Statistics Central Tendency: Mean, Median, Mode

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We are going to be talking about central tendency, mean, median, and mode.0002

Mean, median, and mode are what most people think about when they think about statistics especially descriptive statistics.0009

When I ask my students what is descriptive statistics about, they are like mean, median, and mode.0015

Sometimes they also say standard deviation.0021

I know this is largely what sticks on people’s minds.0024

We are going to be talking about these three measures of central tendency.0029

Then we are going to spend a little bit more time on mean.0032

We are going to review the summation symbol Sigma (Σ) and we are also going to talk about the different formulas0035

for population mean versus sample mean.0045

Then we are going to talk about some Excel examples and finally we are going to compare these measures of central tendency.0048

Central tendency, basically the idea is we want to be able to summarize a distribution and we have learned about0058

some different ways you could summarize it like the shape.0066

For example you say it is a uniform shape, it is a nice way to summarize a distribution.0068

Shape is just one dimension that you could summarize a distribution on.0075

A way that you can describe it on.0080

Central tendency is another dimension that you could describe it on.0082

The so central tendency you could think of as the middleness.0086

How do you say that is the middle of a distribution?0090

There is a couple of ways.0094

When we talk about mode, we are going to be talking about most frequent value as sort of this center member.0096

In the case with 1, 2, 2, 3, 4, 5, 7, 2 is the most frequent value and because of that we are going to say okay that is one way0107

we could describe the center median is literally the middle value.0120

In order to find the median it often helps to line up your distribution in order.0130

We have the least on this side and the greatest on this side.0137

We would just count to the center so since there is 7 numbers here, we know that is an odd number so here we just pick the middle value, the median.0142

Let us talk about mean.0157

Mean is what people often call the average, the average number, the average value.0159

With mean we actually care what each value means, what is the extent of each value.0168

Because of that what we are going to do is add up, sum up all the values in our distribution,0179

in our sample and divide by the number of values you have in your sample.0190

Here we are just going to add all these up so here is a 5, 10, and another 10, 20, 24.0203

24 ÷ 7 and that will give you that mean.0211

This is probably something like 3 and 3/7.0219

That is our mean.0225

I have changed that same distribution just slightly, all that I have done is I have added an extra number and let us see what changes.0229

The way we describe the central tendency of our distribution change?0239

Let us see.0244

Remember mode is most frequent, does our most frequent value change?0246

In this case, no.0255

2 is still our most frequent value.0257

What about the median, the middle value? has that changed?0262

Here we have 8 numbers in our distribution and because of that there is no exact middle point.0268

The middle is in between two numbers.0277

In the case what you do is you take those two numbers in the middle and you divide by 2.0280

You add them up and you divide by 2.0287

Basically you find the average of the two middle values or average of two middle values.0288

In this case it would be 2 + 3 ÷ 2, something like 2, 2 ½ or 2.5.0304

That would be our median.0313

Notice that our median has changed a little bit because we added something over the smaller side of our distribution, our median has shifted over.0315

Let us think about the mean or the average.0325

Let us add them up and see.0331

0, because I added a 0 the actual value of the sum does not change, that is 24.0335

We are just dividing by 8.0344

Our mean is now just 3 instead of 3 and 3/7.0348

Once again because we added a number that is small in our distribution, our mean has shifted over a little bit towards the smaller side.0354

That is basically the ideas of central tendency.0368

Before we go on to talk about the actual formula, the formal algebraic notation for mean, I want to talk a little bit about the summation symbol.0374

What we want is some kind of symbol that will tell us add up all the numbers.0385

I do not care how many numbers you have, add them all up.0389