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13 - ANOVA Basics - The Grand Mean — Transcript

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0:00hello welcome back to statistics we're
0:03talking about analysis of variance we're
0:05gonna talk about the basics here so this
0:08lesson and the next several lessons will
0:09be going through the components of the
0:11calculations one little piece at a time
0:13so the first piece is called the grand
0:15mean it's the easiest piece to
0:17understand is very very simple before we
0:20get to that let's look at this problem
0:22what we're going to do is end up solving
0:24this problem by hand I know in the
0:26previous section we talked about school
0:27districts and testing that was great to
0:29keep in your brain just because
0:31everybody has some experience with
0:32grades but this is a real problem so
0:34below are the ages that females get
0:37married in New York Texas and Oregon so
0:40you say we have New York here Texas and
0:42Oregon and we have some numbers written
0:45down here so these are the ages that
0:48girls get married so New York we have
0:50one person that got married at 18 years
0:52old one in 19 years old and so on so
0:55everybody you ask there's ten people
0:56that we sampled in New York 10 people we
0:59sampled in Texas and 10 people that we
1:00sampled in Oregon so this is another
1:03example the population that we're
1:05studying is the entire New York female
1:07population the age that they get married
1:09the population here is all the females
1:11in Texas the population here is all the
1:14females in Oregon but we can't sample
1:15everybody what we want to do because you
1:18can't do that because there's not enough
1:19money or time what we want to do is
1:22perform an analysis of variance test to
1:24see if the average age of marriage in
1:26these three states are equal so again
1:29analysis of variance three or more
1:31populations in this case three but we
1:33could have Louisiana and Kentucky and
1:35Florida if we wanted to we could
1:36continue doing testing and doing doing
1:38them all do them all at once but we
1:41cannot know the population average we
1:43can't sample all girls in Texas all
1:45females and figure that out it's just
1:47too costly so we sample 10 from each of
1:49these guys and we want to figure out
1:50from that information at a point one
1:53level of significance which means 90%
1:55level of confidence if this data
1:57indicates that the average age of
1:59marriage in these three states are the
2:01same or not now I have the numbers here
2:03because the equations I'm going to write
2:05down are going to have subscripts 1 2 3
2:07so when you see a number one it means
2:09New York number two mean
2:11Texas number 3 means Oregon you should
2:12label your data that way as well all
2:15right so first thing we need to do is
2:17write down the null and alternate
2:18hypothesis and I already told you that
2:20it's always the same for these problems
2:21but let's write it down anyway the null
2:23hypothesis is that the average age that
2:25females get married in New York C mu
2:28number one this is population 1 is equal
2:32to the average age of the females in
2:34population 2 which is equal to the
2:36average age in population 3 that's the
2:37null that's the currently accepted
2:39hypothesis that we think is true what
2:42would then be the alternate hypothesis
2:43well it just means that at least one of
2:48these means is different and I want to
2:57remind you we did talk about it in the
2:58last section I want to remind you though
3:00that even at the end of this test I'm
3:02really not gonna know which of these
3:04means is different if any of them are
3:06different I'm just gonna know that one
3:07of them is different you have to do
3:09different testing beyond the scope of
3:11ANOVA to figure out which one of them is
3:13different alright so the first thing
3:15we're going to talk about is the concept
3:16of the grand mean so let me write that
3:18down
3:18grand mean because here's the deal
3:25I've got sample data from population one
3:27sample data from population to sample
3:29data from population three so what I'm
3:30going to end up doing is I'm gonna
3:32average this data and I'm gonna get a
3:34number that's going to be the sample
3:36mean from population one then I'll
3:38average all of this and get a sample
3:39mean from population 2 then i'll sample
3:42average all of that and i'll get a
3:44sample mean from population 3 so i'll be
3:46comparing those three sample means
3:48together i'll be comparing them but how
3:50am I really gonna know if any of them
3:52are different I mean what were you doing
3:53in the previous section where we're
3:54looking at the test scores me I graphed
3:56it so you could visually see it but what
3:59were you really doing you were looking
4:00at those three scores and comparing them
4:02to each other but really what you were
4:03doing is comparing them to a common
4:05baseline the common baseline that you're
4:07going to end up using to compare each of
4:10these averages to is what we call the
4:12grand mean basically you have all of
4:15this data here so if you if you take the
4:18mean of all of each of these sample
4:21means let's say you get a sample mean
4:22from this data sample mean from this
4:23data sample
4:24from this day to see have a mean from
4:26New York mean from Texas and a mean from
4:27Oregon if you average all three of those
4:30means together you get what we call the
4:32grand mean so I'm gonna write it down it
4:35is the mean of the sample means right so
4:47I'm gonna say this a few different ways
4:48it's a really simple concept but the
4:50reason I'm saying it this way is because
4:51your book is going to describe to you
4:53it's gonna say the grand mean is the
4:55mean of the sample means and a lot of
4:57people look at that and say what does
4:57that mean well I'm telling you what it
4:59means you average up your data here and
5:01get an average we call it a sample mean
5:03average this together we get another
5:05sample mean average this together get
5:06another sample mean but we need a common
5:08baseline to compare everything to so we
5:11take all of these three means and
5:12average them together that's what we say
5:15it's the mean of the sample means now
5:16when you mathematically do that when you
5:18take this and average it with this an
5:20average it with this other sample mean
5:21what you're really doing really is just
5:24averaging all of the data together
5:26that's really what the grand mean is so
5:29some books make it really difficult to
5:30understand that concept but basically to
5:32calculate this grand mean all you do is
5:34you add all the numbers together you end
5:37up adding them all together and you
5:39divide by the total number of samples
5:41you have across everything that's the
5:43grand mean it's exactly the same answer
5:45as if you take the individual sample
5:47means that you calculate and average
5:48them together so either way you want to
5:50think about it you're going to get the
5:51same number right now I'm telling you
5:53this in words because now I'm gonna
5:55write the equation down and the
5:56equations scare a lot of people so just
5:58bear with me I'm writing them down so
6:00that you can learn how to read them so
6:01the grand mean is denoted like this X
6:06double bar now the single bar notice the
6:08sample means have a single bar right
6:11because when you think about it this guy
6:13is going to give you a sample mean X bar
6:161 this guy is going to give you a sample
6:19mean X bar from population 2 this is
6:21gonna give you x-bar population 3 single
6:24bar means sample means from each
6:26individual population grand mean double
6:29bar is the grand mean it's the grant in
6:31the mean of everything so let me write
6:33the equation down it looks ugly but it's
6:35not that hard so here you have a sick
6:37I'll explain all this in a second I is
6:39equal to 1 up to K open up a bracket and
6:44we have another Sigma here inside this
6:47bracket looks really complicated don't
6:48worry about it it's not complicated J is
6:50equal to 1 up to n sub I over X I comma
6:57J and I'll close this guy out and now
7:01you're taking all that stuff and you're
7:02dividing it by another Sigma if that was
7:05already enough in sub I ok the reason I
7:10took a minute to write down the grand
7:12mean and explain what it is in words is
7:14because if I give you this first most
7:16people just fall asleep and they say
7:18this is difficult I can't understand it
7:20I'm not a math person but it's really
7:21simple to understand I already told you
7:23the grand mean is just the average of
7:25all this data all of it right this
7:27equation is the way that you
7:29mathematically write that down so I'm
7:30gonna go through it because even though
7:32I could skip over it what's gonna happen
7:34is we're gonna come across more and more
7:35of these things more and more of these
7:37equations in the next sections you need
7:38to learn how to read these things
7:40because if you deny yourself that then
7:42you're just not going to understand a
7:44nova ever so just stick with me for a
7:45second alright first of all I need to
7:49explain a few things from algebra right
7:51so this Sigma here this Big E this means
7:54you're adding things up this one means
7:56you're adding things up and this one
7:58means you're adding things up now I have
7:59to write a few things down off to the
8:01side before you'll really understand
8:02what it's saying
8:03K this value of K ok is the number of
8:09populations in this case 3 right so
8:15there's 3 populations so when I actually
8:17calculate it it's gonna be I go goes
8:19from 1 up to 3 if I had 16 populations
8:22and I would be going from 1 up to 16 all
8:25right then you have this guy here and
8:28this guy in some I appears in two
8:30different places down here so I'll write
8:32this down n sub I is the sample size
8:39sample size
8:42of if-- population right so we have a
8:48population of New York we took ten
8:50samples the sample size here was ten
8:52Texas we took the same number ten sample
8:55so the sample size was tenth here we
8:57took ten samples the sample size was ten
8:59so I is just a letter that represents a
9:02number so the way you really what I'm
9:04really trying to get across to you is in
9:05sub one was ten because we took ten
9:08samples and sub two was ten because we
9:11took ten samples in sub 3 is 10 because
9:15we took ten samples but again when we do
9:17a Nova I'm doing it with the same number
9:19of samples now but you know I could take
9:21seven samples from New York eight
9:23samples from Texas ten samples from
9:25Oregon in sub 1 in sub-2 and sub-3 the
9:28sample sizes would just be different all
9:30right so now it's time to try to to
9:34figure out what this is really telling
9:36us okay we have an outer in an inner
9:39right so what we do is we work out to
9:41end first I goes from 1 up to K which is
9:453 3 populations so let's say I starts
9:48out being 1 so the value of I is 1 so
9:51that means we go in here and we put a 1
9:52everywhere where the eye is so forget
9:55about the bottom for now on the top what
9:57are you adding together right J which is
10:00another variable starts from 1 and it
10:03goes up to n sub I which means it's just
10:05sampling over the 10 samples from
10:07population 1 because remember I is 1 to
10:10start out with I starts out being 1 so
10:12we just let the second variable J go
10:15from 1 up to the total number of samples
10:17from population number 1 and this Sigma
10:21means I'm adding together what's inside
10:23here is the all the different values
10:25basically all you're doing is this is
10:27telling you that you're just going to
10:28add up all the values from population
10:30number one because and I didn't write
10:32this down yet I'll write it down right
10:35here X IJ is the jafe sample
10:42from the ithe population so as an
10:50example I'm just going to give you a
10:52couple of examples okay x11 from
10:57population one sample number one
10:59population one sample number one is just
11:0218 okay
11:04x2 one the first letter here is the
11:08population that you're on this is the
11:10sample from the second population sample
11:12number one is second population as here
11:15sample number one is also 18 in just one
11:19more example x3 one from the third
11:22population sample number one is 21 so
11:27now you have all the pieces in place
11:29basically what's going on here is you
11:31start out with the outer loop you set I
11:33equal to 1 and then you come in and put
11:35I equal to 1 here and I equal to 1 here
11:38so your sampler is you're summing you're
11:40adding up that's what the Sigma means
11:42you're adding up all of the values from
11:44population number one across J which is
11:47the samples from population number one
11:49so I won one I won - I won three I won
11:53four I won five I won six I won seven I
11:56won eight I won I won ten so you're
11:59adding all those together that's what
12:00the Sigma means right once you're done
12:02you've gone up from one up to the
12:04complete you know sample size that you
12:06had for that population then what
12:08happens you pop out and you increment I
12:11you go to the second population so now I
12:13is 2 so 2 is here so now you're adding
12:16up from here going J from 1 up to the
12:19sample size of population 2 means you're
12:21going to add up all of these guys X 2 1
12:24X 2 2 X 2 3 X - 4 X - five - six - seven
12:28- eight - nine - ten you're just adding
12:31up all those values because you're going
12:33from this value up to the sample size of
12:35population - and then you do the same
12:36thing when you increment I to population
12:393 I 3 1 I 3 2 I 3 3 3 4 3 5 3 6 3 7 3 8
12:443 9 3 10 so you see what you're doing
12:47this entire ugly mess up here just means
12:50that I've added up all of these plus all
12:52of these plus all of these which is what
12:55I told you the grand mean was going
12:56do is gonna add up everything but then
12:59you have to divide by something right
13:01notice what's on the bottom you don't
13:04have anything on the top in the bottom
13:05it means you just sum all of the total
13:08number the sample size of the eigth
13:10population so population number one has
13:13ten samples population two has ten
13:15samples population three has ten samples
13:17you're not summing up the values in
13:19there you're just summing up how many
13:20there are so 10 plus 10 plus 10 is 30 so
13:24you see what's going on here the top
13:25here is you're adding up all of these
13:28things then all of these things to that
13:30and then all of these things to that you
13:32add all of them up and you divide by the
13:33total number of samples that you have
13:35across all the populations that's what
13:37we call the grand mean I could have
13:38skipped that I know I spent some time on
13:40it but I wanted you to understand it
13:42because you're gonna see this kind of
13:43notation here that's why I say it's the
13:45Jade sample from the population so
13:48population 1 then increment 1 2 3 4 5 6
13:517 add all of those up go to the next
13:53population population to increment to 1
13:562 3 2 3 2 4 2 5 and so on and then
13:59population 3 add everything up and
14:01divide by the total number so now that
14:03you have all that down we're gonna
14:06actually do it
14:07let's go ahead and calculate that for
14:09this data that we have so I've stripped
14:11away the problem statement and I've left
14:12just the raw data here essentially all
14:16right another way to write that I know
14:18you're probably sick of math but the
14:20grand mean the way I wrote it before was
14:22very compact this is another way to
14:24write it that's exactly the same thing
14:25if I blow it out for this particular
14:27problem I'm going to sum J is equal to 1
14:30up to 10 X 1 J so across population 1 I
14:36just increment the elements there let me
14:38get it all down here and I'll explain J
14:40is equal to 1 up to the sample size of
14:43population 2 X 2 J and then I'm going to
14:46add to that J is equal to 1 up to 10 of
14:50X 3 J all that's happening on the top is
14:54this is adding up all the elements from
14:56this population this is adding up all
14:58the elements from this population this
15:00is adding up all the elements from this
15:01population because in each case I'm
15:03running up from 1 up to the sample size
15:053 1 3 2 3 3 3 4 so on - 1 - 2 - 3
15:09to four so on 1 1 1 2 1 3 1 4 so on
15:12adding them all up on the bottom you
15:15know we already said that it was the sum
15:17of the total number of samples that you
15:18have so really on the bottom it's gonna
15:20be 10 plus 10 plus 10 which of course
15:24you know is the 30 so you're basically
15:26summing up all the samples divided by
15:28the total number of samples that's what
15:30the grand mean is so to be absolutely
15:32clear the grand mean is drumroll please
15:3618 plus 19 plus 20 plus 21 plus 22 plus
15:4423 plus 18 plus 19 plus 20 plus 21 okay
15:52all of that is what this is doing is
15:55just adding up all of those elements but
15:56to that I have a second summation right
15:59after it which is 18 plus 19 whoops plus
16:0320 plus 16 plus 20 plus 21 plus 20 plus
16:1118 plus 19 plus 17 plus 13 okay and then
16:17I add to that from the last part 21 plus
16:2022 plus 17 plus 18 plus 22 plus 19 plus
16:2821 plus 20 plus 18 plus 23 so I add all
16:34that stuff up and I divide by 30 now I
16:39agree I wrote a lot of stuff out there I
16:41probably didn't have to write all that
16:42but there's a reason why I'm writing it
16:43all out out there so what you're gonna
16:45end up getting there I think I do have
16:48enough room on this page is X double bar
16:52is going to be 584 that's what you get
16:55on the top divided by 30 so let's go and
16:59round over here with a little bit more
17:00room X double bar the grand mean in this
17:02case is nineteen point four six six six
17:08six seven how many decimals you choose
17:11to carry is up to you I'm carrying a lot
17:13because I want to make sure that I don't
17:14have any rounding errors as I go through
17:16here so this is the grand mean I'll
17:18circle it in red and ultimately I could
17:21have probably skipped about 90% of this
17:23hole
17:23lesson because ultimately the grand mean
17:25is just simply the average of all of
17:27this data together I could have just
17:29told you to do that but I want you to
17:30understand where the equation comes from
17:32and how to read them because we're going
17:34to be doing other equations for other
17:35calculations and you need to kind of get
17:37used to seeing summation symbols but
17:39ultimately that's what this one was
17:40doing this guy 19 four point four six
17:43six six six six seven is basically a
17:46representative average of the average
17:48age of marriage of females across all
17:50three of these states so later on when
17:52we calculate the sample mean of each guy
17:54separately and we're gonna have to
17:56compare it to a baseline this is going
17:58to be the baseline we're going to
17:59compare them to to figure out if any of
18:01them are deviating or varying with
18:03respect to any of the other guys so
18:05we're going to be comparing them to this
18:07grand mean essentially it's what we're
18:08going to be doing so make sure you
18:11understand this we're going to be
18:12calculating different parts of this
18:14problem as we go through it we're going
18:15to be taking getting these sample means
18:18comparing them to the grand mean that we
18:20just calculated and they're performing
18:21the rest of the of the ANOVA testing
18:24this was actually the easiest part to
18:26understand there are some other parts
18:27they're a little difficult to understand
18:29so I'm going to try to break it out
18:30slowly for you make sure you understand
18:31it I wrote all those numbers down just
18:33so there's no confusion about how I'm
18:35calculating it by hand then when you
18:37start using Excel you'll know exactly
18:39what it's doing there won't be no
18:40question marks for you so make sure you
18:42understand this follow me on to the next
18:43lesson and we'll take apart a nova to
18:45the next calculation so the next part of
18:47the puzzle time make sure you understand
18:49how to do it

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