Mode vs Median

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.

Sort the values from smallest to largest and the median is whatever sits at position $\frac{n+1}{2}$ (n plus 1, over 2), where $n$ is how many values there are. When $n$ is odd that position lands on an actual observation. When $n$ is even it lands between two of them and the median is their average, so the median of an even-sized set need not be a number that appears in the data at all. It is commonly written $M$.

Take the seven quiz scores 12, 15, 15, 18, 22, 24, 91. They are already sorted and $n = 7$, so the median is the value in position 4: $M = 18$. The mean of the same seven scores is 28.14. Now change the 91 to 27. The median is still 18, while the mean drops to 19. That is resistance: the median only counts how many values sit on each side of it and ignores how far away they are, so moving one extreme value cannot shift it by more than one position.

With an even count the averaging step kicks in. For 2, 6, 8, 10 the median is $(6 + 8) / 2 = 7$, a number that appears nowhere in the list.

The most common error is reporting the position instead of the value. For those seven scores the answer is 18, not 4; position 4 only tells you where to look. Two related slips are taking the middle of the list before sorting it, and averaging the smallest and largest values, which gives the midrange rather than the median.

The median is also the 50th percentile and the center of the five-number summary. The quartiles are the medians of the halves on either side of it, with the median itself left out of each half under the TI-84 convention this site uses: for the seven scores, 12, 15, 15 gives $Q_1 = 15$ and 22, 24, 91 gives $Q_3 = 24$. Comparing the two centers is also a quick read on shape: here the mean of 28.14 sits far above the median of 18, which usually signals either a right skew or a single high value dragging the mean up. In this set it is the second of those, since the 1.5 IQR rule puts the upper fence at $24 + 1.5(9) = 37.5$ and flags 91 as an outlier.