Mode

The mode is found by tallying rather than by arithmetic: count how many times each value or category occurs, then report the one carrying the largest count. It has no standard symbol, and unlike the mean it works on categorical data, because counting never requires the values to be numbers.

In the data set 4, 5, 5, 5, 7, 8, 9 the counts are one 4, three 5s, one 7, one 8 and one 9. The largest count is 3, so the mode is 5. Answering that the mode is 3 is the classic mistake: 3 is the frequency, and the mode is the value that has it.

A data set can have no mode, one mode, or several. In 2, 4, 6, 8 every value occurs exactly once, the counts are all tied, and the convention is to say the set has no mode rather than to call all four values modes. That case is routine with measured quantities such as heights or reaction times, where exact repeats are rare. In 1, 1, 4, 7, 7, 9 both 1 and 7 occur twice, so the set has two modes.

Once data are grouped the mode becomes the modal class, the tallest bar of an equal-width histogram, because individual values have been absorbed into bins. That is also why bimodal describes a shape, two clear peaks separated by a dip, rather than a tie in the counts. Two peaks in a histogram of real data almost never have exactly equal heights, and a bimodal shape usually means two groups have been mixed together.

The mode is the only measure of center that survives nominal data such as favorite color or blood type, which is where it earns its keep. For quantitative data AP Statistics leans on the mean and the median instead: the mean uses every observation, and the median uses the position of every observation in the sorted list, while the mode can be created or destroyed by a single repeated value and says nothing about the rest of the distribution. In 4, 5, 5, 5, 7, 8, 9 the mode of 5 happens to equal the median, but nothing forces that.