There are three “measures of central tendency”, three numbers we use to describe what is a
“typical” data value: the mean, the mode, and the median.
First, the mean.
The mean is what most people call the “average”: You add up all the scores, and then divide
by how many scores there are.
For example, to find the mean of the scores shown, we add up all the scores, and divide
by the number of scores.
A little “n” is the letter that gets used for “number of scores”.
So we add all the numbers together, and then divide by 8.
You need to be careful when you put this in your calculator: Add all the numbers, then
press “equals”, then divide by 8; or you can put the numbers on top in brackets.
If you don’t, your calculator will only divide the 15 by 8, because of order of operations.
That’s not what we want.
In this case, the scores add to 122, which divided by 8 gives us a mean of 15.25.
I do want to show you a different way to write this.
x-bar equals sigma x over n.
x-bar is the accepted convention for the mean.
X-bar is the mean.
That big sigma there is a Greek letter capital “s”, “s” for “sum”.
“Sigma x” means “the sum of the x values”, in this case the sum of the scores.
So “x-bar equals sigma x over n” is just a shorter mathematical way of writing “the
mean of x is equal to the sum of the scores over the number of scores”.
You should try writing it this way if you feel comfortable doing so.
Okay, so if we have the mean, why do we need the mode or median?
Well, consider this situation.
There are five houses that get sold on one street.
Their selling prices are shown.
What’s the mean?
Calculate it now, I’ll wait a few seconds for you to do so.
(Add the prices, and divide by the number of houses sold…)
You should have got $309 000.
So let me ask you now,
Is this a good representation of a typical house price on this street?
Is this a good “average” value for the price of a house on this street?
No, clearly it isn’t.
It’s not typical at all!
Sometimes, for some data sets, the mean just doesn’t help us to describe a typical data
value.
So, let’s look at the mode and the median.
The mode is the most common score, the most frequent score.
If there are two scores that both occur the same most number of times, then we just say
that there are two modes: The distribution is bimodal.
(Yes, there might even be more: trimodal and so on.)
If all the scores occur equally often, then there is no mode: The distribution is said
to be amodal.
“Amodal” means “no mode”.
So the mode is easy.
What about the median?
The median is the middle score, when the scores are arranged in order.
That’s important: You have to sort the data before you can find the median.
I’m always amazed—and disappointed—when my students forget to sort their list of numbers
before they try to find the one in the middle.
Now if there’s an odd number of scores, then the median is the single score in the
middle.
But if there’s an even number of scores, then there is no single score in the middle.
So what we do is we choose the number half-way between the two scores either side of the
middle.
I’ll show you some examples in a moment, but I do also want to show you the formula
for the median.
The median is the n-plus-1-over-2-th score.
I’ll say that again: it’s the n-plus-1-over-2-th score.
This formula gives you the position of the median, not the median itself.
It tells you where to find the median—again, once you’ve sorted the numbers into increasing
order.
What’s that plus-1 about?
Well, you’ll see how it works in a moment.
It’s just a little trick that gets us a more useful result.
If the formula gives you a whole number, then that’s where to find the median: it’s
that score.
But if it gives you a half, like “seven and a half”, then you’ll know you have
to get the number half-way between the seventh and the eighth scores.
It’s not as confusing as it sounds; let me show you.
First let’s try a list with an odd number of scores.
So the median is going to be the one score in the middle.
But first, we have to sort the list of numbers.
There we go.
Now, the median is the number in the middle of the list.
That’s the 4.
The median is 4.
But what about when there’s an even number of scores.
This time the median is going to be half-way between the two numbers in the centre.
First, sort the scores.
Always remember to sort the scores if you have to find the median.
Next, look in the middle.
Because there’s an even number, we have to look at the two scores in the middle, here
16 and 18.
So the median is half-way between 16 and 18, which is 17.
(If you get stuck, you can always add the two numbers together, and then divide by two,
like I’ve done here.)