Comparing Values from Different Normal Distributions

Why Standardize SAT and ACT Scores?

A raw SAT score and a raw ACT score use different scales, so comparing the numbers directly does not tell you which score is higher relative to its own test’s distribution. As in Interpreting a z-Score in Context, a z-score describes a value’s location in standard-deviation units. Using each test’s own mean and standard deviation puts the scores on a common standardized scale for comparing relative positions.

For example, an SAT score of 1200 and an ACT score of 28 cannot be compared by treating 1200 as numerically greater than 28. Those numbers are expressed on different scales. Instead, calculate an SAT z-score using the SAT model and an ACT z-score using the ACT model. Then compare the signed z-scores: the larger one indicates the higher standardized position relative to its own model mean.

This comparison is about relative position, not a conversion from one test scale to the other. A higher z-score does not establish that a student has greater overall ability, that the tests measure exactly the same skills, or that the student would receive a particular score on the other test. It only identifies which score is higher relative to the mean and spread of the specified model.

A Reliable Method for Comparing the Scores

For each score, identify the observed value \(x\), the mean \(\mu\), and the standard deviation \(\sigma\) from the model for that particular test. Do not use the SAT mean with an ACT score or the ACT standard deviation with an SAT score. Calculate each z-score separately:

After calculating both scores, compare their signed values. A larger signed z-score always means a higher standardized position. If both z-scores are negative, the larger one is still higher: for example, \(-0.4\) is greater than \(-1.2\). In this case, both scores are below their respective means, and the score with \(z=-0.4\) is closer to its mean in standard-deviation units. It is not correct to say that either score is “farther above” its mean.

When a z-score is positive, that score is above its own distribution’s mean. When it is negative, the score is below its own mean. The absolute value gives the distance from the mean. These descriptions help explain the comparison, but the direct comparison of standardized positions uses the signed values, not just their absolute values.

Worked Example: Comparing a Score Above Each Mean

Worked Example: Comparing Two Scores Below Their Means

Worked Example: Equal Standardized Positions

What the Comparison Can—and Cannot—Tell You

A z-score comparison is most useful when the means and standard deviations refer to appropriate reference groups for the question. A model based on one group or test version may not describe another group or version. Before interpreting a result, be clear about which model is being used and what population or setting its parameters represent.

The calculation itself standardizes a score by subtracting a mean and dividing by a positive standard deviation. A normal model is useful here because it describes the score distribution and provides a context for relative location. But calculating z-scores does not require finding a normal-curve area. A z-score alone does not give the percentile of a score or the probability of getting a score at least that high. Those questions require a separate probability calculation.

Even when the models are reasonable, the standardized comparison is limited to the distributions named in the problem. For example, one score may have a higher standardized position in its test’s reference distribution, but that does not guarantee the student would have the higher score if both tests were taken again. It also does not account for differences in test content, testing conditions, or the individual’s strengths across subject areas.

Keep the distinction between a higher position and a larger distance clear. If \(z_1>z_2\), the first score has the higher standardized position. If both are positive, the first score is farther above its mean. If both are negative, the first score is closer to its mean and less far below it. If one is positive and one negative, the positive score is above its mean while the negative score is below its mean.

Common Mistakes and AP Exam Tips

• Comparing raw scores across different scales. An SAT score and an ACT score use different scoring scales. Standardize each with its own model before comparing relative positions.
• Using the wrong parameters. A z-score must use the mean and standard deviation for the same test distribution as the observed score. Mixing SAT and ACT parameters gives an invalid comparison.
• Confusing higher position with farther above the mean. A larger signed z-score always means a higher standardized position, but a score with a negative z-score is below its mean. Use “farther above” only when the z-score is positive.
• Comparing absolute values instead of signed scores. Absolute values describe distance from a mean, not whether the position is higher or lower. For instance, \(z=-0.5\) is higher in standardized position than \(z=-1.2\), even though both are below their means.
• Claiming equal z-scores mean equal performance. Equal z-scores indicate the same number of standard deviations from each model’s mean in the same direction. They do not establish that the tests are interchangeable or measure identical abilities.
• Turning a z-score into a percentile or probability. A z-score gives standardized location. Do not report an area or likelihood unless you actually calculate it and the question asks for it.
• Leaving out context. A complete comparison names the test scores, their respective models, and which score has the higher standardized position. A bare statement such as “1.5 is greater than 1.4” does not communicate what the comparison means.

For full-credit communication, show each calculation with its own model parameters, compare the signed z-scores, and interpret the result in context. If the larger score is negative, say it has the higher standardized position or is less far below its own mean—not that it is farther above the mean.

Key Takeaway

To compare SAT and ACT scores relative to different normal distributions, calculate each z-score with that test’s own mean and standard deviation. The larger signed z-score represents a higher standardized position; the sign identifies whether the score is above or below its own mean. This describes relative location in the chosen models, not equivalence between test scales or a complete measure of ability.