#1239 What are the mean, median, and mode?

What are the mean, median, and mode? The mean, median, and mode are all ways of calculating an average, but they are different, and they have different uses.

The mean is what most of us mean when we say the average. To get the mean of a set of numbers, you add them all together and divide them by the number of numbers that you have. For example, the mean of 12, 13, 14, 15, 16 is all of them added together and divided by five, which is 14. The median is the number that comes in the middle of the set of numbers when you list them in order. For example, the mean of 12, 13, 14, 15, 16 is the middle number, which is 14. The mode is the figure that is the most frequently used. It doesn’t have to be a number and you can find the mode of anything. Our previous example doesn’t have a mode because all of the numbers are only repeated once. Let’s try this. The mode of 12, 13, 14, 14, 15, 16 is 14 because it has the highest frequency.

So, why do we have different types of averages? Well, they have different uses and, in fact, according to Wikipedia, there are 18 different types of means, medians, and modes. Among those 18 we have the harmonic mean, the Winsorized mean, and the Turkey median. Together, all of these averages are called central tendency, and they are all ways of describing sets of data.

The mean just gives the exact average of all the data, which can be great, but doesn’t always work. For example, let’s say we have a room with 7 people in it. They are all talking about their salaries. Joe earns $23,000, Sally earns $32,000, Tom earns $38,000, Lucy earns $43,000, Bob earns $51,000, Susie earns $62,000, and Rachel earns $71,000. What is the mean of all the salaries in the room? We add up all of the numbers and divide by 7, which is $45,714.28. That seems fine and in this case the mean works perfectly well.

However, then there is a knock on the door and an eighth person enters. “Hi,” he says. “My name is Jeff Bezos.” “What’s your salary?” Joe asks him. “Well,” Jeff says, “I don’t really have a salary, but I’m worth $227.5 billion.” “Hmm,” all the others say. So, now that Jeff Bezos has joined the party, what is the mean salary? We add up all of the numbers and divide by 8. The salaries of the first seven came to $320,000, plus Jeff’s salary, gives us $227,500,320,000, and an average of $28,437,540,000. Thanks to Jeff Bezos, the average salary in that room is now $28 billion. The mean only works when there are no outliers because they can seriously skew the data. To deal with that, you can either ignore the outliers and work out the mean, or you can use the median. In this case, you line up all of the numbers and take the middle value. We can’t have a middle value of even numbers, so we take the middle two numbers ($43,000 and 51,000), add them, halve them, and that is our median. In this case, it is $47,000. And that is a good value. Of our eight people, four have a salary that is under $47,000 and four have a salary that is over $47,000.

Mode is another type of average, but it looks at the frequency of values. It wouldn’t work in the above example with salaries because none of the numbers were repeated, but it works very well in data sets where the data is not only numbers. For example, if you ask 300 people what their favorite type of dog is, you can’t take a mean of that. You’d end up with a fraction of a labrador. You can’t find a median either because there is no way to put all of the dogs in order. So, you take the mode. You can count how many times each dog breed was mentioned, and the mode is the most common one. The mode is used when the data cannot be ordered meaningfully. One of the biggest uses of the mode is in market research. And this is what I learned today.

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Sources

https://byjus.com/maths/difference-between-mean-median-and-mode

https://www.dictionary.com/e/average-vs-mean-vs-median-vs-mode

https://www.bbc.co.uk/bitesize/articles/zj6nb7h#zmf8mbk