Mean, median, mode and range

In statistics we want to describe what our data looks like. So we use different descriptors, which each tell us something about our data. The mean, the median, the mode and the range are the four you'll use the most. The usual hurdle is that they feel like four versions of the same thing, when actually each one answers a different question.

When do I use this?

Whenever you have a data set and want to summarise it in a number or two. Every descriptor tells us something different about the data set, and that is useful when we investigate our data. Some descriptors describe both grouped and ungrouped data, where others only describe one of them.

We use the shoe sizes of a class of 20, the same data set as in the rest of statistics:

$$\{38, 38, 38, 38, 39, 39, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 45, 46\}$$

The mean

The mean of a data set describes the middle value of all the observations. For example, for the observations 16 and 17 the mean is 16.5.

We can find the mean of a data set by dividing the sum of all the observations by how many observations there are:

$$\text{mean} = \frac{\text{sum of the observations}}{\text{number of observations}}$$

For the shoe sizes we add all 20 sizes together and then divide by how many observations we have. The sum is

$$38 + 38 + 38 + 38 + 39 + 39 + 39 + 39 + 39 + 40 + 40 + 40 + 41 + 41 + 41 + 42 + 42 + 43 + 45 + 46 = 808$$

so the mean is

$$\frac{808}{20} = 40.4$$

The mean of this data set is therefore 40.4. It's a fraction with a big numerator, and dividing out is all there is to it.

The minimum and maximum value

As the names suggest, the minimum value of a data set is the smallest value we have, and the maximum value is the largest. In the shoe-size data set the minimum value is 38, and the maximum value is 46, because that is the largest observation we have.

The range

The range shows the difference between the largest and the smallest observation in the data set. The difference between two numbers is just one subtracted from the other, so it's the maximum value minus the minimum value:

$$\text{max} - \text{min}$$

For the shoe sizes that gives $46 - 38 = 8$.

The mode

The mode is the number in our observations that occurs the most times. In the shoe-size example we can see that size 39 occurs 5 times, which is the highest number of times any size occurs. So 39 is our mode. If you've made a frequency table, the mode is the row with the biggest frequency, and in a bar chart it's the tallest bar.

The median

The median is the number that lies in the middle of our observations. Always remember to have sorted the observations from smallest to largest before you look for the middle.

If the number of observations is odd, there is one number that lies exactly in the middle of the sorted data set. We can find it by crossing out one number from each end, over and over, until we hit the middle.

In our example the number of observations is even, so when we cross out from both ends we end up with two numbers in the middle:

$$\{\cancel{38}, \cancel{38}, \cancel{38}, \cancel{38}, \cancel{39}, \cancel{39}, \cancel{39}, \cancel{39}, \cancel{39}, 40, 40, \cancel{41}, \cancel{41}, \cancel{41}, \cancel{42}, \cancel{42}, \cancel{43}, \cancel{45}, \cancel{46}\}$$

Here we would take the mean of the two middle numbers to find the median, but since both of them are 40, the median is just 40.

Worked example: all four at once

Take a smaller data set so you can see every step: the marks $\{7, 3, 9, 3, 8\}$.

1. Sort it first: $\{3, 3, 7, 8, 9\}$.
2. Mean: $\dfrac{3 + 3 + 7 + 8 + 9}{5} = \dfrac{30}{5} = 6$.
3. Median: five observations is odd, so cross out one from each end twice and the middle one is $7$.
4. Mode: $3$ occurs twice, the others once, so the mode is $3$.
5. Range: $9 - 3 = 6$.

Four descriptors, four different numbers, and each one is telling you something true about the same five marks.

Common mistakes

• "The median is the middle of the list as it was written down." Only after sorting. Always order the observations from smallest to largest first.
• "With an even number of observations there is no median." There are two middle numbers, and the median is the mean of those two. If they're the same number, that number is the median.
• "The mode is the biggest number." The mode is the most frequent number, not the largest one. Here the largest is 46, the mode is 39.
• "The range is the largest value." The range is the difference, max minus min.
• "The mean must be one of the observations." It usually isn't. Nobody in the class has shoe size 40.4.

Related

These are the descriptors under statistics: organising and describing data. The next descriptors are the quartiles, which split the sorted data at 25%, 50% and 75% instead of only at the middle, and from the quartiles we can spot outliers.