Z-Scores and Percentiles Explained

A raw score on its own tells you very little. A 115 on an IQ test, an 80 on an exam, a 4.2 on a rating scale: until you know the average and how spread out the scores are, you cannot say whether any of them is high. A z-score fixes that by expressing a score as a number of standard deviations from the mean. This guide explains it using the examples that come up in intro psychology, where it appears in the research methods unit and again in the unit on intelligence testing. It follows the cards in Encodr's free Intro Psychology deck.

The two ingredients: mean and standard deviation

Descriptive statistics summarize the data that were collected, and the two most useful summaries here are the mean (the average) and the standard deviation (how far scores typically sit from that average). A small standard deviation means scores cluster tightly around the mean. A large one means they are spread out.

One caution from the course: the mean can mislead. If ten people list their yearly incomes and one is a billionaire, the mean is pulled far toward the extreme value while the median, the middle score, barely moves. The median is the average less affected by extreme scores.

The z-score formula

z = (x - mean) / standard deviation

The z-score is the distance of a score from the mean, measured in standard deviations. A z of 0 is exactly average, a positive z is above average and a negative z is below it. To go the other way, x = mean + z x standard deviation.

The IQ example

Intro psychology gives you the perfect example to practice on. Modern IQ scores are scaled to follow a bell curve with a mean of 100 and a standard deviation of 15. That makes the conversions easy:

• A score of 115 is 15 points above the mean, so z = (115 - 100) / 15 = 1. It is one standard deviation above average.
• A score of 85 is 15 points below, so z = -1, one standard deviation below.
• A score of 130 gives z = (130 - 100) / 15 = 2.

This works because IQ tests are standardized and normed. Standardization means giving the test the same way to everyone, and norming means building the scoring scale from a representative sample so that an individual's score can be compared with a population. The scale is built so the average comes out at 100 with a standard deviation of 15. The course also covers the Flynn effect, the rise in average scores across generations, which is a reminder that a score only means something relative to the group it is normed on.

From z-score to percentile

A percentile tells you the share of scores that fall below yours. On a normal curve, each z-score has a fixed percentile. You can look it up in a z-table or use the z-score calculator.

So a 115 sits at about the 84th percentile: it beats roughly 84 percent of scores. A common mistake is to confuse percentile with percent correct. A person at the 84th percentile has not answered 84 percent of questions correctly. They have scored higher than 84 percent of the people in the comparison group.

The 68-95-99.7 rule

For a normal distribution, the share of scores within a given distance of the mean is fixed:

The course ties the first of these to the IQ scale: a score of 85 or 115 is one standard deviation from the mean and within the range of about 68 percent of people. On the IQ scale, about 95 percent of people fall between 70 and 130, and about 99.7 percent between 55 and 145.

The number 1.96 shows up so often because it cuts off the middle 95 percent exactly: 95 percent of a normal distribution lies within 1.96 standard deviations of the mean. That is why a z beyond plus or minus 1.96 corresponds to a two-tailed p-value below .05, the conventional threshold for statistical significance in psychology. Keep in mind the course's reminder that significance does not measure size or importance.

Comparing scores on different scales

Z-scores let you compare things that are not on the same scale. Say a student scores 80 on a test where the class mean was 70 and the standard deviation was 5, and 85 on another test where the mean was 80 and the standard deviation was 10.

• Test A: z = (80 - 70) / 5 = 2
• Test B: z = (85 - 80) / 10 = 0.5

The 80 is the stronger performance, even though 85 is the larger number: it is two standard deviations above its class average (about the 97.7th percentile) versus half a standard deviation above (about the 69th).

Where this breaks down

The percentile conversions assume the scores follow a normal distribution. If a variable is badly skewed, such as incomes, a z-score still tells you the distance from the mean in standard deviations, but the percentile table will give wrong answers. The z-score is also only as good as the mean and standard deviation behind it, which is why the norming sample matters for tests like the IQ scale.

If your data come from measurements with their own uncertainty, a different tool applies: the error propagation calculator shows how uncertainty carries through a calculation.

Practicing

Z-score problems are three-step procedures, which makes them ideal for retrieval practice: close the notes, write the formula, work the example. Why quizzing beats review explains the evidence. The same course also covers how memory fades, in the forgetting curve, which is a useful companion to a stats unit. For the full list of topics, see what's in intro psychology, and for a study plan see how to study for intro psychology.