Why Standardize Scores Before Comparing Them?
In Z-Scores: Standardizing a Value, you learned that a z-score describes how many standard deviations a value is above or below its group’s mean. That idea is useful when comparing an SAT score with an ACT score: the tests use different score scales, so comparing the raw numbers directly does not describe which score is farther above its own group’s average.
Instead, calculate a z-score for each score using the mean and standard deviation for that test’s reference group. The results are unitless, so they can describe relative standing on a common standard-deviation scale. For example, if the SAT z-score is greater than the ACT z-score, the SAT score is farther above its reference-group mean in standard-deviation units.
This comparison is about position relative to each test’s own reference group. It does not convert an SAT score into an ACT score, establish that the tests measure exactly the same things, or show that the person has a particular amount of knowledge. The reference-group summaries must fit the question: a score should be compared with summaries for an appropriate group taking that test, not with the other test’s summaries.
A Method for Comparing Two Test Scores
Suppose a person has an SAT score \(x_S\) and an ACT score \(x_A\). Let \(\mu_S\) and \(\sigma_S\) be the mean and standard deviation of the relevant SAT reference group, and let \(\mu_A\) and \(\sigma_A\) be the corresponding summaries for the ACT reference group. If the supplied summaries describe full reference populations, calculate:
If the summaries are sample statistics instead, use the sample mean and sample standard deviation for each test, as in the earlier tutorial. Either way, each score must be paired with the mean and standard deviation from the same test and reference group. A standard deviation must be greater than zero so that the division is defined.
After calculating both z-scores, compare their values and signs. A positive z-score is above that test group’s mean, a negative z-score is below it, and zero is at the mean. If one z-score is larger, that score is higher relative to its own reference group. If both are negative, the less negative z-score is closer to its group’s mean and therefore represents the higher relative position of the two.
- Use the mean and standard deviation for the same test as the score being standardized.
- Use reference groups that are relevant and reasonably comparable for the question, such as groups defined in a consistent way.
- Use summaries and scores that refer to the same test version and scoring context when that information matters.
- Do not treat z-scores as score conversions or as proof that the tests measure identical skills.
This is a descriptive comparison, not an inference procedure. There is no Large Counts condition or Normality condition required simply to calculate and compare z-scores. However, a z-score alone does not supply a percentile or probability. Do not turn a z-score comparison into a percentage claim unless you have appropriate data or a model for doing so.
Worked Examples: Comparing SAT and ACT Scores
Worked Example: One Student’s SAT and ACT Scores
A fictional student has an SAT score of 1320 and an ACT score of 29. For this example, suppose fictional reference-group summaries are an SAT mean of 1050 with standard deviation 150, and an ACT mean of 21 with standard deviation 5. Compare the student’s relative standing on the two tests.
State the comparison. We want to determine which score is farther above its own reference-group mean in standard-deviation units. The summaries are hypothetical and are provided only for this example.
Plan. Calculate an SAT z-score using the SAT mean and standard deviation, and an ACT z-score using the ACT mean and standard deviation. Both scores are being compared with their respective fictional reference groups. No shape condition is needed to make this descriptive comparison.
Do: calculate the SAT z-score. The SAT score is 270 points above its reference-group mean, since \(1320-1050=270\). Dividing by the SAT standard deviation gives:
Check: \(1.8(150)=270\), and \(1050+270=1320\), so the standardized difference matches the score’s distance above the SAT mean.
Do: calculate the ACT z-score. The ACT score is 8 points above its reference-group mean, since \(29-21=8\). Dividing by the ACT standard deviation gives:
Check: \(1.6(5)=8\), and \(21+8=29\), confirming the ACT calculation.
Conclude in context. The SAT score is 1.8 standard deviations above the fictional SAT reference-group mean, while the ACT score is 1.6 standard deviations above the fictional ACT reference-group mean. Because \(1.8>1.6\), the SAT score is higher relative to its own reference group in this example. The difference between the z-scores is \(1.8-1.6=0.2\) standard-deviation units on this standardized scale. This does not mean the SAT score is 0.2 score points higher than the ACT score.
Worked Example: The ACT Score Has the Higher Relative Position
A second fictional student has an SAT score of 1180 and an ACT score of 27. Suppose the relevant fictional SAT reference group has mean 1100 and standard deviation 160, while the fictional ACT reference group has mean 20 and standard deviation 4. Which score is higher relative to its own group?
Calculate the SAT z-score. The SAT score is \(1180-1100=80\) points above its group’s mean. Standardizing gives:
The check is \(0.5(160)=80\), and \(1100+80=1180\). Thus, the SAT score is 0.5 standard deviations above its reference-group mean.
Calculate the ACT z-score. The ACT score is \(27-20=7\) points above its group’s mean. Standardizing gives:
The check is \(1.75(4)=7\), and \(20+7=27\). Thus, the ACT score is 1.75 standard deviations above its reference-group mean.
Compare and conclude. Since \(1.75>0.5\), this student’s ACT score is higher relative to its own fictional reference group than the SAT score is relative to its group. The raw scores, 1180 and 27, cannot be compared by size because they are on different scales. The comparison comes from the z-scores, not from subtracting or dividing the raw SAT and ACT scores.
Worked Example: Equal Z-Scores on Different Scales
A fictional student earns an SAT score of 1260 and an ACT score of 28. Suppose the fictional SAT reference group has mean 1080 and standard deviation 120, and the fictional ACT reference group has mean 22 and standard deviation 4. Compare the relative positions.
Standardize the SAT score. The difference from the SAT mean is \(1260-1080=180\) points:
The check is \(1.5(120)=180\), and \(1080+180=1260\).
Standardize the ACT score. The difference from the ACT mean is \(28-22=6\) points:
The check is \(1.5(4)=6\), and \(22+6=28\).
Interpret the match. Both scores are 1.5 standard deviations above their respective fictional reference-group means. They have equal z-scores, so they have the same relative position as measured by standard deviations from those means. The raw differences are not equal: the SAT score is 180 SAT points above its mean, while the ACT score is 6 ACT points above its mean. The z-score comparison expresses each difference in units of its own test’s standard deviation.
What the Comparison Can—and Cannot—Say
A larger z-score indicates a higher relative position compared with the mean of that score’s own reference group. This interpretation applies whether the z-scores are positive or negative. For example, \(-0.4\) is greater than \(-1.2\): both scores are below their respective means, but the first is closer to its group’s mean.
A z-score comparison does not establish a precise conversion between test scales. It does not say what ACT score corresponds to a particular SAT score, and it does not guarantee that two students with equal z-scores have identical skills. The tests may differ in content, format, and reference populations. The comparison answers a narrower question: how far is each score from the mean of its chosen group, measured in that group’s standard deviations?
The choice of reference group matters. A score might have one z-score relative to all test takers and a different z-score relative to a narrower group. Before comparing, check who is represented by each mean and standard deviation. If the reference groups are not suitable for the question, the calculations may be correct but the interpretation may not be useful.
Do not assume that a z-score gives a percentile rank. A z-score describes distance from a mean in standard-deviation units; a percentile rank describes the proportion of observations at or below a value. As discussed in Percentiles and Their Interpretation, percentile rank depends on the ordered observations. Without more information about the distribution or the actual data, a z-score comparison alone does not give the percentage of people a score exceeds.
Common Mistakes and AP Exam Tips
- Comparing the raw scores. Saying that 1320 is greater than 29 does not compare relative standing. Calculate a separate z-score for each test first.
- Using one test’s summaries for both scores. The SAT score must be standardized with SAT summaries, and the ACT score with ACT summaries. Mixing the means or standard deviations makes the resulting comparison invalid.
- Reversing the conclusion. The larger z-score represents the higher relative position. If both values are negative, the less negative value is larger and closer to its group’s mean.
- Calling the result a score conversion. Equal z-scores do not make the raw scores interchangeable. They indicate equal distances from their respective means in standard-deviation units.
- Claiming a percentile without evidence. A z-score is not itself a percentile. Do not write that a person scored higher than a particular percentage of test takers unless percentile information or an appropriate model supports that statement.
- Leaving the reference group out of the interpretation. A full-credit statement identifies which score is higher relative to which test group. The mean and standard deviation used for each score should be clear.
A strong response shows both calculations and gives a direct, qualified comparison. For example: “The SAT score is 1.8 standard deviations above the fictional SAT group mean, while the ACT score is 1.6 standard deviations above the fictional ACT group mean. Therefore, the SAT score has the higher relative position in these reference groups.” This states the results in context without suggesting the raw scales are interchangeable.
Check Your Understanding
Use each score’s own reference-group summaries. Show the z-score calculations and state what the comparison means.
- A student has an SAT score of 1200. The fictional SAT group mean is 1080 and its standard deviation is 120. Find and interpret the SAT z-score.
- The same student has an ACT score of 26. The fictional ACT group mean is 20 and its standard deviation is 4. Find the ACT z-score. Which score has the higher relative position?
- Two scores have z-scores of \(-0.3\) and \(-1.1\). Which score is higher relative to its own reference group? Explain using the signs and values.
- What does it mean if an SAT score and an ACT score have equal z-scores? Name one conclusion that the equality does not support.
- Why is it important to check which test takers are represented by the reference-group mean and standard deviation?