What is the Mode in Math?

When it comes to mathematics and statistics, there are many variables, equations, and values that are all closely related. The mode in math is certainly no exception. It is most often thought of as being in the same circle as the mean and median of a data set, which is perfectly understandable as all three (mean, median, mode) are often used at the same time to describe a particular data set. That said, learning about the mode is a fairly simple and straight forward endeavor.

Simply put, the mode is a numerical value that tells you which number, or even numbers, occurs the most frequently in a data set. This is very different than the mean, which is the average of a set of numbers. For example, if you have the numbers 0,1,3,6,10, the mean (the average) would be 5 ((0+1+3+6+10)/4). The mode is also completely separate from the median, which tells you the midpoint of a set of numbers. In our above quick example, the median of {0,1,3,6,10} is 3, as that number has an equal amount of data points before and after.

How Do You Calculate the Mode?

Calculating the Mode is simple and straightforward—with a couple of minor tweaks for some instances. You count the data points and find out which set has the most. 

That’s it. Really. 

It's almost as easy as calculating the square footage of a space. 

Let’s look at a very straightforward example.

Let’s say we have a classroom of kids, and they are all going to grow bean plants for a science project. There’s nothing particularly hard about this, or even some grand experiment going on. You just want to help them grow cute little plants and knock their socks off when they see that they can do this. To that end, you buy fourteen plastic cups, one for each child in your class, and fill them with dirt. You also have a plethora of beans to plant. So, when it’s time to get to work, you don’t pay particular attention to how many seeds each cup gets. Some get two beans. Some get three. Some get four…A few might have even gotten five.

You don’t really think about this until it’s time to bury the seeds. You glance into each cup and start counting. What you find are the following in terms of the numbers of beans in each cup:

{2,4,4,3,2,3,2,2,2,5,5,3,2,2}

Now, you want to know what number of beans per cup is the most frequent. This isn’t the average because the average would be 2.92, and you didn’t put fractional beans into cups.

You could simply tally the numbers, but it might be easier to see this if we rearrange our data set from least to most:

{2,2,2,2,2,2,2,3,3,3,4,4,5,5}

You can then easily count how many times each number appears. What we get is that we have seven 2’s, three 3’s, two 4’s, and two 5’s. The mode, therefore, is two. This will also play into statistics nicely because you can also say that given a random cup, a child is most likely to have received two beans in his cup.

Calculating Special Cases

Ah, but what if there’s a tie? Yes, this is probably the most common question when it comes to determining what the mode is. You’ll see that nothing changes in how we get to our answer. So let’s look at another quick example:

Let’s say you have a standard deck of cards, complete with 52 cards total, and you want to play bridge with three other people. Everyone is dealt a hand of 13 cards. For giggles, you decide to determine what the mode is for your hand. Just because. To keep things even simpler, we’ll pretend you weren’t dealt any Jacks, Queens, Kings, or Aces, but if you want bonus points in this scenario, you can just assign them a value like Jack = 11, Queen =12, King =13, and Ace = 1 (this is done in computer programming all of the time).

Your hand looks something like this:

2 of clubs, 5 of spades, 4 of hearts, 7 of hearts, 7 of diamonds, 7 of clubs, 5 of spades, 4 of diamonds, 8 of hearts, 9 of spades, 3 of clubs, 8 of clubs, 8 of diamonds

What’s the mode regarding the number dealt? Well, let’s first strip out the suits, so we get:

{2,5,4,7,7,7,5,4,8,9,3,8,8}

Then let’s rearrange our numbers to make counting easier:

{2,3,4,4,5,5,7,7,7,8,8,8,9}

We see that there are both three sets of 7s and three sets of 8s. As such, the mode of this data set is 7 and 8.

We can also determine the mode of the suits too. We had:

{clubs, spades, hearts, hearts, diamonds, clubs, spades, diamonds, hearts, spades, clubs, clubs, diamonds}

Rearranged:

{clubs, clubs, clubs, clubs, diamonds, diamonds, diamonds, hearts, hearts, hearts, spades, spades, spades}

We see here we got four clubs, and three of all the other suits. The mode in terms of suits, therefore, is “clubs.” Of course, that doesn’t sound very math-like since it’s not pure numbers. So if we wanted to shuffle it through a computer program for whatever reason, we’d likely assign a value to each suit, so that clubs were 1, diamonds were 2, hearts were 3, and spades were 4. That, of course, is a little beyond the scope of this tutorial, but that’s the general idea.

Alright, but what if everything only occurs once and it’s a giant tie?

So this is the last and final rule that has to be obeyed. To have a mode in a data set, whatever data point you are looking at MUST occur at least twice. So while the data set:

{1,2,2,3,4} has a mode of 2

The data set:

{1,2,3,4,5,6,7,8,9,10}

does NOT have a mode, as each number only occurs once.