Why a List Is Allowed to Have Two Modes

Two values can tie for most frequent

The list 3, 3, 5, 5, 7 has two 3s and two 5s. Both appear more often than 7, and they appear equally often, so the list has two modes, 3 and 5. Nothing in the definition requires a single winner. Forcing one mode, by taking the lower value or the higher one, is a choice you would be adding, not a result the list gave you. Report both. If a form has one blank labeled mode, write 3 and 5 in that blank rather than flipping a coin between them.

The mean of this list is 4.6, because the sum is 23 and the count is 5. The median is 5. Neither of those is a substitute for the modes, and the average of the two modes is 4, a number that never appears. A reader who hears 4 will look for papers that scored 4 and will find none. Keep the modes as a pair. Say the list is bimodal at 3 and 5, with one score of 7 sitting off those peaks, so the shape is visible in a single clause.

A class that clusters on two scores

Imagine 20 quiz scores: 9 students score 70, 9 score 90, and 2 score 80. The two modes are 70 and 90, each with frequency 9. The sum is 9 times 70, plus 9 times 90, plus 2 times 80, which is 630 plus 810 plus 160, or 1,600. The mean is 80. Only 2 of the 20 papers actually scored that mean. A histogram would show two peaks with a thin bridge. Teaching to the average of 80 misses both groups. Split the feedback: one note for the 70s and one for the 90s.

The Mode Calculator will list both 70 and 90 when their counts match and exceed the other scores. If a later makeup moves one student from 70 to 80, the frequencies become 8, 3, and 9. Then 90 is the only mode. A one-paper change ended the tie. That is a reason to report frequencies beside the mode, not the label alone. Write 90 as the mode, with 9 papers, and mention the 8 papers at 70. Otherwise a tied peak that almost won disappears from the summary entirely. Keep the unit, the count, and the name of the formula in the same sentence as any figure you copy into a report.

Do not average the modes into a new score

Averaging 70 and 90 produces 80, which is the mean's neighborhood, not a mode. The mode is the value that occurs most often, and an average of two such values is a different statistic wearing the wrong name. In the class of 20, 80 occurred twice, less often than either peak. Calling 80 the mode would be false. If you want one number for a form that cannot take two, report the median or the mean and label it correctly, and attach a note that the score distribution has two peaks.

A second temptation is to pick the mode closer to the mean and drop the other. On a 70 and 90 tie, neither is closer to a duty to be the single mode. They tied. Inventing a tie-break from the mean imports a different question, about distance from the equal share, into a question about frequency. If a rule must break ties, write the rule down before you see the list. Alphabetical order, lower value, or higher value can be rules. Secret preference for the peak you hoped to see cannot.

No mode, one mode, and a flat list

The scores 1, 2, 3, 4, and 5 each appear once. Every value is tied for most frequent, which is another way of saying the list has no distinctive mode. Some software still returns all five values, and some returns a blank. Read the documentation, then say in your own note that no score repeated. Do not fill the blank with the median, which is 3, and call it the mode. A mode that is really a median will confuse anyone who later counts the frequencies and finds that 3 appeared once.

A nearly flat list deserves the same caution. Suppose 12 scores on a 1-to-5 survey are 3, 3, 3, 2, 2, 4, 4, 1, 5, 3, 2, 4. The value 3 appears four times, and 2 and 4 appear three times each. The mode is 3, but the lead is a single response. Report the frequency: 4 of 12, about 33 percent. A mode with a one-response lead is a weak peak. If you are choosing a product color from a mode like that, look at the runner-up before you order a full run of a single shade.

Labels have modes even when means do not

Commute methods for 30 staff might be bus 14, car 10, and walk 6. The mode is bus. There is no honest mean of the labels bus, car, and walk, because those words do not add. Coding them as 1, 2, and 3 and averaging the codes produces a number with no commute attached. Resist that coding unless the codes are a real scale you are willing to defend. For categories, the mode and the full frequency table are the summaries that stay meaningful. Percents help: bus is about 47 percent of these 30 trips.

A category mode can still be a poor guide to action. If 14 ride the bus and 16 do not, bus is the mode and it is also a minority of the staff. Ordering bus passes for everyone because bus was the mode overshoots the data. Report the mode and the share. A useful sentence is that the most common commute is the bus, used by 14 of 30 people, while 16 use something else. The second clause stops a planner from treating the winning label as if it were the whole group.

Uneven peaks still need the runner-up

Twelve students score 8, four score 6, and four score 7, on a ten-point check. The mode is 8, with a frequency of 12 out of 20, which is 60 percent. That is a real peak, not a tie. The other 8 students, 40 percent, scored 6 or 7. A sentence that says only that the mode is 8 makes the check sound more uniform than it was. Add the frequency and the count below the mode. Then a tutor knows that a majority hit 8 and that a sizable group did not.

Compare two sections before you declare a common mode. Section A peaks at 8 with 12 of 20. Section B peaks at 6 with 11 of 18. There is no single mode of the course until you pool the papers or you deliberately summarize sections. Pooling, 12 plus any 8s in section B, may move the mode or create a tie. Compute the frequencies on the pooled list if the question is about the course, and keep the section modes if the question is about each room. Write which list you counted so the peaks can be audited.

Open the worksheet

The matching page is the Mode Calculator. It opens with a worked example, names the formula, and keeps the steps next to the result. Use it when you want the arithmetic on your own numbers rather than the argument on this page.