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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
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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
1:08I'll feel that life is much too short
1:10and the world is much too large for us
1:12to miss the chance to connect when we
1:14can number three if you like the video
1:17please give it a thumbs up share it with
1:20classmates or colleagues or put it on a
1:22playlist that does encourage me to keep
1:24making them for you on the flip side if
1:27you think there's something I can do
1:28better please leave a instructive
1:30comment below the video and I will take
1:32those ideas into account when I make new
1:35ones and finally just keep in mind that
1:37these videos are meant for individuals
1:39who are relatively new to Stats so I'm
1:42just going over basic concepts and I
1:45will be doing so in a slow deliberate
1:48manner not only do I want you to know
1:51what is going on but also why and how to
1:55apply it so all that being said let's go
1:58ahead and get started
2:02so this video is the next in our series
2:03about simple linear regression in our
2:06last three videos we talked about the
2:08very basics of regression and introduced
2:10other fundamental concepts like the
2:12algebra of lines General patterns to
2:14look for on Scatter Plots and the leas
2:16squares method in this video we're going
2:19to learn to evaluate how well a
2:21regression line fits the data it models
2:25it is important to note that regression
2:27model is unique to the data it
2:29represents
2:30adding or changing data points will most
2:32certainly change the regression model
2:35the model is also only valid for the
2:37range of data points under analysis it
2:40is not proper to extrapolate above or
2:43below the data being
2:45evaluated so once a regression line is
2:48calculated how much better is it than
2:51using only the mean of the dependent
2:53variable alone to answer that question
2:56we will have to calculate the sum of the
2:57squared residuals or errors much like we
3:00did in the previous video however this
3:04time we will do so using the regression
3:06line not the dependent variable mean
3:10line and finally we will be able to
3:13quantify the fit of the regression model
3:16using a simple Ratio or percentage
3:19called the coefficient of determination
3:21so if you are new to regression or are
3:23still trying to figure out exactly what
3:25it even is this video is for you so sit
3:29back relax and let's go ahead and get to
3:34learning so we can think of this video
3:36as a story we'll call it a tail of two
3:40lines now in the first few videos we
3:42worked with this graph and data over
3:44here on the left remember in this case
3:48we only had the dependent variable which
3:50is the tip amount at the restaurant so
3:53we had six meals and then we had the six
3:56tip amounts that are graphed on the Y
3:58AIS in terms of their dollar
4:00amounts now we figured out the mean of
4:03these six tips was $10 so we went ahead
4:06and put a line a horizontal line at
4:09$10 then we found the errors which is
4:12the distance from the mean line to each
4:16observed data point and then we squared
4:19all of those errors and then added them
4:22up when we did that we came up with an
4:25ssse or sum of squared errors of 120
4:30now this is a very important point with
4:33only the dependent variable the only sum
4:36of squares is due to error that's all we
4:40have to work with therefore it is also
4:44the total sum of squares and the maximum
4:48sum of squares for the data under
4:50analysis so for these six specific data
4:53points the total sum of squares will
4:56never be more than
4:58120 so since the only source of error we
5:01have is the SSE it's also the total sum
5:05of squares in this case the same thing
5:08so SST is also 120 again because in this
5:12case they're the same
5:15thing now in the previous video we
5:17actually went ahead and calculated the
5:19regression line using the least squares
5:21method so we had the same six data
5:23points the graph looks different because
5:25they're graphed differently and then of
5:26course we have our regression line now
5:29in this video we're going to do the same
5:32thing we did over here on the left we're
5:34going to find the errors and square them
5:37and then add them up but of course we
5:39will not be using the mean of the
5:41dependent variable to do that we're
5:43going to be using the regression line to
5:45do that now with both the independent
5:48variable and the dependent variable the
5:50total sum of squares Remains the Same
5:55it's 120 still and it will always be for
5:58these six points but ideally the error
6:02sum of squares or the SS will be reduced
6:07significantly because our line fits the
6:11data better so the difference between
6:14the SST of 120 and the SSE which we will
6:18calculate is the SSR or the sum of
6:22squares due to the regression we did
6:25this is very much like a Nova in a way
6:27we'll talk about that as we go forward
6:29Ward but the SST will always be 120
6:32we're going to calculate the ssse based
6:34on the residuals from the regression
6:36line and then the difference between the
6:38SST and SS e will be the SSR or the sum
6:43of squares due to regression take a step
6:46back and look this is always what we are
6:49doing in simple linear regression we are
6:52comparing a regression model over here
6:54on the right that we found using the Le
6:57squares method to the case where we only
7:00have the dependent variable and the line
7:03is flat over here on the left now if the
7:07model on the right we calculate looks a
7:09lot like the one on the left and the sum
7:12of squared error is very high then our
7:15regression model doesn't really do a
7:17whole lot for us the idea is to reduce
7:20the ssse by having a line that fits the
7:23data better because the better the line
7:26fits the data the smaller the residual
7:29uals or the errors will
7:32be so this is our first case you can go
7:35and look at this again if you'd like but
7:37we just took the residuals of the errors
7:38squared them and then added them up this
7:41was
7:42120 so having only the dependent
7:44variable the best prediction for the tip
7:46of the next meal is just the mean of the
7:48tips which in this case is
7:50$10 since the mean line is flat its
7:53slope is zero so beta sub 1 equals 0
7:57again this is just a revieww
8:00now in the last video we actually
8:02calculated the regression line so here's
8:03a regression equation at the top and
8:06here's another one remember that Excel
8:08had a little bit different number for
8:10The Intercept than we did just because
8:11of rounding but basically they're the
8:13same thing so a few things our slope is
8:16not zero our slope is
8:1901462 and remember we interpreted that
8:22as for every dollar the meal amount
8:25increases the tip is expected to
8:28increase by about 15 cents that's the
8:31slope of
8:3301462 then we also had our intercept
8:36down here on the left which really for
8:37this problem isn't very meaningful now
8:39we also had the centroid so remember the
8:42centroid is the intersection of the
8:43means of both variables so the mean meal
8:47amount was $74 and the mean tip amount
8:49was $10 and the centroid will always
8:53fall on the least squares line and that
8:56can be very helpful if you have to find
8:58out other things about your
9:01graph so now that we're done reviewing
9:04let's go ahead and get to sort of the
9:05heart of the matter of this video but to
9:07do so we have to calculate the predicted
9:10values for each meal amount so over here
9:13on the left we have the observed total
9:15bill which in like the case the first
9:17meal was $34 and then we have the
9:20observed tip so that was the tip that
9:22was actually received by the waiter so
9:25the first one was $34 the tip that the
9:27waiter got was $5 $18 meal headed a tip
9:31of $17 so on and so forth now in this
9:35middle column we have our actual
9:36regression equation so 01462 xus
9:410.81
9:4388 now that is what we're going to use
9:46to create the predicted tip amount using
9:49our regression equation for each of our
9:52meals over here in the red on the
9:57left all we do is substitute each meal
10:01amount in for X because again that's our
10:04independent variable so we go ahead and
10:07evaluate all of these equations and that
10:10will give us our predicted tip amount
10:12over here on the right so again to
10:14substitute the meal amount into the
10:16regression equation and that will give
10:18us our predicted tip amount based on the
10:24regression so substitute over evaluate
10:27and now we have our predicted
10:29tips so for this meal that was $34 the
10:33waiter received
10:35$5 now our regression equation would
10:39have predicted a tip of
10:42$415 let skip down we had a meal that
10:46was
10:46$88 the waiter or waitress received
10:50$8 Now using our regression equation we
10:53would predict a tip of $12
10:57about5 so you can see the obser tip
10:59amount is not always or usually is not
11:03the predicted tip amount that is given
11:05using the regression equation so we have
11:08a discrepancy here we have the observed
11:10tip amount and we have the predicted tip
11:13amount and the difference between those
11:15two is going to be our
11:20error so here is our data again so the
11:22Diamonds the orange diamonds represent
11:24our actual observed values for the tips
11:27that the waiter received so here's our
11:29regression equation now the purple dots
11:33are the predicted tip amounts for those
11:36meal amounts as you can see they're not
11:38the same as what we observed so for the
11:41first one we had $415 that we would
11:43predict the tip to be for the second one
11:45we had a predicted value of
11:47$664 for a tip and so on and so forth on
11:51up to the
11:54ground now we have to find out what the
11:57error is or what the residual R so again
12:01that is just the distance between the
12:03predicted value and the observed value
12:07because that's quote how far off our
12:10regression is from the actual data so
12:12it's just the distance between the
12:14predicted and The
12:16observed so now we have to find out what
12:18those errors are and as you can imagine
12:21it's simple subtraction it's just the
12:22difference between the observed and the
12:25predicted that we calculated so for our
12:27first meal $34 we have observed tip
12:29amount actually in the restaurant of $5
12:32our regression predicted a
12:35$415 tip approximately so the difference
12:38between those two was
12:410.849 or about
12:4385 and that is the error for that meal
12:47amount and then we do that for all six
12:49meals very simple just
12:53subtraction now of course as we do with
12:56other errors we have to square those
12:58differences so the difference of
13:008495 for the first meal if we square
13:04that residual it is
13:060.721 7 for the second meal the
13:09difference is
13:122.37 we go ahead and square that and
13:15it's
13:29we sum those up we have an SS e that is
13:3430.75 so for our regression model that
13:38we came up with our SSE is a little bit
13:41over 30 so
13:4530.75 so on our graph we take each
13:49residual and then we Square it just like
13:53this so we take the residual and square
13:55it so these are areas we have here and
13:59then just like we did before we go ahead
14:02and add those
14:06up now let's compare the sum of squared
14:09errors from our two models now remember
14:12in the first model where we only had the
14:14dependent variable which was the tip
14:16only we had a sum of squared errors or
14:18SSE of 120 and again remember that's
14:22also the SST for that model so
14:26120 now for our regression model we have
14:29the DV and the IV to the dependent and
14:32independent variable where the tip
14:34amount is a function of the meal amount
14:37now we have a suos squared errors or SS
14:40of
14:4430.75 see how this works that's the
14:47entire point of simple linear regression
14:51to create a model to create a line
14:54through our data that reduces the ssse
14:57as much as it possibly can
15:01so if we actually make these physically
15:04the same scale side by side we can see
15:06that we put all those sses for the first
15:09model together it's 120 and we do the
15:11same thing for our regression model and
15:13it's
15:1530.75 so you can actually see the
15:17physical difference in the scale of the
15:20errors so when we conducted the
15:23regression the ssse decreased from 120
15:27to 30.0
15:30075 that is
15:3230.75 of the sum of the squares was
15:36explained or allocated to the error in
15:40the regression model so instead of
15:42having 120 as the error as in the first
15:45case we reduced that to
15:4830.75 because we reduced the distance
15:52from the regression line to the data
15:54points where did the other 89.8 n25 go
15:59well
16:0189.95 is the sum of squares due to our
16:07regression SST equal SSR plus
16:11SS in this case SST is always 120 and
16:15that equals 89.9 25 +
16:2130.75 so here are our two models again
16:25so the first one our ssse was 120 and of
16:28course the s s and SST are the same for
16:30that model so the SST is also
16:33120 now in our second regression model
16:36that we did our SST again is still 120
16:39but now the SS is
16:4330.75 because of the reduced distance
16:46between the regression line and the
16:49observed data points as it compares to
16:51the one over here on the left so doing
16:54Simple subtraction we had 120us 30.75
16:58that gives us our SSR or sum of squares
17:01due to regression and that's
17:0389.9
17:0725 so now that we have the SST the SSR
17:11and the ssse we can go ahead and talk
17:13about the coefficient of
17:15determination so how well does the
17:18estimated regression equation fit our
17:22data now this is where regression begins
17:25to look a lot like a Nova the total sum
17:29of squares is partitioned or allocated
17:33into SS and SSR and hopefully at some
17:38point I'll do a separate video on the
17:39relationship between regression and
17:41Anova but it's all about allocating or
17:45partitioning the total sum of squares
17:48into two or more things in this case SS
17:51and
17:52SSR so the total sum of squares is split
17:56some of it goes to SSR some of it goes
17:59to
18:00SS so if the SSR is large it uses up
18:05more of the SST and therefore SS is
18:10smaller relative to the total the
18:13coefficient of
18:14determination quantifies this ratio as a
18:18percentage so the coefficient of
18:21determination has the variable R 2
18:25equals SSR / SST so it's the sum of
18:29squares regression divided by the total
18:32sum of
18:34squares and if you actually do this in a
18:37statistical software package regression
18:39will actually give you back an anova
18:42table just like you saw when you did an
18:45anova so we're going to have SS and mean
18:48squares and an F statistic and a
18:51significance level and everything just
18:53like we did in an NOA and of course in
18:56this case this is the actual Anova for
18:58the data what we're working with we can
19:00see that the f is very large and
19:03significance is
19:060.258 which is significant at the 05
19:09level so there is a relationship between
19:12linear regression and an NOA because
19:14it's all about partitioning or
19:16allocating the total sum of squares into
19:19different components and then we measure
19:22the ratio of those components to figure
19:25out whether or not the model is
19:26statistically significant same
19:30idea so how do we interpret the
19:32coefficient of determination so again
19:35it's SSR over SS in this case it's
19:3989.95 ID 120 so we go ahead and do that
19:43Division and we come up with 0.74 93 or
19:4974.9 3% cuz remember the 120 is the
19:53total so
19:558992 of the 120 that's a percent
19:59and that is
20:0074.9 3 so we can conclude that
20:0474.9 3% of the total sum of squares can
20:09be explained using the estimated
20:11regression equation to predict the tip
20:14amount the remainder so
20:1925.7% is due to error and again that's
20:23based on this regression equation we
20:25used where X is the dollar amount of the
20:29bill so it's really about partitioning
20:31the total sum of squares as a percentage
20:34saying hey this percentage belongs to
20:36SSR and this percentage belongs to
20:39ssse and then we can develop a model
20:43that says this fits well or does not fit
20:45well in this case it is a good fit so a
20:50coefficient of determination of 74.9 3%
20:54is good and we also saw that in the
20:56Innova table which was statistic Bally
20:59significant so it's all about
21:01partitioning the total sum of squares
21:03into SSR and ssse and we want SS to be
21:08as low as possible for the model to be
21:11considered a good fit and for the Innova
21:14table to produce a significant
21:19result so let's put all together on one
21:22graph so you can see here we're dealing
21:24with three data points for each measure
21:28on our xaxis or independent variable so
21:31if we look at this one here sort of to
21:33the right we can see that we have a
21:35orange diamond we have a black circle
21:39and a purple circle so three data points
21:43that represent that meal of $88 I think
21:46it was so remember that the diamond is
21:49the actual observed value the black
21:52circle is the mean of the dependent
21:56variable which was the tips which is $10
21:59the purple circle is the predicted tip
22:02amount based on the regression model so
22:05each meal kind of has three reference
22:07points The observed the mean of the
22:10dependent variable and the predicted
22:13value so the first Square difference to
22:15look at is the SS and I think it's the
22:17easiest to remember so it is the sum of
22:20the squared differences between y sub I
22:23which is the observed value the diamond
22:26minus y hat sub I which is the predicted
22:30value which is the purple circle so we
22:32take the difference of each one along
22:34the graph we Square them and then we add
22:37them all up and that is the SS for our
22:40regression model so it's the distance
22:43between those two and similarly those
22:46two up here so the difference between
22:49the observed the diamond and the
22:50predicted the purple circle now the next
22:54one is SST it's the sum Square total now
22:58it is the sum of the square difference
22:59between each observed value which is the
23:03diamond minus the mean of the Y AIS or
23:07the dependent variable which is y bar so
23:11the difference between those then we
23:13Square it then we add them all up so
23:16that's the difference between the
23:17observed and the mean so the orange
23:19diamond and the black circle so just
23:22like that and just like that there and
23:25of course the final one is SSR
23:29so it is the sum of the squar difference
23:32between y hat sub I that's the predicted
23:35whe is the purple circle minus y bar
23:39which is the mean of the dependent
23:42variable so we find the difference we
23:44Square them and then add them up and
23:46that is this distance here and this
23:49distance here of course we would do this
23:51for each point along the graph so it's
23:55always a relationship between these
23:58three points that represent each value
24:02of the independent variable so in the
24:04middle here in this case a $88 meal
24:07amount we have the observed we have the
24:10mean of the Y's and we have the
24:12predicted and we have three
24:13relationships going on between those
24:15three data points and the square
24:17differences among those three data
24:19points is actually the total sum of
24:21squares package of SST ssse and SSR so
24:26now you can see what it actually looks
24:27like on
24:31graph okay so that wraps up our video on
24:34simple linear regression fit and the
24:37coefficient of determination so remember
24:40this is all about a comparison A Tail of
24:43Two Lines where we had the one graph
24:46where we only used the mean of the
24:48dependent variable and then we had our
24:50regression equation and that graph over
24:53on the right and it's always a
24:55comparison between the two if the
24:58regression model isn't any better than
25:01the mean of the dependent variable then
25:03the model doesn't really give us
25:04anything new and will most likely come
25:07back is statistically insignificant the
25:09SS will be very high and then we will
25:12just move on from there but it is always
25:15a comparison between these two different
25:18models now the coefficient of
25:20determination just tells us how we
25:23allocate the total sum of squares in the
25:25regression model so it's split it's
25:27either SS s e or it's SSR and we hope
25:31for a good fitting model that the SS is
25:34very low and the SSR is very high but in
25:38either case their sum will always add up
25:41to SST whatever that happens to be so I
25:44look forward to seeing you in that next
25:46video I wish you all the best of luck in
25:48your studies and in your work and I look
25:50forward to seeing you again next time
25:55[Music]

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