The ACT is a standardized test that is used for college and university admissions in the United States. According to their web site, the ACT test “assesses high school students’ general educational development and their ability to complete college-level work”. The ACT test has a multiple choice component and an optional writing component. The multiple-choice component covers four skill areas: English, mathematics, reading, and science.
The ACT Inc, the organization that produces the test, provided a document to help students, parents, educators, school administrators and other interested individuals interpret the ACT. This document turns out to be a lesson in normal distribution. What is the lesson?
The lesson is that the numerical summary called standard deviation provides us with important information about bell-shaped distributions (aka normal distributions). If the shape of a distribution is bell-shaped (or one that is reasonably bell-shaped), the standard deviation can tell us how close a data point is to the mean. As discussed in this document, one of the methods for interpreting the ACT scores involves the standard deviation.
The following bullet points can be found in this document (by scrolling to the paragraph with the heading “Comparisons to the Average ACT Score”):
- Another norm-referenced way of interpreting an ACT score is to determine how close it is to the national average ACT score. For example, the average ACT Composite score for the 2001 graduating class (the “national” average) is 21.0. The corresponding standard deviation, a measure of how the scores vary around the average, is 4.7. We know from statistical theory that about 95% of students will score within plus or minus 2 standard deviations of the average Composite score; i.e., between about 12 and 30.
- Therefore, scores that are more than two standard deviations away from the average are fairly uncommon.
- A Composite score of 23, for example, is less than one-half standard deviation above the average score, suggesting that it is above average, but not unusually so.
The above bullet points indicate there is something special about a data value that is two standard deviations away from the mean. ACT scores that are 30 or higher are excellent and scores that are 12 or lower are poor scores. Such scores happen only about 5% of the time. However, data points that are within one standard deviation of the mean are not so special.
The above sentence in bold is tied to the footnote 3 in this document:
- Similarly, we know that about 68% of students will score within plus or minus one standard deviation of the average score.
The above two sentences in bold are two parts of a rule called the empirical rule, which is a short version of the normal distribution. The following is the full statement of the empirical rule centered on the example of composite ACT scores. Recall that the mean of all composite ACT scores is 21 and the standard deviation is 4.7.
- About 68% of students will score within plus or minus one standard deviation of the average score (i.e. between 16.3 and 25.7).
- About 95% of students will score within plus or minus two standard deviations of the average score (i.e. between 11.6 and 30.4).
- About 99.7% of students will score within plus or minus three standard deviations of the average score (i.e. between 6.9 and 35.1).
The 99.7 part of the rule says that practically all composite ACT scores are between 6.9 and 35.1. Only about 0.3% of the scores are outside of this interval (about 3 out of 1000 scores or 30 out of 10,000). Since the bell-shaped curve is symmetric, there are only about 15 scores out of 10,000 that are more than 35.1. So ACT test takers that are three standard deviations above the mean score form an exclusive club! Only 15 out of 10,000 test takers can join.
The 95 part of the rule says that virtually 95% of the ACT test takers will be in the range from 11.6 to 30.4. So only 5 out of 100 scores (50 out of 1000) are outside of this range. Since the bell curve is symmetric, there are only about 25 scores out of 1000 scores that are above 30.4. So achieving a composite ACT score of 30 or above is definitely a big deal. Such scores still form an exclusive club (only 25 out of 1000 test takers can join).
The 68 part of the rule says that 68% (roughly two-third) of the ACT test takers will have scores from 16.3 to 25.7. So roughly 1 out of 3 test takers (or 2 out of 6 test takers) will score above 25.7 or below 16.3. It follows that 1 out of 6 test takers will score above 25.7. A score of 26 is certainly an above average score but is a fairly common score.
So the empirical rule is a quick way to evaluate data in a normal distribution (a bell-shaped distribution). Calculate the number of standard deviations a score or a data point is away from the mean. The empirical rule is like a scale: around one standard deviation (common data point), around two standard deviations (uncommon data point) and around three standard deviations (extremely rare).
The empirical rule also applies in many other settings as long as the shape of the distribution is bell-shaped or close to bell-shaped. The SAT scores, the other standardized test that is commonly used as a college entrance test, also shape like a bell curve. The mean is 1509 and the standard deviation is 312 (the version described here has a maximum score of 2400). So the 68%, 95% and 99.7% intervals are:
- About 68% of SAT test takers will score between 1197 and 1821.
- About 95% of SAT test takers will score between 885 and 2133.
- About 99.7% of SAT test takers will score between 575 and 2400.
So an SAT score of 1700 is above average but is less than 1 standard deviation from the mean, thus not an exceptional score. Achieving a 2100 is certainly good reason to celebrate since it is around two standard deviations above the mean. On the flip side, getting a score of 900 is nothing to be proud of (this score is only better than about 2.5% of the scores). A score of 2350, though technically not a perfect score, is essentially way up there (only about 15 out of 10,000 scores are better).
Here is the statement for the empirical rule (also known as the 68-95-99.7 rule) in the general setting. The rule applies whenever the shape of the distribution is bell-shaped (or reasonably bell-shaped).
- About 68% of the data will fall within plus or minus one standard deviation of the mean.
- About 95% of the data will fall within plus or minus two standard deviations of the mean.
- About 99.7% of the data will fall within plus or minus three standard deviations of the mean.
Write down the 68%, 95% and 99.7% intervals as in the ACT and SAT examples. To gauge the significance of a given data point, see where the data point lands. If the data point is near the endpoints of the 95% interval, it is far from the mean (how to interpret this is based on the context of the problem). If the data point is near the endpoints of the 99.7% interval, then the data point is extremely rare (the odds of happening are about 3 out of 1000).