How to use the simulator
- mean μ: 50 to 150 in steps of 1 (starts at 100).
- std dev σ: 5 to 30 in steps of 1 (starts at 15).
- raw score x: its range follows the other two sliders and always runs from to . The step is 0.5 when σ is 15 or less, so every whole-number score in range is reachable; for larger σ it is σ/30 (1 when σ = 30).
- region: chooses which area is shaded. Area to the LEFT of x (percentile) shades everything below x, Area to the RIGHT of x shades everything above it, and Area within ±z of the mean shades the band symmetric about μ.
The readout box shows the Raw score x, the signed z-score to two decimal places, and the Shaded area as a percentage to one decimal place. On the graph, tick marks sit at the mean and at 1, 2 and 3 standard deviations either side. Each tick is labelled with its score and its z value (z=−3 to z=3). A dashed line marks the mean and a dark line marks x.
Reset returns to μ = 100, σ = 15, x = 100 and the left-area region. That is the Wechsler IQ scale.
Try this first: with the defaults, choose the ±z region and move x to 115. The z-score reads +1.00 and the shaded area is 68.3%, which is the first part of the empirical rule. Switch to the left region and the shaded area reads 84.1%, so the same score sits at about the 84th percentile.
The formula
A z-score measures how many standard deviations a raw score is above or below the mean:
- is the raw score, is the mean and is the standard deviation.
- A positive z is above the mean, a negative z is below it, and is exactly at the mean.
- Rearranged, , which converts a z-score back to a raw score.
In a normal distribution the mean, median and mode are equal and the curve is symmetric. The empirical rule gives the approximate share of scores in each band:
- About 68% fall within of the mean.
- About 95% fall within .
- About 99.7% fall within .
The percentile rank of a score is the percentage of scores at or below it, which is the left-hand area. Z-scores let you compare results from tests with different means and spreads.
Worked example
Problem: Scores on a memory test are normally distributed with a mean of 70 and a standard deviation of 10. Maya scores 85 and Leo scores 62. Find each z-score and percentile.
Step 1: Maya's z-score. .
Step 2: Maya's percentile. Set μ to 70, σ to 10 and x to 85, with the left region selected. The readout shows z-score +1.50 and Shaded area 93.3%, so Maya is at about the 93rd percentile. Switch to the right region and the area reads 6.7%: only about 7% of test-takers did better.
Step 3: Leo's z-score. . Move x to 62 with the left region selected and the shaded area reads 21.2%, so Leo is near the 21st percentile.
Step 4: Compare across tests. On a different test with a mean of 80 and a standard deviation of 6, a score of 89 also gives . That is the same relative standing as Maya's 85, even though the raw scores differ. You can check it in the sim: μ = 80, σ = 6, x = 89 gives the same 93.3%.
Common mistakes on the AP exam
- Dropping the sign. A score below the mean has a negative z. A z of −1 and a z of +1 lie on opposite sides of the curve.
- Using 68% for one side. The 68% covers both sides of the mean. Between the mean and is only about 34%, so a score at is at about the 84th percentile (50% + 34%).
- Confusing percentile with percent correct. The 90th percentile means scoring at or above 90% of test-takers, not answering 90% of the questions correctly.
- Applying the empirical rule to skewed data. The 68–95–99.7 pattern holds only for normal distributions. In a positively skewed distribution the mean is pulled toward the long right tail, above the median.
- Mixing up spread and centre. Changing σ in the sim widens or narrows the curve but leaves the centre where it is. A larger σ means scores are more spread out, not higher.
When the AP exam uses this
Calculating and interpreting standard deviation, z-scores and percentile rank is part of Science Practice 3 (data interpretation) in AP Psychology. The normal curve also appears in the Unit 2 cognition material on intelligence, where standardized tests such as the Wechsler scales are normed to a mean of 100 and a standard deviation of 15. Expect questions on the percentage of people within a range, or on what a z-score says about a score's position. Free-response questions can also include data you must interpret this way.