# Mode

In a set of numbers, the mode is the number that occurs most often. In other words, it measures how frequently a particular number occurs within a set of numbers. It is a type of average, like its counterparts, the median and mean.

To find the mode of a set of numbers,

1. List the set of numbers in ascending or descending order, including any duplicates
2. Count the number of times each number in the set of numbers occurs. The number that occurs the highest number of times is the mode of the set of data

Example

Below are a few examples of the different outcomes we can encounter when finding the mode of a set of data:

Case 1 - one mode:

1, 3, 3, 4, 7, 8

Mode(s): 3

Case 2 - more than one mode:

1, 3, 3, 4, 7, 7, 8

Mode(s): 3, 7

Case 3 - no mode:

1, 3, 4, 7, 8

Mode(s): none

### Mode vs mean and median

Although they are all types of averages, mode, mean, and median are most effective when used in certain situations.

The mode and median are both useful when the data is highly skewed, meaning that there are very small or very large numbers that shift the average of the data such that the arithmetic mean wouldn't represent the average very well.

On top of this, the mode can be used in sets of data where the data is skewed in such a way that the median is not representative of the sample, or even when the mean or median may just not make as much sense in the context of the data.

Examples

1. Let's say that there are 50 students in a class, and each of the 50 students was given 50 jellybeans. 24 of the students in the class ate all but 1 jelly bean, 2 students ate 25 and have 25 jellybeans left, and the remaining 24 ate 0 jellybeans, so still have all 50.

If we were to check the median number of jellybeans each student has left, we would find the median to be 25 jellybeans. This is not representative of the class because only 2 students out of 50 have 25 jelly beans. Every other student has either 1 jellybean or 50 jellybeans left. Depending how we are studying the data, this could be useful information, but if we want to know whether the kids tend to save their jellybeans, eat some of them, or eat them all, the median would not be useful information.

On the other hand, the mode would tell us exactly what we need to know; the kids either like to eat most of their jellybeans or save them all. Few will eat just some jelly beans.

2. The mode can be especially useful when dealing with variables with integer values with large sample sizes. Say we're looking at how many arms that a population of 100,000 people have on average and that 0.5% of the population has had an amputation.

0.005 × 100,000 = 500

So, 500 people have fewer than 2 arms, and significantly fewer people could have a mutation that results in them having more than 2 arms, so:

• The mode would be 2
• The mean would be some non-integer value less than 2, since it should still be very close to 2
• The median would also be 2

Whether a measure of data is meaningful depends on what we are studying. In this example, if we want to know how many arms a typical person has, both the mode and median would be sensible to present, since it would mean that most people in the population have 2 arms.

On the other hand, if we knew the mean were 1.985 arms, it would be useful in that it tells us that even though most people have 2 arms, and the average is very close to 2, that not everybody has 2 arms. This is a simpler example, but in more complex cases it could tell us that there's some factor worth studying that the mean or mode may not have told us.

### Did you know?

Unlike median and mean, mode can be used with more than just numbers or ordered values. The data sample can also be something like the names of the people at an event. So, you could use the mode to find out that the most common names of people attending the event were Peter and Sally!