The median is the middle term. Median = Middle. It sounds a bit like the word ‘medium’, which is also in the middle.
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The median income in a country is the income of the person who, if you lined everyone in the country up by income, would be exactly in the middle of the line. The median is different from the mean, and tells us different things. The mean can be skewed by outlier results, like huge incomes, but the median is less skewable, so to speak.
But if it’s that simple, what’s the point of a whole video on it?
Well, depending on the context, there are three different ways of calculating the median, each one more advanced than the next, and you need to know all of them! Well, at the very least it would be cool to know them. Like, in a desert it is useless to know how to calculate the median. But it’s great for tests like the GRE and GMAT.
Let’s start with the finding the median in a situation that you are probably most familiar with.
The disordered list:
Find the median of: 4, 17, 12, 7, 13, 7, 8
You might know that to find the median you need to put the numbers in ascending order. But many people forget to do this in the rush of the test and just pick the middle number on the screen, which is the number 7 in our list. Don’t be that student. Instead, reorder the list first:
4, 7, 7, 8, 12, 13, 17
Now we can visually spot the middle number simply by looking at the list and seeing that it’s 8, or by manually crossing off the biggest and smallest numbers in pairs, until we are left with one or two numbers remaining. Here, we are just left with 8. But what if we are left with two numbers?
Take this list:
60, 62, 67, 64, 65, 69
What is the median here?
Did you remember to reorder the list?
You get: 60, 62, 64, 65, 67, 69. As there are an even number of terms, we don’t get one obvious middle number. If we cross off the biggest and smallest numbers in pairs we get 64 and 65 left over in the middle. In this situation, simply find halfway between the two numbers. Halfway between 64 and 65 is just 64.5, so 64.5 is the median.
But if you can’t easily find halfway between two numbers, for example 37 and 72, just add the two remaining numbers up and then divide the answer by two. 37 + 72 = 109 and half of 109 is 54.5 (because half of 100 is 50 and half of 9 is 4.5). So, halfway between 37 and 72 is 54.5
I have now dealt with the first way of calculating medians, in the situation of a simple disordered list.
But did you know there are two other situations in which you might be asked to find a median?
The consecutive list:
You might be asked to find the median of a consecutive list, for example:
What is the median of all the positive odd integers less than 75? This is a consecutive list because the difference between all the terms is the same, we just go up by 2 each time to get to the next odd number.
For this, there is a simple formula to find the median: (Biggest term + smallest term)/2
The biggest odd term beneath 75 is 73 and the smallest is 1. 73 + 1 = 74. 74/2 = 37. So the median of all the positive odds less than 75 is 37.
But I can tell you something even more amazing, which very few students realise. In any consecutive list, whether that’s consecutive integers like 5, 6, 7, 8; consecutive multiples of 5, like 60, 65, 70; or consecutive odd or even numbers, there is one amazing fact to remember.
The median will always equal the mean.
That’s right. No need to add up all the terms and then divide by how many terms there are. No.
Once you have found the median of a consecutive list, you have also found the mean!
This has saved me a huge amount of time on several occasions.
Applied to our example, we can say that the arithmetic mean, or average, of all the positive odds numbers less than 75 is 37, as that was what the median was.
There we have it. Everything you need to know about the median for real life, and for the GRE and GMAT tests. I said almost in the title just so I don’t get sued by an applied mathematician complaining that I didn’t talk about pseudo-medians or marginal medians.
If you found this at all helpful, please do interact in some way! Everything helps!
I offer private GRE / GMAT tutoring online at a fixed rate of $140/hr. Please get in touch via the email below, or through my tutoring website: www.gretutorlondon.com/
Enquiries: philip@gretutorlondon.com