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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
0:24before 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
1:05nice 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
2:01so this is the first video about a new
2:04topic the analysis of variants or more
2:07commonly known as Anova in our last set
2:10of videos we learned how to compare the
2:13variances of two populations using the F
2:16ratio and F distribution in the context
2:20of Anova it is important to remember
2:23that the F ratio is simply a ratio of
2:26two variances F ratios are a central
2:30part of Anova so if you are unsure what
2:33the f ratio is you may want to go back
2:36and watch those
2:37videos Anova allows us to move Beyond
2:41comparing just two populations with
2:44Anova we can compare multiple
2:47populations and even subgroups of those
2:50populations we can investigate how two
2:53groups interact with each other
2:56quantitatively many experimental
2:58research designs use an NOA for these
3:01very reasons now in this video we will
3:04not be doing any calculations looking at
3:07any formulas or testing any hypothesis
3:11many students I've worked with over the
3:12years learn a Nova in class without
3:16actually knowing what it is or why it
3:19even exists so this video offers a solid
3:23conceptual Foundation about an NOA using
3:26illustrations and Graphics so if you are
3:29new to Anova
3:30or are still trying to figure out
3:32exactly what it is this video is for you
3:36so sit back relax and let's go ahead and
3:39get to
3:42work so the first and most obvious
3:45question is well why an Nova so up to
3:49this point we have been comparing two
3:52populations so the independent samples T
3:54Test which are two random samples We
3:57compare or the Matched sample T Test
4:00where each measurement is maybe the same
4:02person or the same machine something
4:04like that but of course limiting
4:06ourselves to the comparison of two
4:09populations as well limiting the world
4:12is much more complex than just two
4:15things what if we wish to compare the
4:17means of more than two
4:19populations what if we wish to compare
4:22populations each containing several
4:25sublevels or groups well enter another
4:29NOA so an NOA the acronym comes from the
4:34phrase analysis of variance so a NOA
4:38greatly expands what we were able to do
4:42in
4:45statistics so suppose we want to compare
4:48three sample means to see if a
4:51difference exists somewhere among them
4:55so our first sample mean is up here in
4:56the blue xar sub one xar bar sub two as
5:00our second sample mean here in the pink
5:02distribution and then our third one is
5:04down here in the green so each sample
5:08will have its mean and its own
5:12distribution so what we are asking is do
5:15all three of these means come from a
5:19common
5:21population so is one mean so far away
5:25from the other two that it is likely not
5:28from the same population as those other
5:31two or are all three so far apart that
5:36they all likely come from unique
5:39populations so you can see we're talking
5:41about sort of the relative distance
5:43between these means now the variance of
5:47these distributions is also important
5:49we'll talk about that later but for
5:51right now let's focus on the relative
5:53distance between these means and whether
5:57or not we could conclude that they come
5:59come from the same overall
6:04population so here is our first mean so
6:07xar sub one with its distribution xar
6:10sub two with its distribution and then
6:13xar sub3 with its
6:16distribution now let's say we take all
6:19the data points in all three of those
6:22samples and we put all those data points
6:24into a common larger distribution so
6:28we'll put that by Behind these three now
6:31we're asking oursel is where is each
6:34mean relative to the overall data set
6:38sort of in the background you can see
6:40that xar sub one Falls pretty much right
6:43down the middle xar sub 2 that mean is a
6:47bit to the right so you can see the red
6:49arrow denotes how far it is away from
6:52the mean of the larger sort of combined
6:55population what about xar step 3 well it
6:59is a bit to the left so you can see that
7:01red arrow denoting how far it is away
7:05from the mean of the overall sort of
7:07combined
7:10population now look at this example so
7:13xar sub one is where it was sort of
7:15right at the middle xar sub 2 is a bit
7:18to the right but now look where xar sub3
7:21is our third sample mean it's way over
7:25to the left so we might conclude that
7:28this mean the third one in the green is
7:32too far away from the others it's too
7:34far away from the mean of the larger
7:36group to be considered as part of that
7:39larger population it's kind of the
7:42Oddball so xar sub 2 is about this
7:44distance you can see the red arrow but
7:47xar sub3 is way over to the left so is
7:51this sort of the Oddball distribution
7:54sort of the weird one sort of the one
7:56that doesn't belong in the same
7:58population as as the other
8:02two now look at this case here we have
8:05xar sub one that's pretty much right
8:06down the middle again but now xar sub 2
8:09the second sample mean is way over to
8:12the right so you can see that distance
8:14is pretty far away and look at xar sub
8:17three its way over to the left so we
8:21might conclude that each one of these
8:24sample means belongs to its own
8:27population only X X bar sub One belongs
8:31to this distribution in the background
8:33xar sub2 May belong to one that's off to
8:35the right and xar sub three May belong
8:38the one that's off to the left so you
8:41can see that the means are in very
8:42different locations relative to the
8:45overall mean there in the
8:50background so the null hypothesis in
8:53these type of problems in an noas is
8:56whether or not these three sample means
8:58come from from the same population now
9:01remember the sample mean is a point
9:03estimator of the population mean so our
9:06no hypothesis is that mu sub 1 = mu sub
9:102 = mu sub3 which is another way of
9:14expressing that these three means come
9:17from the same overall
9:21population now remember we're not asking
9:23if they are exactly equal we're asking
9:26if each mean likely came from the same
9:30larger overall
9:33population so in an NOA this idea is a
9:37very specific and important idea we call
9:41this the variability among or between
9:44the sample means so each sample mean is
9:48a certain distance from the mean of the
9:51overall population in the background and
9:54we know that that is an expression of
9:56variance sort of the distance of the
9:59sample mean from the overall mean in the
10:01back so this variability between the
10:05sample means is something I really want
10:07you to keep in your mind as we proceed
10:10this is between
10:14variants so we could test all of these
10:17sample means using pairwise T tests so
10:21here are our three sample means so xar
10:24sub 1 xar sub 2 and xar sub3 now we'll
10:28block out this part of this little
10:30Matrix here because in this triangle we
10:33have comparisons to themselves and
10:36repeat comparisons so we're only going
10:38to have three that we could actually do
10:40so in this first one we could compare
10:43mean one here in the blue to mean 2
10:47which is there in the pink so our null
10:49hypothesis would be xar sub 1 = xar sub
10:532 we could have a t test that tests that
10:56now we could also compare mean one and
10:58mean mean 3 so we would have a t test of
11:01xar sub 1al xar
11:04sub3 now we could also test mean 2 and
11:07mean three so xar sub 2 = xar sub3 now
11:13notice each one of those independent
11:15tests has its own Alpha level so Alpha
11:19Point 05 05 and
11:2305 now the problem with doing all of
11:26these pairwise comparisons is that the
11:29Alpha level is the type one error rate
11:31of course which means 95% confidence but
11:35the error compounds with each T Test so
11:40if we compared each possible pair we
11:43would have .95 * .95 * .95 so our 95%
11:49confidence is now 857 or
11:5585.7% so of course our Alpha is 1 minus
11:59that so 1 - 857 equal. 143 well what is
12:05that1
12:06143 well that is our overall Alpha level
12:11that is our overall type one error rate
12:15so our type 1 error rate went from
12:175% to
12:2114.3% that is why we do not conduct T
12:23tests for every possible pair of means
12:27the error rate compounds and therefore
12:30the test of course has
12:35problems now what's different so if I
12:38stretch each one of these sample
12:40distributions out what
12:42changes well the spread or the variance
12:46of each
12:48distribution so we call this the
12:50variability around or within the
12:55distributions so remember before we
12:57talked about the variance between the
13:00distributions and that was sort of the
13:02distance of each mean from the overall
13:04population in the background so that was
13:07variability between this is variability
13:11within So within each sample
13:17distribution so at its heart a Nova is
13:20really a variability ratio it is a ratio
13:24of the first type of variance we saw the
13:26variability between the means
13:29over the variability within the
13:32distributions that's the second type we
13:34saw when we stretched each sample
13:36distribution out so it's just a ratio it
13:40is between variance divided by Within
13:45variance so remember in the first type
13:48we had an overall mean and then the
13:50distance of each sample mean from that
13:53so you can see that represented on the
13:55top and in the bottom we had the
13:57variability with within the
13:59distributions so their width or spread
14:03side to side so on the top we're talking
14:06about distance from the overall mean and
14:09on the bottom we're talking about each
14:11one's spread or width within its own
14:15sample so distance from the overall mean
14:18divided by sort of the internal
14:22spread so we reduce this to a very
14:25common fraction it is the very
14:29between divided by the variance within
14:33so remember this is a ratio of variances
14:37so of course the F distribution will
14:39come into play here in a bit so just
14:42keep in mind this sort of visual tool on
14:45the top the distance from the overall
14:47mean and in the bottom the internal
14:50spread of each sample
14:55distribution so variance between divid
14:58by variance within now if we put those
15:01together those are the components of the
15:04total variance so for an anova we have
15:09the total variance sort of split into
15:11two parts we have the variance between
15:14the means and then the variance within
15:17each means distribution so we put those
15:20together and we have the total variance
15:23for the entire data set so this is
15:27called partitioning so we're separating
15:30this total variance into its two
15:32component parts and again this is what's
15:35called a one-way in Nova and I'm not
15:37going to go into that right now that's
15:38for the next video but in this type of
15:41problem sort of the basic problem we
15:43have total variance and it's made up of
15:45variance between the means and the
15:47variance within each
15:50sample so here are sort of the nuts and
15:52bolts summary of everything we have here
15:55if the variability between the means
15:58sort of the distance from the overall
15:59mean we saw that in the red before in
16:02that numerator is relatively large
16:06compared to the variance within the
16:09samples So within the actual individual
16:12samples to spread in that
16:14denominator then this ratio will be much
16:19larger than one because the variance
16:22between will be quite large relative to
16:25the variance within that will make this
16:27ratio much larger than one if that's the
16:31case then the samples most likely do not
16:35come from a common population so then we
16:39would reject the null hypothesis that
16:42these means are equal or come from the
16:45same population so if that's the case we
16:49may have one Oddball distribution out to
16:51the side or all three of them may be so
16:54far apart that it would create a very
16:57high ratio because the variance between
17:00would be much higher than the variance
17:05within so what could this look like sort
17:08of in our test in general so if the
17:10variance between is large relative to
17:13the variance within being small we would
17:16reject that null hypothesis that says
17:18all the means are equal so at least one
17:22mean is an outlier sort of off to the
17:24side or they may all three be spread far
17:27apart and each distribution is
17:29relatively narrow so they don't sort of
17:31melt together they are distinct so think
17:34of this as three distinct distributions
17:37that are far apart or one that is sort
17:40of by itself off to the side that would
17:43create a large variance between the
17:47means now if the between variance and
17:50the within variances are similar then we
17:53would probably fail to reject that null
17:55hypothesis so they would appear to be
17:58equal or from the same overall
18:00population so in this case the means may
18:02be fairly close to each other fairly
18:05close to that overall mean and or the
18:08distributions overlap a bit so they may
18:11be a bit harder to distinguish from each
18:13other so if they're very close together
18:16or the variances within is very wide
18:19then they'll sort of melt together and
18:21therefore they will not be
18:24distinct now the other case is where the
18:26between variance is very small
18:29and the variance within is very large so
18:32you can think of this as three
18:34distributions that are very spread out
18:37internally and they not have a whole lot
18:39of distance from each other so they may
18:43be close together and or the
18:44distributions because of very high
18:46variation they're sort of wide and
18:48spread out they sort of melt together
18:51and you really can't distinguish them
18:53from each other as coming from separate
18:56or distinct populations so again this is
18:59a very general look at how we would
19:01interpret an an NOA and it obviously
19:04gets more complicated than this but I
19:06just want to give you a very rough
19:10overview so here's what we're left with
19:13this F ratio remember the F ratio is a
19:16ratio of two variances so we have the
19:20between variance so again the distance
19:23of each mean from the overall mean or
19:26the combined population in the
19:28background as compared to the variance
19:32within So within each sample
19:36distribution now sometimes this is
19:38called the among variance and the around
19:43variance so the thing about an noas is
19:46that I could have three different or
19:48four different stats textbooks and they
19:51all call this something different but
19:53the most common way of expressing it is
19:55the between variance and the within
19:57variance but you may see it as the among
20:00variance the the variance among the
20:03means and the around variance which is
20:06the variance around each sample mean but
20:10it means the same
20:12thing so this is that partitioning so
20:15the variance between they're in the red
20:18arrows plus the variance within they're
20:21in the blue arrows adds up to the total
20:26variance now you will also see a term
20:29called the error or the error variance
20:33well the error variance is another name
20:35for the within there in the blue or the
20:38around so again you might see that in
20:40the textbook and that's one of the
20:42challenges of doing these type of videos
20:44is because stats books like to give
20:46different names to the exact same
20:53thing okay so remember why and Nova so
20:57up to this point we had been been
20:58comparing just two populations so the
21:00independent samples T Test and the match
21:03sample T Test are two examples but
21:05limiting ourselves to the comparison of
21:07two populations is of course limiting
21:10what if we wish to compare the means of
21:12more than two populations what if we
21:14wish to compare the populations each
21:16containing several levels or subgroups
21:20well that's what we have an NOA for
21:22remember an NOA stands for the analysis
21:25of variance now one more thing I want to
21:28hit on before we go on to the end slide
21:30and wrap up remember that we're looking
21:32at the variance between the means as
21:35compared to the variance within each
21:39sample so it's all about variance that's
21:44why it's called analysis of variance
21:46variance between as compared to variance
21:49within so we're always looking at a
21:51ratio of two
21:56variances okay so that wraps up our
21:59first video on the analysis of variance
22:02or an NOA so again I wanted to give you
22:04a graphical representation of what's
22:06going on in an Nova so when you're
22:09working with the table of data and
22:11you're working with your numbers and
22:12your F ratios and things like that you
22:15actually know what is going on in the
22:19background what we're really trying to
22:21do is find any distinctions between
22:24sample means so we may have sample means
22:27that sort of all line up therefore we
22:29could conclude that those probably come
22:31from the same combined population but we
22:35might have one of the means that's sort
22:37of an oddball out by itself so in that
22:40case it probably comes from a different
22:43population off to the side or we could
22:45have three sample means that are very
22:48far apart from each other therefore each
22:51sample mean may come from its own
22:54population sort of in the background so
22:57what we're trying to look look for here
22:58is distinctions or differences among
23:01several means and we are doing that by
23:05looking at two types of variance between
23:07variant and within variance so a few
23:10last words and then we are done if
23:13you're watching this video because
23:14you're struggling in class stay positive
23:16and keep your head up I have faith in
23:19you many other people around you have
23:21faith in you so so should you if you
23:24like the video please give it a thumbs
23:25up share it with classmates or
23:28colleagues feel free to follow me here
23:30on YouTube on Twitter on go+ or on
23:33LinkedIn it's always nice hearing from
23:35you and finally just keep in mind that
23:37the fact that you're on here trying to
23:39learn trying to improve yourself as a
23:41student or as a business person that is
23:44what really matters I firmly believe if
23:46you have the right learning process in
23:48place the results will take care of
23:51themselves so thank you very much for
23:53watching I wish you the best of luck in
23:55your work and in your studies and I look
23:57forward to seeing you again next
24:01[Music]
24:16time

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