Sign In | Subscribe
Start learning today, and be successful in your academic & professional career. Start Today!
Loading video...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Statistics
  • Discussion

  • Download Lecture Slides

  • Table of Contents

  • Transcription

  • Related Books

Bookmark and Share
Lecture Comments (5)

0 answers

Post by Paulette Jones on May 8, 2013

Thanks for your help. You're a terrific teacher. :-)

0 answers

Post by DIntre Smith on August 21, 2012

Keep up the good work! Ever thought about teaching the Probability P/1 Actuary exam?

0 answers

Post by Johnnie Brown on March 26, 2012

You are a God send!!!!

0 answers

Post by Ryan Mulligan on January 26, 2012

Amazing teacher, Taking this course at University and our Russian prof is horrible. This series saved me for Midterms... Cheers!

0 answers

Post by munir eldeeb on February 8, 2011

your explanation is good, but your sloppy on things, like look at that S at 6:50

Central Tendency: Mean, Median, Mode

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Roadmap 0:07
    • Roadmap
  • Central Tendency 1 0:56
    • Way to Summarize a Distribution of Scores
    • Mode
    • Median
    • Mean
  • Central Tendency 2 3:47
    • Mode
    • Median
    • Mean
  • Summation Symbol 6:11
    • Summation Symbol
  • Population vs. Sample 10:46
    • Population vs. Sample
  • Excel Examples 15:08
    • Finding Mode, Median, and Mean in Excel
  • Median vs. Mean 21:45
    • Effect of Outliers
    • Relationship Between Parameter and Statistic
    • Type of Measurements
    • Which Distributions to Use With
  • Example 1: Mean 25:30
  • Example 2: Using Summation Symbol 29:50
  • Example 3: Average Calorie Count 32:50
  • Example 4: Creating an Example Set 35:46

Transcription: Central Tendency: Mean, Median, Mode

Hi and welcome to

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 about 0058

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 way 0107

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

It will be nice if we have a symbol that could do that.0394

Here is the summation symbol (Σ).0397

It looks like that sigma (Σ), upper case.0401

We are going to be using lower case sigma for something else.0409

Upper case sigma and people will write some variable here to represent which variable set you want to take from.0413

Add up all of age.0424

If it is age and I will represent that with letter A, if A = age and I want to get the average age then maybe I will put A here.0427

Sum up all of A.0441

Sometimes the summation symbol is written like this.0445

There is always a little more detail that sometimes they do not show you what you cannot see.0448

Here is often what is hidden underlying this.0456

Usually this is saying add up everybody in your set A.0460

Take them all the way from I, the index from the first one all the way to the last one in your set.0467

Remember how many are in our set is usually represented by n.0477

From 1 all the way to n, 1, 2, 3, 4, 5, 6, 7, all the way to n, whatever n is.0485

Add up A.0493

Here we are going to put a little A sub I.0496

This means this.0499

Here A is age, and let us say A is actually the set 5, 10, and 15.0501

Here is the corresponding I, 1, 2, and 3.0515

What the summation symbol is telling us is if I =1 for this one this index twice to, add up A sub I, the first A.0523

Then add to it A sub 2, which is 10.0537

Then add to is A sub 3, which is 15.0542

That is how the summation symbol works, al the way up to n.0547

N in this case is 3 because there is 3 numbers in our set.0551

Let us do one more example.0556

Here is my summation symbol and now I’m going to say add up x.0561

Let us say x is number of books read this summer.0570

X is going to be 2, 4, 6, 8, 10.0586

We now that it is saying go from I sub 1 all the way to n, n in this case is 5.0595

All the way to I here, this is x sub 1, x sub 2, x sub 3, x sub 4, x sub 5.0609

Here, this is telling us go all the way from x sub 1 to x sub 5 and add them all up.0625

This is just a formal algebraic notation to say add up all the numbers in your set no matter how big or small your set is.0634

Now that you know the summation symbol we can start creating the formula for mean, samples, and population.0648

Let us think about the mean for the sample.0659

The mean of the sample is always represented by the symbol x bar.0662

How do we calculate x bar?0671

We have to add up all the numbers in our set and divide by the number of items in our set.0674

We know that number of items is n.0680

We could use our summation symbol.0685

We know that it is the summation symbol but all of x.0693

This is one way that you could write it and that is a very simple way.0698

It is implicit but it is telling you go to i=1 all the way to n.0701

But just for our purposes, I’m just going to put in the hidden stuff just to show you as well.0708

It is implicitly saying go from I =1 all the way up to n, however many n is and because of that out of each x sub i.0716

X sub 1, x sub 2, x sub 3, all the way to n.0728

That is one way that you could see it but you do not need all of this complicated stuff.0732

That is optional.0738

If you have the distribution of the actual population which is almost impossible to get 0741

but let us say from some reason you want to write a formula for it.0749

Actually you do want to write a formula for it because it is going to be handy if we do not.0753

How would we write the formula for that?0760

For population, we do not call the population mean x bar, instead we call it mu.0762

Our population mean is called mu.0771

Here we want to add up all of the x but here we use an upper case X because we are saying draw nail from the population distribution.0774

Instead of the lower case n, we are going to write upper case n because upper case N means how many values0787

you have in your population not just your sample.0796

I’m going to put in the invisible stuff, go from X sub 1 all the way up to N.0802

Here I’m going to put that index.0813

There you have it.0815

This is the population formula and this is the sample formula.0817

Although their sums like differences and notations, for instance here we use the Greek letter, 0824

we use the roman letter, here we use upper case, here we use lower case.0831

Except for I, I just means index.0836

It just means a little counter or pointer to each thing in the set at a time.0838

I is just, you could use j or whatever you want nut we usually use I for index.0844

Other than those little characteristics, the mean is the same.0850

It means add up all values in your set and divide by however many you have in your set.0855

I want to point out one other thing here.0862

Here one thing you could see is that if you have the mean of your sample.0866

If you happen to have x bar and you multiply n to each side.0874

I’m going to multiply n to each side invisibly here.0879

I’m going to multiply n.0883

Then you will get the sum of all of your x.0885

That is just algebraic transformation.0892

I have not done anything to change the formula.0896

The formula stays the same.0898

This little trick is going to come in handy later on.0900

Just keep that in mind in the back of your head.0905

Let us do some Excel example.0910

Here we see that this is asking us about our data once again from our 100 friends.0914

It is asking us find the mode, median, and mean in Excel for height as well as male height.0924

Just to remind you, here is our frequency distribution that we looked at in previous lessons.0931

We thought that this was a bimodal distribution, if you just consider height of everybody.0937

This one is actually a little bit taller because we have to add up all these little guys on.0948

They are always asking us to do is find the mode, median, and mean in Excel.0963

Excel is going to make life a little bit easier for us.0968

We do not have to put them in order.0970

We do not have to spend time adding them up.0972

Excel will do it automatically for us.0975

Let us open it up to Excel.0978

Here is our data once again and remember it is asking us the height.0981

I’m going to go to our height variable, our height column.0985

It is height in inches and I’m going to click on the height sheet.0991

Here I have put in some labels for us.0998

Height, find the mode, median, and mean.1002

Just the height of males, find the mode, median, and mean.1004

Excel makes it easy for us with their functions.1010

Their functions for mode is simply mode.1013

We are just going to go and select our data to find our mode from.1019

Excel will go ahead and count which is the most frequent.1023

I’m going to close my parentheses and hit enter.1029

It turns out that our most frequent mode is 64.1032

Let us find the median.1038

Excel makes it easy for us once again, it is just =median.1040

If you are ever at lost for how to find formulas, one thing I do is it over.1045

I look it up on the Excel help or function help and they have a whole bunch of list of functions categorized into different types.1053

It is pretty easy to find things that you need.1063

We should have saved whatever our data is but we could just drag it again.1068

Here is our data.1074

I’m going to close my parentheses and hit enter.1080

Here we see that the median is different from the mode.1084

The median is actually 66.1087

That is the middle.1089

But the most frequent values is 64.1090

Now let us calculate the mean.1094

In Excel, mean would not mean anything.1097

Instead, you have to type in average.1102

Let us put in our data, and I’m going to close in my parentheses.1110

Hit enter.1119

What we find is our average or mean is 67.1120

It is not 66 or 64.1127

Remember height was bimodal distribution.1129

One of the things that we see here in this bimodal distribution is that mode, median, and mean are not necessarily of the same value.1138

Remember how does males, when we looked at it before, this was actually approximately normal distribution.1151

Normal distribution for male, we are going to find that more in detail later.1162

Normal distribution basically means it has a one month in look.1166

It has axis of symmetry, that is also the mode.1173

It is unimodal, symmetrical, and the mode of inflection is about the size of a standard deviation.1178

Let us look at height of males.1190

The mode, median, and mean.1192

In order to find that just for males, one thing we may want to do is sort our data.1195

We probably want to sort it so that all the males height are grouped together.1200

I have already sorted it for you but I’m just going to color the heights of males.1203

I’m just going to color this blue so that we can remember to ourselves these are the heights of males.1209

We could just use the same formulas, mode.1223

I’m only going to select these blue ones and I’m going to close my parentheses.1231

For males, the mode is 69 and I’m just going to copy control c so that I do not have to go back to my data all the time.1240

Let us put in median and I’m just going to command v, copy and paste in my data, hit enter.1257

How do you know? The mode and the median are the same number.1268

Finally I am putting in average and I’[m just copying my data and here we find that the average is also the same.1274

The mean is also the same.1285

That is largely what you find from normal way of distribution.1286

You find that the mean, median, and mode are the same values.1291

That is one thing handy about a normal distribution.1295

That is our Excel example.1302

Let us contrast median and mean, two of the measures of central tendency.1307

One of the things about median that you should know is that it is handy because it is less affected by outliers.1315

Means are more affected by outliers and you could think about why.1322

They are more affected because they are actual extent of the values matter.1328

If you have a very large number, that value is exactly how much that value is, it gets added in.1343

Because of that means are more affected by outliers or one extreme score.1352

Medians are less affected even if you add in one extreme score, it does not usually change the median by too much.1357

Now let us talk about the relationship between the parameter, the parameter mean, and the statistic mean or the parameter median or statistic median.1366

The sample mean or x bar is actually the best predictor of the population mean or mu.1377

Usually in medians though, we do not use the median of a sample which is not very many people use 1386

that in order to predict the median of the population.1398

Not many people use that to predict this and largely it is because the relationship is less stable than this one.1428

We are going to learn more about why this relationship holds later on in the lessons.1436

Different types of measurements, it will be better to use median or mean for different types of measurements.1442

For mean, usually you want to use any type of measurements that are either ratio or interval.1455

Those are going to be best for using mean as an indicator of central tendency.1466

Median is best for measurements that are ordinal.1474

For nominal measurements, median and mean do not mean very much.1485

For nominal measurements, you may want to use mode.1492

Which distribution you want to use this with? 1497

Median is most frequently use for describing the center of skewed distributions.1500

That is what you want to think about when you think about median.1508

When you think of mean, this can be used for a lot of distributions but for skewed distributions the mean will be a little bit off.1512

For all others.1520

The mean is going to be pretty flexible for us.1524

We are going to be using that quite a bit.1527

Let us move on to an example.1532

Sometimes people think that mean, median, and mode is pretty easy.1535

It is true.1539

It is pretty easy to calculate but there are going to be problems 1541

that do not necessarily ask you just to straight up calculate the mean, median, and mode.1544

But asks you to think about it more flexibly.1548

Here is an example of that.1553

There were 9 people in a room who made an average salary of 40,000 per year.1556

When someone walks in who makes 84,000 per year, what happens to the mean?1561

Here you do not know each individuals salary of these 9 people.1567

All you know is that their average is 40,000.1573

Then you know that somebody also walks in who makes 84,000.1578

Can you calculate the new mean of the 10 people in this room?1582

Yes you can.1588

Here is how.1589

The previous mean of the sample, we call it x bar, was 40,000.1591

I will just write 40 for now and later will have in to add that k.1599


Since there were 9 people in the room, what we do not know is this.1608

I going from 1 all the way to 9.1620

What we do not know is the sum?1625

We do know the average ends up being 40 for this 9 people.1629

Remember there is that algebraic transformation that we do.1634

I can multiply both sides by n and get the value of this.1639

I do not know each individual value but I know what the value if it is all added up.1645

If I do 40 × 9 then I will get the sum for 9.1650

I will get that sum automatically.1657

In order to get the mean for everybody in the room, this is going to be my x bar of the 9 people.1661

What about my x bar for 10 people?1670

What I need is the sum of all 9 of my people and add in 84, and divide that whole thing by 10.1674

Because this is now the sum of all 10 people.1693

I have the sum of 9 + my last guy.1698

Because we know this guy it is 40 × 9 ÷ 10.1703

I’m just going to use just random Excel sheet to help me do this calculation.1719

You could do this in your calculator, in our head.1737

Here is (40 × 9 + 84) ÷ 10.1740

I get 44.4.1751

My new mean is 44.4.1753

Before our mean was 40,000, now it is 44,400.1759

That is our new mean.1765

This is what you mean by it is not just only a straight forward calculation of the mean, this is what I want you to think 1769

about what the mean is made up of.1775

It is made up of two pieces, the sum of all the values and it is divided by n.1777

It is those two pieces.1785

Can you play with these two pieces?1787

Here is another example.1792

The mean of x and y is 20, the mean of x, y, z is 17 ½ , what is the value of x?1797

The mean if x and y, it is just x bar for two of these guys.1808

Goes from I all the way up to 2.1820

That is equals 20.1830

But the x bar of sub 3, the 3 of these values is I goes from 1 all the way to 3.1834

X sub I / 3.1846

This is 17 ½ 1850

In order to get this, it will be nice to know (x + y + z) ÷ 3 = 17 ½ .1854

We actually have x + y, because this is actually x + y.1871

That is simply x + y = 20 × 2.1882

I’m just going to multiply 2 on each side to get 40.1889

We already have x + y.1895

I will put in my 40 + z / 3 = 17 ½ .1898

I will just multiply both sides by 3 and then subtract 40.1909

I will just do that here, 17.5 × 3 – 40 = 12.5.1929

Z=12.5 or 12 ½ .1945

That makes sense because here when it is just x and y, the mean was higher.1951

When we added z in, the mean became lower so we know that z must have been something low to drag down the mean.1957

It is indeed 12.5.1965

Next it says the average number of calories in a BK burger at their 15 burger menu is 700 calories.1972

The average number of calories of 12 burgers in Mc Donald’s burger menu is 670.1980

What is the average calorie count when combining across BK and Mc Donald’s menus?1989

Here what we could do is get the sum of the BK calories and the sum of all the Mc Donald’s calories and1997

divide by the total number of burgers when we combine across their menus.2007

For the BK burger menu it is 15 burgers and their average is 700 calories.2014

700 × 15 should give us the summed up number of calories.2027

Let us add that to the sum of all the Mc Donald’s calories, 670 × 12.2037

To get the average, we have added all the calories up.2049

To get the average we want to divide by number of total burgers we are talking about now which is 15 + 12.2053

Remember you could always do it in your head, on a paper, calculator, whatever you want.2074

I’m just going to do it in Excel just to show you.2079

You need to make sure about your parentheses, that is (700 × 15 + 670 × 12) ÷ 27.2086

That is something and 2/3.2112

I forgot, I cannot keep this up at the same time.2125

686 2/3 is the average number of calories when we average all the burgers on both Burger King and Mc Donald’s menus.2128

Here is example 4.2147

It says create an example set where n =5, where the median is greater than the mean.2148

Modify that set so that the mean is greater than the median.2154

Let us start off by just putting down 5 slots.2158

Basically when looking for the median here, what we want the mean to fall somewhere down here.2167

In order to pull that mean downward what we would do is simply create more numbers down here.2178

These numbers to be farther away from the median.2191

Let us make them 0, and we will make this 5 to make that distance farther.2195

We will make these 6 and 7.2203

The median is easy, we already have them it is 5.2208

It is in the middle.2211

What is the mean?2214

Let us add these up and divide by 5.2216

5 + 6 = 11 + 7 = 18 ÷ 5 = 3 3/5.2218

3 3/5 or 3.6 that is going to fall below 5.2233

Here we have the median being greater than the mean.2242

5 is greater than 3.35.2248

Let us modify that set so that the mean is greater than the median now.2250

What I would do is I just think about the mean as being more influenced than the actual little number in here.2261

I could probably keep a lot of these the same but all we want to do is weight down one side.2268

If I do that let us see what happens.2276

The median obviously does not change but let us see what happens to the mean.2279

Does the mean change?2286

11 + 20 =31 ÷ 5 = 61/5 or 6.2.2288

Here we see now the mean is greater than the median.2299

What I have to do is whatever you want your mean, you want to weight one side or the other.2304

Here is a skewed but they are skewed in different ways.2313

That is the end of central tendency.2326

Thanks for joining us on