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0:02[Music]
0:17hello thanks for watching and welcome to
0:20the next video in my series on basic
0:22statistics now as usual a few things
0:25before we get started number one if
0:27you're watching this video because you
0:28are struggling in a class right now I
0:31want you to stay positive and keep your
0:33head up if you're watching this it means
0:35you've accomplished quite a bit already
0:37you're very smart and talented but you
0:39may have just hit a temporary rough
0:41patch now I know with the right amount
0:43of hard work practice and patience you
0:46can work through it I have faith in you
0:49many other people around you have faith
0:51in you so so should you number two
0:55please feel free to follow me here on
0:57YouTube on Twitter on Google+ or on
1:01LinkedIn that way when I upload a new
1:03video you know about it and it's always
1:06nice to connect with my viewers online I
1:08feel that life is much too short and the
1:10world is much too large for us to miss
1:12the chance to connect when we can number
1:16three if you like the video please give
1:18it a thumbs up share it with classmates
1:21or colleagues or put it on a playlist
1:23that does encourage me to keep making
1:25them for you on the flip side if you
1:27think there's something I can do better
1:29please leave a instructive comment below
1:30the video and I will take those ideas
1:33into account when I make new ones and
1:36finally just keep in mind that these
1:37videos are meant for individuals who are
1:40relatively new to Stats so I'm just
1:42going over basic concepts and I will be
1:45doing so in a slow deliberate manner not
1:49only do I want you to know what is going
1:51on but also why and how to apply it so
1:56all that being said let's go ahead and
1:58get started
2:02so this is the next video in our series
2:04about the analysis of variant or an NOA
2:08to make the topic more manageable I've
2:10divided this video into two parts in
2:13part one we will discuss the conceptual
2:15background using graphics and charts and
2:18an example problem and then in part two
2:21we will actually go into Microsoft Excel
2:24to conduct a hand calculation of an Nova
2:28and also solve our examp example using
2:31excel's builtin data analysis tools if
2:34you're watching this video on YouTube
2:37you can download the Excel file using
2:39the link in the description so you can
2:42follow along when we get to part two now
2:46in our last video we took an in-depth
2:48look at the conceptual foundations of
2:51Anova and what it offers us prior to
2:54Anova we were limited to conducting
2:57hypothesis tests about a maximum of two
3:00populations an NOA frees us of that
3:03limitation permitting comparisons of
3:06multiple populations and even subgroups
3:09or what are called blocks of those
3:12populations now there are several types
3:14of Anova in this video we will be
3:17focusing on what is called the oneway
3:19Anova and in a later video we will
3:22discuss the two-way Anova it is
3:25important to note that one way anovas go
3:28by other names such as as the single
3:31Factor Anova and in experimental
3:34contexts the completely randomized
3:36design these are all basically the same
3:39thing now this video is very
3:42comprehensive in part one I will use
3:44several illustrations to make concrete
3:47the abstract Concepts underlying an NOA
3:51in part two we will go into Microsoft
3:54Excel to hand calculate the Innova which
3:57admittedly you may never actually have
3:59to do do so this video offers a solid
4:03comprehensive conceptual Foundation of
4:06oneway in Nova using an example problem
4:09with illustrations and Graphics so if
4:12you are new to an NOA or are still
4:14trying to figure out exactly what it is
4:17this video is for you so sit back relax
4:21and let's go ahead and get to
4:25work so the most obvious question is why
4:27do we have an NOA in the first place so
4:30like I said before to this point we have
4:32been comparing two populations only so
4:35we use the independent samples T Test or
4:37the Matched sample or the paired T Test
4:41of course limiting ourselves to the
4:42comparison of two populations as well
4:46limiting what if we wish to compare the
4:48means of more than two populations what
4:51if we wish to compare populations each
4:53containing several levels or subgroups
4:57well that's why we have a NOA
5:00it allows us to do those things and
5:02remember a Nova is an acronym that comes
5:05from the phrase analysis of
5:11variance so suppose we want to compare
5:13three population means like we will in
5:15this example problem to see if a
5:17difference exists somewhere among them
5:21so we have population one here in the
5:23blue population two here in the pink and
5:27population three here in the green and
5:29each population has its own population
5:32mean so mu1 mu2 and
5:37mu3 so what we're asking is do all three
5:40of these means come from a larger common
5:44population sort of in the
5:47background or is one of the means so far
5:50away from the other two that it is
5:53likely not from the same
5:56population or are all three so far apart
6:01that they all likely come from unique
6:04populations so really it's about
6:07distinction are all three of these
6:09populations common to a larger
6:11population or is one or all three of
6:14them different from one
6:17another now I mentioned before there are
6:20several types of innovas so if you go
6:23into the data analysis section in Excel
6:26you will see all those so you will see
6:28Anova single Factor
6:30a NOA two factor with replication and a
6:33Nova two Factor without replication now
6:37we will get to all those eventually but
6:40for now we're going to be dealing with
6:41the first type so in this video we will
6:44be learning about single factor or
6:47oneway anovas they are the same thing so
6:51if you go into Excel and see this menu
6:54in the data analysis tool I want you to
6:56know exactly which one we're doing we
6:58are doing the Inova
7:00single factor or the one-way
7:04Anova so here is our fictitious problem
7:08so 21 students at the autonomous
7:11University of Madrid the auum in Spain
7:14were selected for an informal study
7:17about student study skills so s first
7:21year 7 second year and S thirdy year
7:24undergraduate students were randomly
7:27selected the students were given a a
7:29study skills assessment having a maximum
7:32score of 100 now as researchers we are
7:35interested in whether or not a
7:37difference exists somewhere between the
7:41three different year levels so we will
7:44conduct this analysis using a one-way
7:47Anova
7:51technique so here is our basic chart so
7:54you can see that we have a column for
7:56each year of students so these are our
7:59columns s or sometimes they're called
8:01groups and I mentioned in the previous
8:03video one of the challenges of teaching
8:06and learning and novas is that different
8:09professors and different textbooks will
8:11call the same thing different things so
8:15sometimes these are called columns
8:17sometimes these are called groups now
8:19why is this called a single Factor Anova
8:22well if you look at this chart this year
8:25of student is a single factor with sort
8:28of several levels to it so that that's
8:31why it's called a single
8:34Factor now within each year of student
8:37we're going to select a random sample in
8:40this case seven students so if you get
8:44the geometry of the chart you can kind
8:47of understand how it's all put together
8:49so we have three columns or groups the
8:52single Factor we're looking at is the
8:54year of the student and within each
8:56group we're going to be taking a random
8:59sample
9:01so here is our actual data so remember
9:04these scores are out of 100 so you can
9:06see the seven year one student scores in
9:08the left the 7e 2 student scores in the
9:12middle and the year three student scores
9:15over here on the right now remember that
9:18these are random samples within each
9:21year or each column each group that's
9:24why in an experimental context this is
9:26called the completely randomized design
9:29so you may see that in your class or in
9:31your textbook or of course
9:35both so I want ahead and color coded
9:38each year so we can keep them distinct
9:41now if you notice at the bottom here I
9:43have xar sub one xar sub 2 and xar sub3
9:48well of course that's going to be the
9:49mean for each column or each group so
9:54each year will have its own
9:57characteristics it will have its own
9:58mean
10:00and its own distribution or its own
10:02variance so I went ahead and colorcoded
10:05those to
10:06match now there will also be an overall
10:09mean which is the mean of all 21 scores
10:14taken together sometimes this is called
10:16the Grand mean I call it the overall
10:19mean same thing so there are several
10:22means under consideration here we have
10:25the mean for each column or each
10:27factored level it's the same thing cuz
10:30our single factor is the year of the
10:32student so we have xar sub 1 xar sub 2
10:35and xar sub3 then of course we have the
10:39overall mean or sometimes it's called
10:41the Grand mean which is the mean of all
10:4421 scores taken
10:48together so the first thing we want to
10:50do is find the mean for each column and
10:53the overall mean over here on the right
10:57so the mean of the Year One scores was
11:0171.7 that's xar sub 1 the mean of the
11:05year 2 scores was
11:087.29 that's xar sub 2 then xar sub3
11:13which is the mean of the year three
11:14scores was
11:1676.5 s now the overall mean or the grand
11:21mean of all 21 of those scores taken
11:24together is
11:2774.5 2 so that's the first thing we want
11:30to do when doing our hand calculation
11:33for the oneway
11:34Anova so you can see our column means
11:37here on the bottom and our overall mean
11:40here on the right now if we take a quick
11:43look at the column means what do we see
11:47well we can see that the 71.7 one over
11:51here for the year one students seems a
11:55bit odd it seems a bit different than
11:57the other two now we don't know if
12:00that's going to be just due to Natural
12:01variation or there's actually something
12:03there that's why we're doing the actual
12:08Anova so let's briefly go back and visit
12:11the idea of variance and its related
12:13concept the sum of squares so since
12:15Anova is by definition the analysis of
12:18variance we should briefly review this
12:20as a concept remember that variance is
12:23the average squared deviation or the
12:27average squared difference same thing of
12:30a data point from the distribution mean
12:34so we take the distance of each data
12:36point from the mean square that distance
12:40add those together and then find the
12:44average that is variance but if we take
12:48out that last step if we take out the
12:50find the averages part we are left with
12:53just the sum of the
12:56squares so we would take the distance of
12:58each data point from the mean Square
13:01each distance and then add them together
13:04if we stop there that is the sum of
13:08squares so the sum of squares is
13:11variance without finding the average of
13:14the sum of the square deviations so it's
13:17the variance without that last step and
13:19that's because sum of squares is a
13:22foundational component of an
13:26NOA so remember the formula for the same
13:29sample variance that's what we just
13:30talked about it is the sum of the squar
13:33deviations divid the sample size minus
13:36one in the sample case so on the top we
13:39have the squared differences so remember
13:42X is a data point minus mu which is the
13:44mean we take that difference and we
13:46Square it we sum all those up and then
13:50in this case we divide by n minus
13:53one so on the bottom we're doing the
13:56averaging of the squared differences
14:00what if we take away the N minus one
14:02part well we're just left with the sum
14:05of the squares so each data point minus
14:08the mean Square it do that for all the
14:11data points add them up and that is the
14:14sum of squares so with the sum of
14:17squares of the difference between the
14:19dependent variable and its mean that's
14:22all the sum of squares
14:25is now when we talk about sum of squares
14:28in the Inova context what we do is we
14:31say this overall sum of squares is
14:34partitioned or split into two parts so
14:38we call the total sum of squares SST
14:42that's sum of squares total now that's
14:45made up of two components the first
14:48component is the
14:50SSC that is the sum of squares of the
14:53columns so the columns the between
14:56variance the treatment sum of squares
14:59that's what we're talking about and
15:00again this will make sense more than a
15:01minute when we look at the actual
15:03problem now the other component is the
15:05SS e which is the sum of squares error
15:10so this is the within or the error sum
15:13of squares so the overall or total sum
15:16of squares in the oneway an NOA is
15:19actually a combination of two things the
15:22sum of squares of the columns sort of
15:24between the columns and the sum of
15:27squares of the error which is actually
15:29the sum of squares within each
15:33column so let's look at each one of
15:35these types of sum of squares using our
15:37actual data so we'll start with SST or
15:41the sum of squares total so what we do
15:44is we find the difference between each
15:46data point so all 21 data points and the
15:50overall mean over here on the right hand
15:52side of 74.5 2 we would then square that
15:57difference and then add all of those up
16:02so we would have 21 squared differences
16:06so here I only circled five but we would
16:08do this with all 21 so it's the
16:11difference between each data point and
16:14the overall mean Square it and then add
16:18them all up that is the sum of squares
16:24total so if we look at this in the
16:27actual distribution we have all 21 of
16:30our data points put together and then we
16:33have the overall mean of 74.5 2 in the
16:37distribution there so what we're doing
16:40is we are finding the distance of each
16:42data point to the overall mean and then
16:46of course we Square it so we would have
16:4921 squar deviations now here's the thing
16:53when we take a distance and we Square it
16:56guess what we get
16:59we literally get a
17:01square so when we say sum of
17:05squares we actually mean that literally
17:09so we take every distance from the point
17:11to the mean we Square it so we would
17:14have 21 individual Square distances on
17:18this graph if we actually did it and
17:20then we sum up the area of all those
17:24squares now I only mention that because
17:27it's going to become an important part
17:28later on when we talk about regression
17:32because sum of squares is a very
17:34important part of regression as well so
17:37sum of squares is literally squar
17:39distances when added together or the
17:42bunch of squares added
17:46up now let's look at SSC which is the
17:50sum of squares of the columns so this is
17:53the column or the between sum of squares
17:58so in this case we find the difference
17:59between each group mean and the overall
18:04mean Square those deviations and add
18:08them up so the
18:10SSC is the difference between the
18:12individual column means and the overall
18:15mean so in this case we'll have three so
18:18each individual column mean and the
18:21overall mean so remember the
18:24SST or the sum of squares total was the
18:28relationship between each individual
18:29data point and the overall mean well the
18:33SSC or the sum of squared columns is the
18:36relationship between each column mean
18:40and the overall
18:42mean so that would look like this so the
18:46SSC is the distance from each mean to
18:50the overall mean so you can see that
18:52there in the red and again in this case
18:55we would literally have squares so the
18:57squared distance with the square
19:00deviation add them all
19:04up now the third type is the
19:07s that is a sum of square error or the
19:11within sum of
19:13squares so this is actually about the
19:15individual distribution around each
19:19column mean we're not dealing with the
19:22overall mean in this case so we would
19:25find the difference between each data
19:27point and its own column
19:31mean Square each
19:34deviation and then add them up so for
19:37example we would have 21 Square
19:39deviations in this case because we would
19:42take each column
19:44score and then find the relationship
19:46between the overall column mean so the
19:50difference between each data point and
19:52its corresponding column mean find the
19:54difference Square it add them up and
19:57then we do the same thing for each
20:00column that is SS e or the sum of
20:04squares
20:06within so what we're looking at here is
20:09the distribution within each column so
20:11where each data point Falls within each
20:16column so this is what these three types
20:19of sum of squares look like when we put
20:21them all together so in the first case
20:24we have SST remember that is each
20:27individual score so all 21 scores as
20:30they relate to the overall mean so we
20:32find the difference between each score
20:34the overall mean Square it add them
20:37up the SSC is a relationship between
20:41each column mean there at the bottom and
20:44the overall mean so we find the
20:47differences Square them add them
20:50up then the
20:52SSE is the relationship between each
20:55individual score and its own column mean
20:59at the bottom so we find the difference
21:01between those two square it add them all
21:04up so you can see the three different
21:06ways we're looking at the sum of squares
21:09and the cool thing is is that the SSC
21:12the sum of squares columns plus the SS
21:15sum of square error adds up to the SST
21:21or the total sum of squares so you can
21:24see all the ways the data is sort of
21:27parsed when we do the oneway
21:32Anova so this is an exaggerated view of
21:35the relationship between our three
21:36column means so you can see that group
21:39one or the first year students had a
21:41mean of 71.7 one the year 2 students had
21:44a mean of 75.2 n there in the pink and
21:47the year three students had a mean of
21:4976.5 s there in the green and I've kind
21:52of exaggerated the distance of group
21:56one so you can see that each
21:59distribution or each column has its own
22:01distance to the overall mean so what
22:04we're trying to figure out here is is
22:07the firste student distribution those
22:09scores kind of an outlier or sort of an
22:12oddball distribution that's what we're
22:15checking for when we're doing the oneway
22:18in NOA now remember in the previous
22:20video I talked about why we cannot do
22:23three pairwise tee tests so we could
22:27compare grou group year 1 to year 2 year
22:301 to year 3 and year 3 to year 2 that
22:34would be three individual T tests we
22:36cannot do it that way because the error
22:39the type 1 error compounds each time we
22:42do that and we end up with a type 1
22:43error rate of over 14% so we have to do
22:47the Innova method when looking at this
22:49type of
22:52problem okay so that ends part one of
22:55our video on the oneway in Nova so again
22:58I I want you to show how the data is
23:00actually arranged when we actually do
23:02the calculations so we're looking at
23:05many comparisons within our chart so
23:07we're comparing the column mean to the
23:09overall mean we're comparing each data
23:11point to the overall mean and we're
23:13comparing each data point in a column
23:15with that columns overall mean and again
23:18it's all about variance so the square
23:22deviation is a measure of variance so
23:26the sum of squares is a way of
23:27quantifying that variance so now that we
23:30have a comprehensive understanding of
23:31what the oneway an NOA is and how the
23:34data points relate to each other let's
23:36go ahead go into Excel and do this
23:39calculation by hand now I have some
23:42pre-made formulas that we can actually
23:43just sort of paste into Excel but I will
23:46walk you through the actual computation
23:49of all those arrows and diagrams we had
23:51in the previous slides so let's go ahead
23:53and look at part two
23:57[Music]

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