When you look at a set of data, there are a number of ways that you can interpret the information. Depending on how you want to analyse the given values, there are various averages that you can calculate, such as the median, mean and mode.
The mode is a useful way to measure central tendency in categorical data sets. It can be used to find out significant information about a set of values as it demonstrates how popular or frequent certain categories appear.
You may need to calculate the modal average for a data set if you want to analyse a data set, so we've compiled this guide to help you find the mode, as well as the advantages and disadvantages of this type of average.
The mode is the most frequently occurring value in a data set. In cases where there are no duplicated values, a data set won't have a modal value. This makes the mode the only average that can sometimes have no value in a given data set. There may also be more than one mode in a set of values if there are multiple categories that have the same frequency.
In some instances, the mode will be different from other average values in a set of data. Many people consider the mode to be the easiest average to calculate as it can be found by listing values in a data set without the need to add, subtract or divide the values.
You can find the modal value by placing the numbers from a data set in ascending or (or less commonly) descending order. Any repeating values should be placed next to each other.
For example, you may be presented with a data set listing the shoe sizes of students in a class:
4,7,4,5,8,8,6,5,8,8,5,7,7
To find the most common shoe size, you can list the numbers in order, from smallest to largest, as follows:
4,4,5,5,5,6,7,7,7,8,8,8,8
The most common shoe size in this data set is 8, as it appears four times, which makes it the modal value. The least common shoe size is 6, as it appears only once. Size 4 appears twice, whilst sizes 5 and 7 appear three times each.
Another way to display the results is to put the values into a table, along with their frequency, as shown below:
| Shoe size | 4 | 5 | 6 | 7 | 8 |
| Frequency | 2 | 3 | 1 | 3 | 4 |
In the above example, there is a clear modal value as size 8 appears more times than the other shoe sizes. If there were no size 8 results, sizes 5 and 7 would both be the modal values because they share the next highest frequency.
You can easily calculate the mode by listing the given values in ascending order. This will show you which values appear the most often without the need for any additional calculations.
The mode isn't affected by extreme values, as other types of averages are. For example, you may be presented with the following data set:
3,3,4,5,6,23
In the given example, the last value is an anomaly because 23 is far larger than the other results listed. This would affect the mean average, as this involves adding the numbers together and dividing the total by the number of values.
With 23 included in the mean calculations, the average would be 7.3. Without the 23 value, the average would be 4.2, which demonstrates the effect an anomaly can have on the mean results. However, the modal value would still be three, whether the 23 value was included in the data set or not.
The mode is inconsistent as some data sets will have two modal averages while others will have three or more modal averages. In addition, there won't be a modal average if none of the values in a set of data appear multiple times. This is different from mean, median and range averages, which can always be calculated, no matter the frequency or volume of the values.
The mode is also unstable if there is a small data set. It is also not representative of all the values featured.
Unlike other types of averages, a data set can have multiple modal values. The values that occur most frequently should all be referred to as the mode, even if this means you will have more than one modal value in your data analysis.
When a data set has two mode averages, the set is considered bimodal. Data sets that have three modal averages are called trimodal, while data sets with four or more modes are referred to as multimodal.
Below is a selection of data sets, along with explanations of how you can identify the modal value from them.
Example 1
A dice is thrown fifteen times. The numbers below show how often the die fell on each number:
| Number on die | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency | 0 | 4 | 2 | 4 | 3 | 2 |
The values can be written as followed:
2,2,2,2,3,3,4,4,4,4,5,5,5,6,6
In the above example, the dice fell on both the number 2 and number 4 four times each, whilst the dice only fell on 5 thrice, 3 and 6 twice and no times on 1. This means that 2 and 4 are the modal values in this set of data points.
Example 2
A knitting group collects the ages of their members to see the age demographic that they most appeal to.
| Member age | 18-24 | 25-34 | 35-44 | 45-54 | 55-64 | 65 and over |
| Frequency | 2 | 4 | 12 | 8 | 13 | 14 |
The mode is the age group with the highest frequency. In this case, that is the '65 and over group' with a frequency of 14 So the modal age group in this data set is '65 and over'.
Example 3
In some instances, the categories in a data set may not be numerical. In the example below, the data set shows how popular each colour is in a class of 30 children:
| Color | Blue | Red | Yellow | Pink | Green | Orange | Purple |
| Frequency | 5 | 6 | 8 | 4 | 5 | 0 | 2 |
The mode is the eye color with the highest frequency. In this case, that is yellow with a frequency of 8. So the modal eye color in this data set is yellow.
Along with the mode, you can also work out the mean and median averages from a data set.
Mean
The mean is the most common way to work out an average from a set of data. To do so, you need to multiply the numbers on the dice by the frequency that they appear before dividing that total by the total frequency. In the example below, you would need to divide by 15 as this is the number of times that the die was thrown.
For example, if you wanted to calculate the mean for Example 1, you would do the following:
| Number on dice | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency | 0 | 4 | 2 | 4 | 3 | 2 |
(1x0) + (2x4) + (3x2) + (4x4) + (5x3) + (6x2) = 57
57 ÷ 15 = 3.8
Therefore, the mean for this data set is 3.8.
Median
To calculate the median from a data set, you need to place the values in ascending order and find the middle value. If there are two middle values, you need to calculate the mean of them.
For example, if you wanted to calculate the median for Example 2, you would do the following:
| Member age | 18-24 | 25-34 | 35-44 | 45-54 | 55-64 | 65 and over |
| Frequency | 2 | 4 | 12 | 8 | 13 | 14 |
The total frequency (or the number of members in this example) is 53. The value halfway between 0 and 53 is 27. At this point, you need to calculate which age group that entry would fall into if the set were placed in ascending age order (from 18-24 years to 65 and over). In the given example, the median age group is the 55-64 category, as this is where the 27th entry would fall.
The mean, median and mode are three different ways that you can interpret a data set by finding different averages. The mode is the most frequently occurring value in a data set. In some cases, you may find three modes or more or no modes if values don't appear multiple times.
To find the mode, you need to list the values in ascending order to see if any of the values appear more than once. The values that appear most frequently are called the modal average.
