Mean, median, mode, and range on the ACT

Mean is the sum divided by how many numbers there are, median is the middle value after you sort the list, mode is the value that appears most often, and range is the largest value minus the smallest. That is the entire topic. What costs students points is never the definitions, it is a mechanical step skipped in a hurry.

What each word means

• Mean. Add every value, then divide by how many values there are. This is the one most people call the average.
• Median. Sort the list from smallest to largest, then take the middle value. If the count is even, average the two middle values.
• Mode. The value that appears most often. A list can have two modes, or three, or no mode at all when every value appears the same number of times.
• Range. The largest value minus the smallest value. It is one number, not a pair of endpoints.

Statistics and Probability makes up 12 to 15 percent of the Math section, and these four ideas are the backbone of that category. Since the ACT does not give you a formula sheet, they need to live in your head, which is fine, because they are short.

One list, all four measures

Take this set: 12, 3, 9, 14, 7, 3.

• Sort first: 3, 3, 7, 9, 12, 14. Do this before you do anything else.
• Mean: the sum is 3 + 3 + 7 + 9 + 12 + 14 = 48, and there are 6 values, so the mean is 48 divided by 6, which is 8.
• Median: with 6 values there is no single middle, so take the third and fourth, which are 7 and 9. Their average is 16 divided by 2, which is 8.
• Mode: 3 appears twice and nothing else repeats, so the mode is 3.
• Range: 14 minus 3 is 11.

Finding a missing value when you know the mean

This setup shows up constantly, and it turns on one move: a mean is really a total in disguise. If the mean of n numbers is m, then the sum of those numbers is n times m.

Say Priya's first four quiz scores are 92, 85, 90, and 79, and she wants a mean of 88 across five quizzes. The five scores must total 88 times 5 = 440. Her current total is 92 + 85 + 90 + 79 = 346. So she needs 440 minus 346, which is 94.

It runs backward too. Seven numbers have a mean of 12, so their total is 84. Remove the value 18 and the remaining six total 66, for a new mean of 66 divided by 6, which is 11.

Weighted averages

When groups are different sizes, you cannot average the averages. One class of 20 students has a mean score of 78, and another class of 30 students has a mean of 88. The first class contributed 20 times 78 = 1560 points, the second contributed 30 times 88 = 2640 points, and together that is 4200 points across 50 students. The combined mean is 4200 divided by 50, which is 84, not the 83 you would get by averaging 78 and 88.

Weights can also be percentages. If homework is worth 20 percent of a grade and tests are worth 80 percent, a student with a 95 on homework and an 85 on tests earns 0.20 times 95 = 19, plus 0.80 times 85 = 68, for a final grade of 87.

Common traps

• Taking the median without sorting. The single most common error on this topic. In the list 14, 2, 23, 8, 5 the value sitting in the middle as written is 23, but sorted it reads 2, 5, 8, 14, 23, so the median is 8. Sort every time, even when the list looks orderly.
• Forgetting the even-count rule. The average of the two middle values may not appear in the list at all.
• Assuming exactly one mode. Check whether a second value ties for most frequent before you answer.
• Reporting the range as two numbers. The question wants the difference.
• Letting an outlier fool you. For 4, 5, 6, 7, 200 the mean is 222 divided by 5, which is 44.4, while the median is 6. One extreme value drags the mean and leaves the median alone.

A short self-check

Write down any six numbers, deliberately out of order and with one repeat, then compute all four measures in under a minute. Next, cover one of your numbers and ask what it would have to be for the mean to hit a round target. thirty-six quizzes bury these measures inside word problems the way the test does, which is the harder version of the skill.

Once these feel automatic, the natural next stop is ACT probability problems, since the two topics share a category and often share a question set. For the wider view of that category, see statistics and probability on the ACT, and when the data arrives as a graph instead of a list, see reading charts and tables on ACT Math.

This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.