For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

### 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
- Roadmap
- Central Tendency 1
- Central Tendency 2
- Summation Symbol
- Population vs. Sample
- Excel Examples
- Median vs. Mean
- Effect of Outliers
- Relationship Between Parameter and Statistic
- Type of Measurements
- Which Distributions to Use With
- Example 1: Mean
- Example 2: Using Summation Symbol
- Example 3: Average Calorie Count
- Example 4: Creating an Example Set

- 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

### General Statistics Online Course

### Transcription: Central Tendency: Mean, Median, Mode

*Hi and welcome to www.educator.com.*0000

* 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 formulas*0035

*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 values*0787

*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 www.facebook.com 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 www.google.com 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

*$40,000.*1604

*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 and*1997

*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 www.educator.com.*2328

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