In mathematics and statistics, the mode is often considered a kind of average and spoken about alongside the mean and the median.
In this article, which is part of our series on averages, we'll be looking at averages, the mode, how you find the mode, and the pros and cons of using it.
What Are Averages?
Averages are used in maths to generalise datasets by giving them a value based on the values in the dataset. Basically, an average should give you a value that's representative of all of the values within the dataset.
We're familiar with averages in our everyday lives as they're used in everything from academia (for grades and scores), sports and betting, household spending habits, temperature, travel times, and population demographics. You've likely seen a value today that was actually an average.
The Different Kinds of Averages
There are three or four main kinds of averages depending on who you ask or what you're taught: mean, median, mode, and range. The range, as the name suggests, indicates the full range of a dataset so it doesn't really provide the same kind of information as the other three.
The mean, median, and mode all attempt to represent the most typical value for a set of data, albeit in different ways.
The Mean
The mean is often what people are referring to when they say “average” and this is calculated by dividing the sum of the values by the number of values. This will give you a typical value and can be very useful for statistics.
However, the mean can be skewed by extreme values and very high or very low values in datasets can result in the mean not being truly representative of the majority of values.
The Median
The median is sometimes used as an alternative to the mean in instances where outliers or extreme values could skew the mean.
The median is simply the middle number in a sorted dataset. To find the mean, you arrange all your values numerically and select the one in the very middle of the list.
For datasets with an odd number of values, there's only one value in the middle. For datasets with an even number of values, the true middle lies between two values, which means you need to take the mean of these two values. This is essentially the halfway point between the two values.
The Range
The range is simply the difference between the highest and lowest values in the dataset and doesn't give any more information than that. The range is sometimes just referred to as the spread and while quite different to the other averages, it can be quite useful.
For one, if you took the average temperature for a place that you were looking to visit, you could get a good idea of what kinds of clothing you should pack. However, if the range indicated that over a given period the temperature varied by 30 degrees, you'd know that it's worthwhile bringing clothes for hot and cold days!
How Do You Find the Mode?
The mode is the most commonly occurring value in a dataset, which makes it quite easy to find. You just have to count the frequency or number of times each value occurs in your dataset.
This is quite time-consuming if you have a lot of values and are doing it manually, but mathematically, it's very simple as you're just counting.
Let's take the weather in Brisbane this week (at the time of writing) as an example.
- The highs each day are 27, 26, 28, 31, 29, 32, and 31.
- It'll make things easier if we put the values in order: 26, 27, 28, 29, 31, 31, 32.
- Every value occurs just once except 31, which occurred twice. This is our mode.
However, if we were to include an extra day, with a high of 28 degrees, our dataset is now 27, 26, 28, 31, 29, 32, 31, and 28.
- Let's put them in order again: 26, 27, 28, 28, 29, 31, 31, 32.
- In this example, we have every value occurring once except for both 28 and 31. Unlike the median, we don't take the mean of these two values. Instead, this dataset has two modes and is known as “bimodal”.
If every value only occurs once, then the dataset has no mode.
With more than two modes, the dataset is multimodal.
With one mode, the dataset is unimodal.
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What are the Advantages of the Mode?
The mode is a useful average as it's very simple and easy to understand. The maths behind it is nothing more than a certain value that appears more frequently than any other. The mode is very good for non-numerical data.
Unlike the mean, the mode won't usually be affected by outliers and extreme values and for datasets where there are clusters or groups of commonly occurring values, it can be particularly useful.
In multimodal datasets, for example, you can see if there a certain patterns or phenomena that repeat within the data. Averages like the mean and median, which only offer up one value, won't recognise repeating patterns in distribution like the mode would.
What are the Disadvantages of the Mode?
The mode isn't flawless, though, and the fact that datasets can have zero, one, or multiple modes means that it's not as precise as the mean or the median.
The mode also won't necessarily change if the dataset subtly changes. Lots of very close values won't be reflected by the mode, especially since it only takes one value to occur once more than any other to become the mode and those values have to be the same.
These subtle changes also mean that the mode doesn't fully reflect the entire dataset as a whole. With the mean, adding or removing values from the dataset will change the average (unless the new value is the average) so you know that every value has been used.
The simple fact that there may not be a mode means that it can't really be applied to certain datasets.
When Should You Use the Mode?
Now that you're aware of the pros and cons of using the mode, you need to think about when it should be used.
The mode is particularly good for categories and non-numerical data. For example, you could use the mode in a dataset of people's favourite colours.
Imagine you surveyed your classmates or colleagues and asked them their favourite colour and had the following results: Red, Blue, Green, Blue, Yellow, Blue, Red, Green, Green.
As always, it's useful to organise the data. We can't sort colours numerically, but we can sort them alphabetically: Blue, Blue, Blue, Green, Green, Green, Red, Red, Yellow.
By counting the frequency of each response, we can see that both blue and green are the modes for this dataset, making it bimodal.
With this example, the most popular colours are blue and green. Knowing this, you could make sure that the decorations for your next event reflect this.
The mode also won't be affected by extreme values so even if somebody didn't take your survey seriously and put “tartan” as their favourite colour, you'd still know that blue or green would be good colours to go with.
Getting Help with Averages
As you can see, there are lots of applications for different averages, but it's not always clear which one you should use and which will give you the best information.
Whether you need help understanding averages, learning more about maths, or passing a statistics exam, there's plenty of help out there.
While averages might work well for numbers, you don't want a one-size-fits-all approach to learning about maths and statistics. Private tutors can provide tailored sessions that are designed with you in mind and not just a typical or average student.
You can learn what you want in a way that works for you and with so many tutors out there, there's something for everyone and every budget.
On the Superprof website, for example, there are maths tutors all over Australia and around the world and even if you can't find a suitable tutor locally, there are online tutors and for subjects like maths and statistics, online math tutoring can be just as effective as face-to-face tutoring.
For those on a tight budget, group tutoring is a way to spread out the cost of a tutor's time as every student is contributing to the cost of the sessions.
Don't forget that a lot of the tutors on the Superprof website offer their first session for free so you can always try a few out and decide which of them are right for you. If you live in Melbourne, for example, all you need to do is search on your browser for "maths tutor melbourne" followed by Superprof, and you will have access to an array of math tutors in the city.
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