The Modal Class

Grouped data has no single mode

Thirty people count the text messages they send in one day, and the counts are grouped into classes: 0–9 messages, 10–19, 20–29 and 30–39. The table keeps only the frequency of each class, the number of people whose count fell in it.

The mode of a list is the value that occurs most often. But the table no longer shows any single value. It says that 12 people sent from 20 to 29 messages, not whether they all sent 23 or each sent a different number. So no single value can be named as the mode.

Each class holds ten possible counts, and the frequencies 4, 9, 12 and 5 add up to the 30 people.

The class with the largest frequency

What the table does show is which class holds the most values. The largest frequency is 12, so the class 20–29 holds more people than any other class. That class, 20–29 messages, is the modal class.

The classes here all have the same width, 10 messages, so they can be compared by frequency alone. On a histogram of this table the bars are equally wide, and the modal class is the one with the tallest bar.

The row 20–29 has the largest frequency, 12, so 20–29 messages is the modal class.

The answer is the class

The modal class is 20–29 messages. It is not 12. The 12 is the frequency, the reason this class was chosen; the answer is the interval itself.

It is not a single number such as 25 either. The midpoint of the class is useful for estimating a mean, but the modal class says only that the most crowded interval is 20 to 29 messages. It does not say which number of messages inside it was most common.

The class that contains the median

A grouped table can also say which class the median is in. The median is the middle value of the ordered data, and with n values it sits in position $\frac{\left(n+1\right)}{2}$(n + 1)/2. Here n = 30, so it is in position 15.5, halfway between the 15th and the 16th value.

The classes are already in order, so keep a running total down the frequency column: 4, then 4 + 9 = 13, then 13 + 12 = 25, then 25 + 5 = 30. The first two classes hold the 1st to the 13th values, and the class 20–29 holds the 14th to the 25th. The 15th and the 16th are both in 20–29, so the median is in the class 20–29.

For these 30 people the modal class and the class containing the median are the same, but they answer different questions, and often they differ. For another 30 people, the frequencies are 11, 8, 6 and 5. The modal class is 0–9, with 11 people. The running total is 11 after the first class and 19 after the second, so the 15th and the 16th values are in 10–19, and the median is in the class 10–19.

The largest frequency is in 0–9, but the running total passes 15 and 16 in 10–19. The modal class is 0–9, and the median is in 10–19.

Unequal widths: compare frequency density

When the classes have different widths, the largest frequency can mislead, because a wide class collects more values just by being wide. Another group of 24 people is grouped into 0–9 messages with 4 people, 10–29 with 14 people and 30–39 with 6 people. The class 10–29 holds twenty possible counts, twice as many as the others.

Compare how crowded each class is instead, with its frequency density: frequency ÷ class width. For 10–29 it is 14 ÷ 20 = 0.7 people for each message, which is 7 for each ten messages. For 0–9 it is 4 ÷ 10 = 0.4, and for 30–39 it is 6 ÷ 10 = 0.6. The class 10–29 has the largest density, so it is still the modal class, but by a much smaller margin than 14 against 6 suggests.

Sometimes the density changes the answer. A survey of 100 people finds 30 in 0–9 messages, 50 in 10–29 and 20 in 30–39. The class 10–29 has the most people, 50. But the densities are 30 ÷ 10 = 3, 50 ÷ 20 = 2.5 and 20 ÷ 10 = 2. The people are most crowded in 0–9, so the modal class is 0–9. On a histogram drawn with frequency density, the modal class is again the tallest bar.

Drawn with frequency as the height, the wide class 10–29 looks by far the largest, because it is both tall and twice as wide.

Drawn with frequency density, 3, 2.5 and 2 people per message, the tallest bar is over 0–9. That is the modal class.

The gold class always holds 10 values, and the class 10–20 holds 12. Switch to frequency density and narrow the gold class to a width of 5: its density is 10 ÷ 5 = 2, more than 12 ÷ 10 = 1.2, so its bar is the tallest and it is the modal class, though it holds fewer values. Widen it to 40 and its density falls to 10 ÷ 40 = 0.25.