YT YouTube2Text

YouTube2Text → Transcripts → Analysis of Variance (ANOVA) Overview in Statistics - Learn ANOVA & How it Works

Analysis of Variance (ANOVA) Overview in Statistics - Learn ANOVA & How it Works — Transcript

by Math and Science · 4,518 words · 680 segments · language en · Watch on YouTube

Open in transcriber Summarize with AI Download SRT/VTT

Full transcript

0:01Hello, welcome back to Mastering
0:02Statistics volume seven. What we're
0:04going to do now is start talking about
0:06the concept of ANOVA. Uh this is a topic
0:10that is is in the back of your
0:11statistics book. It's not difficult to
0:13understand, but there's a lot of
0:15components to it. So, I want to give you
0:17a game plan here in the first lesson and
0:18give you an overview of what this really
0:20is.
0:21So, I'm going to map it out for you.
0:22Basically, in this lesson, I'm going to
0:23give you an overview. I'm going to
0:24describe what ANOVA is in the big
0:27picture, right? And then over the next
0:29probably three or four lessons, we're
0:31going to calculate an ANOVA analysis by
0:34hand. It's it's not difficult, it's just
0:36tedious. So, I'm going to show you how
0:38to do it so that you can really
0:39understand all of the formulas that go
0:41into it. Because ultimately, you're
0:43probably going to use a computer program
0:45to do most of the problems for in real
0:47life like Excel. So, Microsoft Excel.
0:49So, what I'm going to do is teach you
0:50the equations and formulas by hand, at
0:52least for the one one long problem,
0:55so that you'll understand what the
0:56computer is doing.
0:58And then, you know, most books just tell
0:59you to use the computer. So, I could
1:01dump you in and just teach you Excel
1:02right now, press a button and get the
1:03answer, but you won't know what's
1:05happening or why it's doing what it's
1:06doing. So, let's just jump into it now.
1:08You need to understand the concept of
1:10what's happening first. The big picture
1:13is that analysis of variance, even
1:15though it's got the word variance in
1:17here, and we'll talk about why it
1:18doesn't in a second, basically, we're
1:20going to be comparing population means,
1:22three or more population means. So, I
1:24have a bunch of things I need to write
1:25down. I promise, if you just kind of
1:28stick with me, even though you have to
1:30invest a little time up front, you will
1:32come out with a really great
1:33understanding about what this thing is
1:35actually about. So, we need to compare,
1:39and you'll see what I mean by compare in
1:41just a second, three or more
1:44uh population means. So, I'll call this
1:46population means.
1:49Now, I need you to think back to
1:50hypothesis testing in in many, many
1:52lessons ago. Recently, we've been doing
1:53hypothesis testing with variances, and
1:56you'll see how that ties in. But, even
1:57before that, we did population means
2:00hypothesis testing with population
2:02means, but we're always comparing two
2:03means, right? Now, we're doing three or
2:06more means, and we're all going to do it
2:08with one test. So, in the past, we might
2:10have done
2:12see if population A or population B,
2:14which mean is larger. We write the null
2:16hypothesis, the alternate hypothesis,
2:18and we have the data, or maybe see if
2:20they're equal or not equal. Now, we're
2:21going to be doing a test a very specific
2:24kind of test with three or more
2:25population means. So, let me just write
2:28down an example of a null and alternate
2:29hypothesis. Basically, the null
2:31hypothesis for these things are always
2:33going to look the same.
2:34Population mean number one is equal to
2:37population mean number two is equal to
2:40dot dot dot because I'm not sure how
2:42many populations you're going to have in
2:44your actual problem. It's going to be
2:45three or more. Um but anyway, they're
2:47it's either going to be equal to
2:50uh the Kth population mean. So, there's
2:51K populations here.
2:56Right? And don't worry, I'm going to I'm
2:58going to nail this down with this very
3:00specific example so that you understand
3:01what I'm talking about. But, basically,
3:03the null hypothesis is that all three,
3:05in case of three or more, however many
3:07you have, they're all going to be equal
3:08to each other. When you see notation
3:10like this, this is equal to this is
3:12equal to dot dot dot equal to the Kth
3:14population, that means if I have three
3:16population means, they're all equal. If
3:18I have 16 population means, they're all
3:20equal. If I have seven population means,
3:23they're all equal. That's the null
3:24hypothesis that all of the means are
3:26equal. So, if that's the null
3:28hypothesis, then the alternate or the
3:30test hypothesis basically means, the way
3:33that you write this down is usually in
3:34words, that at least
3:38and I'm going to write write this down
3:39very specifically, at least one
3:42mean
3:43one of these means that we're studying
3:45differs from the others.
3:49Differs from
3:51the
3:52others.
3:55Okay, so basically every single every
3:58single ANOVA, at least of of the type
4:00we're doing here in this class, that
4:02you're going to do is going to have the
4:04same null hypothesis and the same
4:06alternate hypothesis. So, you don't
4:07really have to do that much thinking on
4:09terms of how to set up the null and
4:10alternate. The null hypothesis is always
4:13the same thing. It means all of these
4:14means are the same.
4:16Okay, the alternate hypothesis is the is
4:18the the test or the claim that the
4:20researcher thinks might be true is that
4:22at least one of these means differs from
4:25the others. So, what we're going to end
4:28up doing as you might be thinking, well,
4:29this thing is called analysis of
4:31variance. Why are we comparing
4:33population means, right? That was my
4:35first question. The bottom line is when
4:37we get into the math, you're going to
4:38find out that we're going to use the
4:39concept of variance in order to study
4:42the means. So, even though we're
4:44actually studying means, the technique
4:46is called analysis of variance because
4:48we're going to be testing how these
4:50means vary with regard to the other
4:53means. So, how does mean number one vary
4:55or differ from all the other means? How
4:57does mean number two vary or differ from
5:00all of the other means? How does mean
5:01number seven, let's say you had 10 10
5:03populations,
5:05population number seven, how does that
5:06differ, right? The populations can be
5:08anything. I could be studying human
5:10males on seven, you know, different
5:13planets if we have a big solar system or
5:15maybe seven different states. Those are
5:17the different populations. Maybe I'm
5:19averaging their IQ or something like
5:21that. And I'm studying and each
5:22different location is a different
5:23population. So, I'll have a mean coming
5:25from one area, mean coming from another
5:27area, mean coming from another area. And
5:30my null hypothesis is that all of these
5:32means are the same. They all have the
5:33same average IQ in the different
5:34populations and so on.
5:36And the alternate is that at least one
5:38mean differs from the others.
5:40So, we're going to be using the concept
5:42of variance to see how each of these
5:44means varies or is different from the
5:47others. Now, as we go through these
5:50calculations here that we're going to
5:51do, I want you to keep one example in
5:53your mind. I know I just mentioned
5:54studying people's IQs, but I have a I
5:57think a better example that you can wrap
5:58your brain around. Let's say that we
6:00have Let's say that we're a school
6:02administrator
6:04and we have, you know, lots of different
6:05schools and we have three schools in my
6:08city and I want to make sure all the
6:10kids are learning the same stuff. So,
6:12I'm going to be testing them, basically.
6:15And so, I have three different schools,
6:16school number one, school number two,
6:17and school number three and I want to
6:19basically see if I want to make sure
6:21that those test scores are coming out of
6:22that school that the kids are learning
6:24the same things in math class, let's
6:25say.
6:26So, as we go through a lot of these
6:28calculations, I want you to keep that in
6:30your mind. So, I'm going to draw a
6:31couple of pictures here to hopefully
6:32make it a little bit easier. So, the
6:34little cloud that I'm drawing here is
6:36the population. This is all the kids in
6:38school number one. So, I'm going to call
6:40it school
6:42number one. See how clever I am? School
6:44number one.
6:45So, this is everybody in school number
6:47one. So, school number one might have
6:482,000 kids in it, right? That's a
6:51population. The cloud is the population,
6:53right? Now, also, I have school number
6:56two cuz I have lots of different
6:57schools. So, I'll call this school
7:00number two.
7:01And then I have school number three over
7:03here.
7:04But, keep in mind
7:06that analysis of variance is used when
7:08you're studying three or more
7:09populations. So, this is a population,
7:11this is a population of different
7:12children, this is a population of
7:14totally different children. But, I could
7:15be studying all the schools in the
7:17country and have a, you know, a bunch of
7:18populations, 10 or 20 or 50 or 1,000
7:21populations, whatever. But, now I'm just
7:22going to keep it simple and we're going
7:24to do an example with three populations
7:26to to make sure you understand the
7:27concepts. Now, what we want to do is we
7:29want to figure out if these kids are
7:31learning the same thing in math class.
7:33So, if we could test every kid in every
7:36one of these schools, if we could test
7:39every kid in school number one, for
7:41instance.
7:43Then we would get an average of their
7:45math test score from school number one.
7:47Now, this is a population. This is maybe
7:49two or three thousand kids, right? So,
7:52whenever I come over here and say from
7:54this population, from everybody here,
7:57what do we get?
7:59If we were to If we could somehow know
8:01everyone's test score, right? Then we
8:04would calculate a population mean.
8:06So, the the the symbol mu is the is the
8:10symbol for the population. But, we don't
8:13have money to test all 2,000 kids in
8:15school one and all 2,000 kids in school
8:17two and all 2,000 kids kids in school
8:19number three. I forgot to put the number
8:21three here. Right? So, we we really
8:23never really know what the population
8:25mean is because even if you say, "Well,
8:27you should test all the kids." I mean, I
8:29can choose a problem where it's really
8:30hard to test everybody. Maybe I'm
8:31studying everybody in the country. Maybe
8:33it's impractical to really give a test
8:34to everybody.
8:36Right? But anyway, if I somehow knew
8:37everyone's score, I would get the
8:39population mean from school number one,
8:41their math score, right? School number
8:43two, I would get a similar similar
8:45population mean. That's their average of
8:47their math exams. And then I would get a
8:49population mean of
8:52of um school number three's test scores
8:54for math class. Now, I'm a school
8:56administrator, so I want to make sure
8:57that everybody's learning the same
8:58stuff. So, my null hypothesis, my my
9:01kind of like my accepted hypothesis, is
9:04that mean number one is equal to mean
9:06number two is equal to mean number
9:08number three. Again, if I had more
9:09schools, it would be more populations
9:11and I would say that they were all equal
9:12cuz that's what I really want. Okay? So,
9:14I want to say that my null hypothesis is
9:17saying that these population means are
9:19all the same. That's what the null
9:20hypothesis is.
9:22And the alternate the test hypothesis or
9:25the or the you know, the the alternate
9:27that you're testing is that at least one
9:29of these guys is different from the
9:30others and that's really important to me
9:31because if school number two has poor
9:34test scores, I want to know about it,
9:36right? Or if any of these schools have
9:37have different test scores.
9:39But anyway, as we said, we cannot test
9:41everybody. So, because this could be
9:4210,000 kids. So, what I really do is I
9:44go inside of school one. So, I'm drawing
9:47a little box inside here. What do I do?
9:48What do you think we do? We've been
9:49doing this stuff a lot. I sample
9:5410 kids.
9:57So, let's say I'm I don't have a lot of
9:59money. So, I only sample 10 kids. I
10:01would like to know the actual population
10:04mean of the math test scores, but I
10:05don't have enough money or enough time
10:07to do that. So, really I just take 10 of
10:09the kids, representative sample from
10:11that population. I put them in a room
10:13and I give them a test. And I take all
10:1410 scores
10:16and I average those 10 scores. What do I
10:18get?
10:19From this calculation
10:21is something called the sample mean
10:24of population one. Right? That's the
10:26symbol here. The When you see X bar,
10:29that means a sample. It means I've taken
10:31a fixed number of people from the
10:32population. I've
10:33measured something, whether it's IQ,
10:35test scores, height, weight, weight,
10:38whatever. Anyway, that's the sample
10:40mean, which is not truth compared to the
10:43population means really what I want to
10:44know, but I can make some inferences
10:46with a sample size of sufficient
10:49sufficiently large. 10 kids is probably
10:51not big enough, but anyway.
10:53That's what I do. And then I take that
10:55and I calculate a sample mean X bar
10:57number one. Then I go over here and I
10:59give the test again.
11:01I sample.
11:05I'll put 10 kids here.
11:07But you'll see that when we do the
11:08ANOVA, it doesn't have to be the same
11:10number of kids. I could only test maybe
11:1220 kids here or seven kids here or
11:13whatever. It all takes everything into
11:15account. And then from that, I get X
11:18bar number two. That's the sample mean
11:20from school number two. And then I do
11:22the same thing over here with school
11:23number three.
11:24Sample.
11:26Again, I'll put 10 kids, but
11:29you can it can be a a number. And then
11:31from that, I get X bar number three.
11:33That's the sample mean. So the the idea
11:35here is
11:36analysis of variance is going to be a
11:38test that I'm going to be basically
11:41applying to the sample data that I got.
11:44The sample, that means the the average
11:46which is from a subset of the
11:47population, not everybody, a subset. And
11:50the analysis of variance test is going
11:52to take into account what these values
11:54are, Xbar one, Xbar two, Xbar three, the
11:56average sample means. It's also going to
11:59take into account how many samples I did
12:01in each population because that's going
12:02to affect things. And then you'll see
12:04exactly what we do in a little bit. And
12:06then it's going to spit out an answer
12:08with a certain level of significance to
12:10tell you if this data that we used
12:13if you can infer from it that the
12:15population means, which are different,
12:17the population means are everybody, are
12:19the same or if they're different, which
12:21is what the null null hype or if one of
12:22them is different, okay?
12:24Now I want to point out to you, the
12:26first thing you might be thinking is
12:27well why don't we just do regular old
12:29hypothesis testing from a long time ago?
12:31Like forget about school number three.
12:32Let's say school number three isn't
12:34here. We can have school number one,
12:35school number two. We could do a
12:37hypothesis test to see if these means,
12:40Xbar one and Xbar two, uh or if these
12:43population means are equal or not based
12:45on the sample data. We've done
12:46hypothesis testing like that.
12:48So we could do that, but then you have
12:50the third school. So then if you wanted
12:51to fold in the information from the
12:53third school, then you'd have to compare
12:54these two means and then you'd have to
12:55compare school number one and school
12:57number three means. And then you'd have
12:59to compare school number two and school
13:01number three means. And then you'd have
13:03a bunch of different combinations cuz
13:04you're doing when you do the regular
13:06hypothesis testing, you're only doing
13:07two populations at a time, comparing
13:09them. So with even with only three of
13:11these guys on the table here, it's a
13:13bunch of different combinations and you
13:14also introduce a lot of errors when you
13:16start doing a bunch of different testing
13:18you know, sequentially.
13:20Then what happens if you have 10
13:21populations? You got a ton of
13:22combinations to do and it gets very
13:24cumbersome really fast. So ANOVA lets
13:27you do three or more population
13:29comparisons at once.
13:30And you can you can draw some
13:32conclusions from that.
13:35Now, I have a few notes on my paper. I'm
13:37not going to write them all down, but I
13:38want to make sure you understand. I'm
13:39going to say them all to make sure that
13:40you that you that I've said them at
13:42least once. First thing is we do not
13:44know what the population means are. If
13:46we knew what the population means are,
13:48which which in this case is that the
13:49math scores from each of these schools,
13:51then we wouldn't have to do any testing.
13:52We would know if they were equal or not
13:54cuz we would have all the information.
13:55We don't know that because there's too
13:57many kids to test in a reasonable amount
13:59of time. Or if you're studying entire
14:00continents or something, it's just
14:02impractical to do it. So, what we do is
14:04we sample a subset of the kids, we
14:06calculate a sample mean, and we study
14:08these guys, and with a level of
14:09significance we draw an inference. So,
14:11what I want to do is give you a couple
14:15of cases of what might come out of such
14:17a test. This is kind of like back of the
14:19envelope just to kind of get you to
14:20understand it. We're not doing the
14:22actual ANOVA testing. But basically,
14:24there's a couple of cases that you can
14:25consider that are really instructive.
14:27So, let's look at case one. And they're
14:28kind of common sense, too, honestly.
14:30What if you look at case one and say
14:32what what could possibly happen if the
14:35following happens?
14:37Okay? So, over here we do the testing.
14:40Here's the score of 100. Here's a score
14:42of 80, let's say, right there. And then
14:44we get the sample means of each one of
14:47those three schools where we sampled and
14:49tested 10 kids in each one of those
14:50schools. So, let's say the first school
14:54comes up with
14:55uh a score of X bar number one 81.
14:59That's the score that they got from
15:01school number two. And then let's say
15:03I'm sorry school number one. Let's say
15:05school number two is really close to it.
15:08X bar number two
15:1179.
15:13And then let's say this one's a little
15:14bit higher. I know I'm not drawing these
15:15exactly right. But you get the idea. X
15:17bar number three
15:2080.5.
15:22So, you can see that this guy, we'll
15:24call this school number one, this is
15:26school number two, this is school number
15:27three. Now, again, this is not the
15:28population mean. This is just from the
15:30the 10 kids in each school that we gave
15:33the test to.
15:34So, basically
15:36the average of each of these test scores
15:38are pretty close. 81, 79, 80.5. To me,
15:41they look pretty close. So, if you
15:43remember back the null hypothesis
15:46is basically that the mean from school
15:48number one of all students, I should put
15:50a colon here, is equal to the mean of
15:53school number two of all students, is
15:55equal to the mean of school number three
15:56of all students. That's the null
15:58hypothesis. The alternate hypothesis is
16:01at least
16:04one mean
16:07different.
16:09So, basically that's my test. Now, of
16:11course, it all depends on the level of
16:12significance, it depends on a lot of
16:14different things, but to me, since this
16:16number is pretty close to this number,
16:17is pretty close to this number, this is
16:19likely.
16:23So, in this case, we would fail
16:26to reject
16:29the null hypothesis. And that's all
16:30you're going to end up doing. You're
16:32going to circle that in your paper one
16:33way or another, just like any other
16:34hypothesis test. Either you have a null
16:36hypothesis and you reject it, or you
16:39have that null hypothesis and you fail
16:41to reject it. In this case, we fail to
16:43reject it because our calculations show,
16:45now, we haven't done the actual test,
16:47but they look pretty darn close.
16:49Now, let's just take another example and
16:51say, well, what would happen in the
16:52following case?
16:54Okay, so we'll look at case
16:56number two. And again, I'm I'm just
16:58doing this back of the envelope. So,
16:59this is 100, this is 80. All right, so
17:01let's say what are some differences. Let
17:03me go flip the page here.
17:06Let's say school number one again comes
17:08in strong here
17:10with an X bar number one of 81.
17:13Let's say school number two is weak,
17:16comes in at X bar number two
17:19of
17:2029.5.
17:22A really low average test score. School
17:24number three
17:26is up here at 80.5.
17:30X bar number three
17:3280
17:33.5.
17:36All right. And again, this is school
17:37one, school two, school three. Now, what
17:40do you think? Just
17:41basically just back of the envelope.
17:42Well, you can say, "Well, school number
17:44one and school number three look to like
17:46they're in line. That's reasonable.
17:47School number two bombed it, right?" So,
17:50this is a drastic example. Anybody can
17:52see this. And of course, I didn't even
17:53draw the bar graph correct. It should be
17:55It should be down here somewhere. But, I
17:56could change this number to, you know,
17:5865 if I wanted to. Whatever. It's
18:00different from the other ones. So, the
18:02null hypothesis in this case, same null
18:04hypothesis as before.
18:06Mean number one, mean number two
18:09mean number three.
18:11And the alternate
18:13is at least
18:18uh one mean
18:21different.
18:23All right. Obviously, to me it looks
18:25like
18:26one of these means, in this case school
18:27number two, is different. So, in this
18:29particular case, this is unlikely.
18:34And so, at the proper level of
18:36significance for your problem, you could
18:38reject
18:40the null hypothesis. I'm just doing this
18:42for you to show you that the ANOVA
18:43testing is going to give you the same
18:45answer as any hypothesis test. Either
18:47you're going to reject the null
18:48hypothesis or you're going to fail to
18:49reject the null hypothesis. Now, there's
18:51a couple of notes that I need to make
18:52sure you understand before I close this
18:54section out.
18:55First thing is we study the
18:57uh the population means of the students
19:00in the school by by sampling, by taking
19:03a subset of them
19:05and and and calculating things and
19:07drawing conclusions based on that,
19:09right?
19:10Um we're going to end up using variance,
19:12the concept of variance, to see how
19:14these means differ from one another. And
19:16it's impossible to explain that to you
19:18without showing you some math, so I have
19:19to do that in the next couple of
19:20lessons, but we're going to use
19:21variance. That's why it's called
19:22analysis of variance, but don't forget
19:24we're studying the means, how the means
19:25are different, okay?
19:27The test here, the ANOVA test, tells us
19:30if one or more of these means are
19:31different, but it does not tell us which
19:34one is different. Now, in this case,
19:36it's obvious, this one's the different
19:38one, right? But it's not going to be so
19:41obvious if you have 10 populations and
19:43one of them differs by just a little bit
19:44statistically.
19:46You're not going to be able to eyeball
19:47it. I've drastically you know, made this
19:49one different so that you could easily
19:50see what I'm talking about, but the test
19:52either tells you if they're all the same
19:54or if one of them's different. It
19:55doesn't tell you mean number one is
19:57different or mean number two is
19:58different. That's beyond the scope of
19:59ANOVA, you'll have to do additional
20:01testing and study to figure that out.
20:03So, that's an important thing. The other
20:05thing I want to allude to, I'm not going
20:06to get into it too much here, but um,
20:09basically the the validity of the test
20:12kind of depends on the data that you
20:13sample to begin with. I mean, we're
20:14doing all of this calculations based on
20:16the 10 kids that we looked at in school
20:18one, the 10 kids that we looked at in
20:20school two, and the 10 kids that we
20:22looked at in school three. So, as we do
20:24the ANOVA test, the number of kids,
20:26number of samples, is going to influence
20:28the outcome,
20:30right? Um, also, we'll find out later
20:32that the quality of the data affects the
20:34outcome. So, if for instance, in school
20:36two clearly was having a bad day, but is
20:39this average of 29 because two of the
20:42kids got a zero?
20:43And maybe maybe eight of the other kids
20:45did great, but two of the kids got a
20:47zero. Maybe two of the kids just ripped
20:48the paper up and threw it in the trash
20:50can. So, you have if you have a bunch of
20:51outlier data, garbage data, out of the
20:54out of the samples from school two, it
20:56can cause the average from the sample to
20:58look weird and different, but it may it
21:01may not be indicative of a really bad
21:03population. What if they had a fire
21:04drill that day? Or what if the teacher
21:06giving the test was had a cold or
21:08something or anything could happen. What
21:09if a meteorite came in through the roof
21:11and disrupted the classroom and
21:12everybody did bad. So, I guess what I'm
21:14trying to say is, yes, you can draw
21:16conclusions, but don't just turn your
21:18brain off. You have to look at the data
21:19and the ANOVA test test does look at the
21:21quality of the data. It does look for
21:23outliers to see if any of that's going
21:26on and it will automatically adjust the
21:28conclusions accordingly. We'll get there
21:30when we get there. I just wanted to
21:31point that out to you. That's the
21:32concept of analysis of variance.
21:35Basically, all we're going to be doing
21:36is studying means, different population
21:38means, and we'll do that by methods of
21:39sampling. We'll take a sample from each
21:41of the populations, calculate a sample
21:43mean, run it through some calculations,
21:45and go from there. So, I'm going to do
21:46those calculations in the next several
21:48lessons and then I'm going to show you
21:49how to use a computer to do it and
21:50you'll see why computers are used
21:52because there's just a lot of numbers to
21:54crunch. So, make sure you understand
21:55this, follow me on to the next lesson,
21:57and we'll get started.

About this transcript

This page contains the full transcript of Analysis of Variance (ANOVA) Overview in Statistics - Learn ANOVA & How it Works by Math and Science, generated from the public captions YouTube serves with the video. The transcript has 4,518 words across 680 segments, with the original timestamps preserved so you can click any line to jump to that moment in the embedded player.

What you can do with it

Use the transcript to take notes, quote the speaker, build a study guide, generate a summary with ChatGPT or Claude via the YouTube Summary tool, or export it as a timed subtitle file with YouTube to SRT. You can also re-open it in the transcriber to translate the transcript into 100+ languages.

Free YouTube transcript tool

YouTube2Text is a free YouTube transcript generator — no signup, no daily limit. Paste any YouTube link and get the full transcript instantly, with timestamps, click-to-jump, translation to 100+ languages, AI prompts for ChatGPT, Claude, and Gemini, and exports to TXT, SRT, VTT, or Markdown.