentile - is the named pitfall of this lesson.

Page: https://tryals.app/practice/psychology-i/the-normal-curve-and-z-scores/order-the-steps-to-standardize-a-raw-score-and-turn-it-into-a

### 2. The empirical rule says about 68% of a normal distribution lies within one standard deviation and 95% within two. Why is this worth memorising rather than looking up?

A. Because the rule applies to every distribution, not only normal ones
B. Because the exact percentages are disputed and the approximation is safer
C. Because statistical software cannot compute these values directly
D. Because it converts a measurement into a judgement of how unusual

**Answer:** D. Because it converts a measurement into a judgement of how unusual

**Why:** It turns a raw number into surprise on sight: one SD is ordinary, two is notable, three is rare. The caution is in the premise, it holds for roughly normal data, and applying it to skewed data like income will badly understate how many people sit far above the mean.

Page: https://tryals.app/practice/psychology-i/the-normal-curve-and-z-scores/the-empirical-rule-says-about-68-of-a-normal-distribution-lies

### 3. A student scores 85 on a test with mean 70 and SD 5, and 130 on a test with mean 100 and SD 15. Both are z = 3. Why does converting to z-scores let you compare performances on tests with nothing in common?

A. Because z-scores convert both tests to percentages, and percentages are always comparable.
B. Because both tests must have been measuring the same underlying ability, or the comparison would be meaningless.
C. Because a z-score reports position relative to its own spread.
D. Because dividing by the standard deviation makes the two scales numerically equal, so the raw scores become interchangeable.

**Answer:** C. Because a z-score reports position relative to its own spread.

**Why:** A z-score asks one question — how far from this distribution's mean, in this distribution's own standard deviations — and that question can be put to any distribution. What makes the comparison possible is precisely that the units are discarded. The caution is about interpretation: a z of 3 in spelling and in the high jump are both 'unusually good for that distribution', and nothing more.

Page: https://tryals.app/practice/psychology-i/the-normal-curve-and-z-scores/a-student-scores-85-on-a-test-with-mean-70-and-sd-5-and-130-on-a