Mean, Median and Mode

There are three kinds of average, and they measure different things. The mean is what most people picture: add every value and divide by how many there are. For 4, 7, 7, 9 and 13 that is 40 divided by 5, which is 8. The median is the middle value once the data is sorted, which here is 7. And the mode is the value appearing most often, also 7 since it occurs twice. All three get called 'the average' in ordinary speech, but in maths they are three different numbers — here 8, 7 and 7 — so read the question carefully because it will name the one it wants. A memory hook: mode and most both start with M-O.

The median needs one step before anything else: sort the data. With an odd number of values, one sits exactly in the middle — sorting 9, 4, 13, 7, 7 gives 4, 7, 7, 9, 13, so the median is 7. With an even number there is no single middle, so average the middle two: sorting 10, 3, 8, 5 gives 3, 5, 8, 10, and the median is 5 plus 8 over 2, which is 6.5. The number one error in this whole topic is forgetting to sort, which would give 13 instead of 7 in that first example. Write the sorted list out rather than sorting in your head. And note the median can be a value not in the data at all, like 6.5 — that is perfectly correct.

The mode is the value that appears most often, and it behaves differently from the other two. A set can have one mode, as in 3, 5, 5, 7, 9 where the mode is 5. It can have two, like 2, 2, 5, 8, 8, 9, which is called bimodal and both values are modes. It can have none at all, as in 1, 4, 6, 9, 12 where nothing repeats — 'no mode' is a valid answer and you should not invent one. And crucially, the mode is the only average that works on categorical data. You cannot average eye colours, but you can name the most common one. That makes it the right choice when the data is words rather than numbers.

Data usually arrives as a frequency table, which changes how you calculate. For the mean you cannot just add the pets column and divide by 5 — you need an extra column of value times frequency. That gives 0, 8, 10, 6 and 4, totalling 28 pets among 20 students, so the mean is 28 divided by 20, which is 1.4 pets. For the median with 20 values you need the average of the 10th and 11th. Counting down the frequencies, the first 4 students have 0 pets and the next 8 take you to the 12th student, so both the 10th and 11th have 1 pet, making the median 1. The mode is easiest: the largest frequency is 8, so the mode is 1 pet.

Choosing the right average matters. Take five house prices of 2, 3, 4, 5 and 86 hundred thousand. The mean is 100 divided by 5, which is 20 — but four of the five houses cost far less than that, so the mean describes none of them. The median is 4, with two houses above and two below, which is a fair description of the group. The difference is that 86 is an outlier, a value far from the rest, and the mean uses every value so one extreme number pulls it hard, while the median only cares about position. So use the mean when there are no outliers, the median when there are outliers or the data is skewed, and the mode for categorical data or when you want the most popular value.