Mean, median and mode: three ways to find the middle of data

Three kinds of average

When people say average, they usually mean the mean. Mathematicians call the mean, median and mode measures of centre: each is a single number that sums up where a data set sits. They answer slightly different questions, which is why there are three.

The mean asks: if everything were shared out equally, how much would each get? The median asks: what value sits in the middle, with half the data below and half above? The mode asks: which value is most common?

How to find each one

Start every problem by writing the data in order from smallest to largest. It makes the median, mode and range easy to read, and counting the sorted list is a check that no value was dropped.

• Mean: add all the values, then divide by the number of values.
• Median: put the values in order and find the middle one. With an even number of values, the median is the mean of the two middle values.
• Mode: find the value that occurs most often. A data set can have one mode, more than one, or none if every value appears equally often.
• Range: subtract the smallest value from the largest. It is a measure of spread, not of centre.

Worked example: an odd number of values

Here the mean is not a whole number and is not one of the data values. That is normal: the mean is a balance point, not a value someone had to score.

Worked example: an even number of values

With an even count there is no single middle value, so the median falls halfway between the two middle ones. It does not have to be in the data set.

Worked example: finding a missing value

Because the mean is the total divided by the count, you can work backwards: mean times count gives the total the values must reach.

Which average should you use?

The mean uses every value, which is its strength and its weakness. One extreme value, called an outlier, can drag it a long way. Take five salaries in thousands of dollars: 30, 32, 35, 38 and 250. The mean is 385 ÷ 5 = 77, which is more than double what four of the five people earn. The median, 35, describes a typical salary far better.

So use the median when the data is skewed or has outliers, as with house prices or incomes. Use the mean when the data is fairly symmetric with no extreme values, as with heights in a class or repeated measurements in an experiment. Use the mode for categories that cannot be added or ordered, such as the most popular shoe size in a shop or favourite colour in a survey.

Comparing the mean and median also tells you about the shape of the data. If the mean is well above the median, a few large values are pulling it up. If it is well below, a few small values are pulling it down.

Common mistakes

These errors account for most wrong answers:

• Finding the median without sorting the data first. The middle of an unsorted list means nothing.
• Picking one of the two middle values instead of averaging them when the count is even.
• Dividing by the wrong count, often after missing a repeated value. Count the sorted list.
• Thinking there is no mode when two values tie. Two values that both appear most often are both modes.
• Giving the range as two numbers, such as 3 to 9. The range is one number: the difference, 6.
• Reporting a mean when an outlier makes it misleading.

Where it fits and how to check your work

In Common Core math, grade 6 students learn that a measure of centre summarises a data set, calculate the mean and median, and choose between them based on the shape of the data. Grade 7 compares two data sets by their centres and spreads, and high school statistics adds standard deviation and the interquartile range.

Check a mean by asking whether it sits between the smallest and largest values; if it does not, something went wrong. Check a median by counting: the same number of values should sit on each side of it. A good habit is to sketch a quick dot plot, since the shape shows you whether the mean or the median is the fairer summary.