PL9fwy3NUQKwaIc7KWc8buW344941GStQL/EuXnSmrCFuE_postprocess.srt

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Last time, I mean Tuesday, we discussed box plot

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and we introduced how can we use box plot to

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determine if any point is suspected to be an

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outlier by using the lower limit and upper limit.

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And we mentioned last time that if any point is

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below the lower limit or is above the upper limit,

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that point is considered to be an outlier. So

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that's one of the usage of the backsplat. I mean,

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for this specific example, we mentioned last time

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27 is an outlier. And also here you can tell also

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the data are right skewed because the right tail

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exactly is much longer than the left tail. I mean

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the distance between or from the median and the

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maximum value is bigger or larger than the

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distance from the median to the smallest value.

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That means the data is not symmetric, it's quite

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skewed to the right. In this case, you cannot use

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the mean or the range as a measure of spread and

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median and, I'm sorry, mean as a measure of

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tendency. Because these measures are affected by

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outcomes. In this case, you have to use the median

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instead of the mean and IQR instead of the range

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because IQR is the mid-spread of the data because

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we just take the range from Q3 to Q1. That means

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we ignore The data below Q1 and data after Q3.

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That means IQR is not affected by outlier and it's

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better to use it instead of R, of the range.

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If the data has an outlier, it's better just to

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make a star or circle for the box plot because

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this one mentioned that that point is an outlier.

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Sometimes outlier is maximum value or the largest

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value you have. sometimes maybe the minimum value.

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So it depends on the data. For this example, 27,

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which was the maximum, is an outlier. But zero is

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not outlier in this case, because zero is above

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the lower limit. Let's move to the next topic,

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which talks about covariance and correlation.

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Later, we'll talk in more details about

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Correlation and regression, that's when maybe

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chapter 11 or 12. But here we just show how can we

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compute the covariance of the correlation

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coefficient and what's the meaning of that value

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we have. The covariance means it measures the

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strength of the linear relationship between two

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numerical variables. That means if the data set is

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numeric, I mean if both variables are numeric, in

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this case we can use the covariance to measure the

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strength of the linear association or relationship

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between two numerical variables. Now the formula

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is used to compute the covariance given by this

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one. It's summation of the product of xi minus x

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bar, yi minus y bar, divided by n minus 1.

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So we need first to compute the means of x and y,

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then find x minus x bar times y minus y bar, then

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sum all of these values, then divide by n minus 1.

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The covariance only concerned with the strength of

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the relationship. By using the sign of the

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covariance, you can tell if there exists positive

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or negative relationship between the two

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variables. For example, if the covariance between

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x and y is positive, that means x and y move In

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the same direction. It means that if X goes up, Y

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will go in the same position. If X goes down, also

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Y goes down. For example, suppose we are

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interested in the relationship between consumption

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and income. We know that if income increases, if

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income goes up, if your salary goes up, that means

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consumption also will go up. So that means they go

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in the same or move in the same position. So for

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sure, the covariance between X and Y should be

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positive. On the other hand, if the covariance

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between X and Y is negative, that means X goes up.

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Y will go to the same, to the opposite direction.

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I mean they move to opposite direction. That means

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there exists negative relationship between X and

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Y. For example, you score in statistics a number

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of missing classes. If you miss more classes, it

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means your score will go down so as x increases y

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will go down so there is positive relationship or

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negative relationship between x and y i mean x

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goes up the other go in the same direction

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sometimes

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there is exist no relationship between x and y In

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that case, covariance between x and y equals zero.

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For example, your score in statistics and your

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weight.

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It makes sense that there is no relationship

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between your weight and your score. In this case,

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we are saying x and y are independent. I mean,

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they are uncorrelated. Because as one variable

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increases, the other maybe go up or go down. Or

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maybe constant. So that means there exists no

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relationship between the two variables. In that

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case, the covariance between x and y equals zero.

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Now, by using the covariance, you can determine

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the direction of the relationship. I mean, you can

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just figure out if the relation is positive or

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negative. But you cannot tell exactly the strength

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of the relationship. I mean you cannot tell if

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they exist. strong moderate or weak relationship

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just you can tell there exists positive or

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negative or maybe the relationship does not exist

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but you cannot tell the exact strength of the

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relationship by using the value of the covariance

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I mean the size of the covariance does not tell

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anything about the strength so generally speaking

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covariance between x and y measures the strength

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of two numerical variables, and you only tell if

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there exists positive or negative relationship,

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but you cannot tell anything about the strength of

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the relationship. Any questions?

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So let me ask you just to summarize what I said so

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far. Just give me the summary or conclusion. of

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the covariance. The value of the covariance

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determine if the relationship between the

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variables are positive or negative or there is no

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relationship that when the covariance is more than

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zero, the meaning that it's positive, the

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relationship is positive and one variable go up,

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another go up and vice versa. And when the

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covariance is less than zero, there is negative

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relationship and the meaning that when one

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variable go up, the other goes down and vice versa

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and when the covariance equals zero, there is no

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relationship between the variables. And what's

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about the strength?

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So just tell the direction, not the strength of

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the relationship. Now, in order to determine both

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the direction and the strength, we can use the

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coefficient of correlation. The coefficient of

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correlation measures the relative strength of the

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linear relationship between two numerical

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variables. The simplest formula that can be used

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to compute the correlation coefficient is given by

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this one. Maybe this is the easiest formula you

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can use. I mean, it's shortcut formula for

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computation. There are many other formulas to

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compute the correlation. This one is the easiest

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one. R is just sum of xy minus n, n is the sample

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size, times x bar is the sample mean, y bar is the

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sample mean for y, because here we have two

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variables, divided by square root, don't forget

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the square root, of two quantities. One conserved

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for x and the other for y. The first one, sum of x

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squared minus nx bar squared. The other one is

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similar just for the other variables, sum y

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squared minus ny bar squared. So in order to

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determine the value of R, we need,

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suppose for example, we have x and y, theta x and

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y.

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x is called explanatory

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variable and

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y is called response variable

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sometimes x is called independent

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For example, suppose we are talking about

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consumption and

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input. And we are interested in the relationship

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between these two variables. Now, except for the

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variable or the independent, this one affects the

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other variable. As we mentioned, as your income

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increases, your consumption will go in the same

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direction, increases also. Income causes Y, or

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income affects Y. In this case, income is your X.

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Most of the time we use

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And Y for independent. So in this case, the

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response variable or your outcome or the dependent

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variable is your consumption. So Y is consumption,

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X is income. So now in order to determine the

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correlation coefficient, we have the data of X and

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Y.

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The values of X, I mean the number of pairs of X

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should be equal to the number of pairs of Y. So if

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we have ten observations for X, we should have the

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same number of observations for Y. It's pairs. X1,

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Y1, X2, Y2, and so on. Now, the formula to compute

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R, the shortcut formula is sum of XY minus N times

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x bar, y bar, divided by the square root of two

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quantities. The first one, sum of x squared minus

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n x bar. The other one, sum of y squared minus ny

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y squared. So the first thing we have to do is to

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find the mean for each x and y.

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So first step, compute x bar and y bar. Next, if

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you look here, we have x and y, x times y. So we

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need to compute the product of x times y. So just

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for example, suppose x is 10, y is 5. So x times y

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is 50.

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In addition to that, you have to compute

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100 x squared and y squared. It's like 125.

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Do the same calculations for the rest of the data

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you have. We have other data here, so we have to

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compute the same for the others.

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Then finally, just add xy, x squared, y squared.

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The values you have here in this formula, in order

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to compute the coefficient.

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Now, this value ranges between minus one and plus

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one.

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So it's between minus one and plus one. That means

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it's never smaller

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than minus one or greater than one. It's between

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minus one and plus one.

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Make sense? I mean if your value is suppose you

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did mistake for any of these computations and R

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might be 1.15, 115. That means there is an error.

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Or for example, if R is negative 1.5, that means

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there is a mistake. So you have to find or figure

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out what is that mistake. So that's simple

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calculations. Usually in the exam, we will give

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the formula for the correlation coefficient, as we

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mentioned before. In addition to that, we will

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give the summation.

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The sum of xy is given, sum x squared and sum y

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squared. Also sum of x and sum of y, in order to

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determine the means of x and y. For example,

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suppose I give sum of xi and i goes from 1 to 10

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is 700, for example. You have to know that the

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sample size is 10, so x bar. is 700 divided by 10,

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so it's 7. Then use the curve to compute the

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coefficient of correlation. Questions? I think

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straightforward, maybe the easiest topic in this

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book is to compute the coefficient of correlation

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or correlation coefficient. Now my question is, do

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you think outliers affect the correlation

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coefficient?

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We said last time outliers affect the mean, the

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range, the variance. Now the question is, do

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outliers affect the correlation?

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Y.

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Exactly. The formula for R has X bar in it or Y

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bar. So it means outliers affect

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the correlation coefficient. So the answer is yes.

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Here we have x bar and y bar. Also, there is

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another formula to compute R. That formula is

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given by covariance between x and y.

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These two formulas are quite similar. I mean, by

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using this one, we can end with this formula. So

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this formula depends on this x is y. standard

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deviations of X and Y. That means outlier will

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affect the correlation coefficient. So in case of

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outliers, R could be changed.

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That formula is called simple correlation

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coefficient. On the other hand, we have population

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correlation coefficient. If you remember last

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time, we used X bar as the sample mean and mu as

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population mean. Also, S square as sample variance

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and sigma square as population variance. Here, R

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is used as sample coefficient of correlation and

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rho, this Greek letter pronounced as rho. Rho is

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used for population coefficient of correlation.

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There are some features of R or Rho. The first one

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is unity-free. R or Rho is unity-free. That means

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if X represents...

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And let's assume that the correlation between X

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and Y equals 0.75.

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Now, in this case, there is no unity. You cannot

255
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say 0.75 years or 0.75 kilograms. It's unity-free.

256
00:20:13,940 --> 00:20:17,840
There is no unit for the correlation coefficient,

257
00:20:18,020 --> 00:20:21,120
the same as Cv. If you remember Cv, the

258
00:20:21,120 --> 00:20:24,320
coefficient of correlation, also this one is unity

259
00:20:24,320 --> 00:20:30,500
-free. The second feature of R ranges between

260
00:20:30,500 --> 00:20:36,740
minus one and plus one. As I mentioned, R lies

261
00:20:36,740 --> 00:20:42,340
between minus one and plus one. Now, by using the

262
00:20:42,340 --> 00:20:48,100
value of R, you can determine two things. Number

263
00:20:48,100 --> 00:20:53,360
one, we can determine the direction. and strength

264
00:20:53,360 --> 00:20:56,940
by using the sign you can determine if there

265
00:20:56,940 --> 00:21:03,980
exists positive or negative so sign of R determine

266
00:21:03,980 --> 00:21:08,040
negative or positive relationship the direction

267
00:21:08,040 --> 00:21:17,840
the absolute value of R I mean absolute of R I

268
00:21:17,840 --> 00:21:21,980
mean ignore the sign So the absolute value of R

269
00:21:21,980 --> 00:21:24,100
determines the strength.

270
00:21:27,700 --> 00:21:30,760
So by using the sine of R, you can determine the

271
00:21:30,760 --> 00:21:35,680
direction, either positive or negative. By using

272
00:21:35,680 --> 00:21:37,740
the absolute value, you can determine the

273
00:21:37,740 --> 00:21:43,500
strength. We can split the strength into two

274
00:21:43,500 --> 00:21:52,810
parts, either strong, moderate, or weak. So weak,

275
00:21:53,770 --> 00:21:59,130
moderate, and strong by using the absolute value

276
00:21:59,130 --> 00:22:03,870
of R. The closer to minus one, if R is close to

277
00:22:03,870 --> 00:22:07,010
minus one, the stronger the negative relationship

278
00:22:07,010 --> 00:22:09,430
between X and Y. For example, imagine

279
00:22:22,670 --> 00:22:26,130
And as we mentioned, R ranges between minus 1 and

280
00:22:26,130 --> 00:22:26,630
plus 1.

281
00:22:30,070 --> 00:22:35,710
So if R is close to minus 1, it's a strong

282
00:22:35,710 --> 00:22:41,250
relationship. Strong linked relationship. The

283
00:22:41,250 --> 00:22:45,190
closer to 1, the stronger the positive

284
00:22:45,190 --> 00:22:49,230
relationship. I mean, if R is close. Strong

285
00:22:49,230 --> 00:22:54,480
positive. So strong in either direction, either to

286
00:22:54,480 --> 00:22:57,640
the left side or to the right side. Strong

287
00:22:57,640 --> 00:23:00,280
negative. On the other hand, there exists strong

288
00:23:00,280 --> 00:23:05,940
negative relationship. Positive. Positive. If R is

289
00:23:05,940 --> 00:23:10,640
close to zero, weak. Here we can say there exists

290
00:23:10,640 --> 00:23:15,940
weak relationship between X and Y.

291
00:23:19,260 --> 00:23:25,480
If R is close to 0.5 or

292
00:23:25,480 --> 00:23:32,320
minus 0.5, you can say there exists positive

293
00:23:32,320 --> 00:23:38,840
-moderate or negative-moderate relationship. So

294
00:23:38,840 --> 00:23:42,200
you can split or you can divide the strength of

295
00:23:42,200 --> 00:23:44,540
the relationship between X and Y into three parts.

296
00:23:45,860 --> 00:23:50,700
Strong, close to minus one of Plus one, weak,

297
00:23:51,060 --> 00:23:59,580
close to zero, moderate, close to 0.5. 0.5 is

298
00:23:59,580 --> 00:24:04,580
halfway between 0 and 1, and minus 0.5 is also

299
00:24:04,580 --> 00:24:09,040
halfway between minus 1 and 0. Now for example,

300
00:24:09,920 --> 00:24:15,580
what's about if R equals minus 0.5? Suppose R1 is

301
00:24:15,580 --> 00:24:16,500
minus 0.5.

302
00:24:20,180 --> 00:24:27,400
strong negative or equal minus point eight strong

303
00:24:27,400 --> 00:24:33,540
negative which is more strong nine nine because

304
00:24:33,540 --> 00:24:39,670
this value is close closer to minus one than Minus

305
00:24:39,670 --> 00:24:44,070
0.8. Even this value is greater than minus 0.9,

306
00:24:44,530 --> 00:24:50,870
but minus 0.9 is close to minus 1, more closer to

307
00:24:50,870 --> 00:24:56,910
minus 1 than minus 0.8. On the other hand, if R

308
00:24:56,910 --> 00:25:01,190
equals 0.75, that means there exists positive

309
00:25:01,190 --> 00:25:06,970
relationship. If R equals 0.85, also there exists

310
00:25:06,970 --> 00:25:13,540
positive. But R2 is stronger than R1, because 0.85

311
00:25:13,540 --> 00:25:20,980
is closer to plus 1 than 0.7. So we can say that

312
00:25:20,980 --> 00:25:23,960
there exists strong relationship between X and Y,

313
00:25:24,020 --> 00:25:27,260
and this relationship is positive. So again, by

314
00:25:27,260 --> 00:25:32,530
using the sign, you can tell the direction. The

315
00:25:32,530 --> 00:25:35,910
absolute value can tell the strength of the

316
00:25:35,910 --> 00:25:39,870
relationship between X and Y. So there are five

317
00:25:39,870 --> 00:25:44,150
features of R, unity-free. Ranges between minus

318
00:25:44,150 --> 00:25:47,750
one and plus one. Closer to minus one, it means

319
00:25:47,750 --> 00:25:51,950
stronger negative. Closer to plus one, stronger

320
00:25:51,950 --> 00:25:56,410
positive. Close to zero, it means there is no

321
00:25:56,410 --> 00:26:00,790
relationship. Or the weaker, the relationship

322
00:26:00,790 --> 00:26:13,240
between X and Y. By using scatter plots, we

323
00:26:13,240 --> 00:26:18,160
can construct a scatter plot by plotting the Y

324
00:26:18,160 --> 00:26:24,060
values versus the X values. Y in the vertical axis

325
00:26:24,060 --> 00:26:28,400
and X in the horizontal axis. If you look

326
00:26:28,400 --> 00:26:34,500
carefully at graph number one and three, We see

327
00:26:34,500 --> 00:26:42,540
that all the points lie on the straight line,

328
00:26:44,060 --> 00:26:48,880
either this way or the other way. If all the

329
00:26:48,880 --> 00:26:52,320
points lie on the straight line, it means they

330
00:26:52,320 --> 00:26:56,970
exist perfectly. not even strong it's perfect

331
00:26:56,970 --> 00:27:02,710
relationship either negative or positive so this

332
00:27:02,710 --> 00:27:07,530
one perfect negative negative

333
00:27:07,530 --> 00:27:14,090
because x increases y goes down decline so if x is

334
00:27:14,090 --> 00:27:19,590
for example five maybe y is supposed to twenty if

335
00:27:19,590 --> 00:27:25,510
x increased to seven maybe y is fifteen So if X

336
00:27:25,510 --> 00:27:29,290
increases, in this case, Y declines or decreases,

337
00:27:29,850 --> 00:27:34,290
it means there exists negative relationship. On

338
00:27:34,290 --> 00:27:40,970
the other hand, the left corner here, positive

339
00:27:40,970 --> 00:27:44,710
relationship, because X increases, Y also goes up.

340
00:27:45,970 --> 00:27:48,990
And perfect, because all the points lie on the

341
00:27:48,990 --> 00:27:52,110
straight line. So it's perfect, positive, perfect,

342
00:27:52,250 --> 00:27:57,350
negative relationship. So it's straightforward to

343
00:27:57,350 --> 00:27:59,550
determine if it's perfect by using scatterplot.

344
00:28:02,230 --> 00:28:04,950
Also, by scatterplot, you can tell the direction

345
00:28:04,950 --> 00:28:09,270
of the relationship. For the second scatterplot,

346
00:28:09,630 --> 00:28:12,270
it seems to be that there exists negative

347
00:28:12,270 --> 00:28:13,730
relationship between X and Y.

348
00:28:16,850 --> 00:28:21,030
In this one, also there exists a relationship

349
00:28:24,730 --> 00:28:32,170
positive which one is strong more strong this

350
00:28:32,170 --> 00:28:37,110
one is stronger because the points are close to

351
00:28:37,110 --> 00:28:40,710
the straight line much more than the other scatter

352
00:28:40,710 --> 00:28:43,410
plot so you can say there exists negative

353
00:28:43,410 --> 00:28:45,810
relationship but that one is stronger than the

354
00:28:45,810 --> 00:28:49,550
other one this one is positive but the points are

355
00:28:49,550 --> 00:28:55,400
scattered around the straight line so you can tell

356
00:28:55,400 --> 00:29:00,000
the direction and sometimes sometimes not all the

357
00:29:00,000 --> 00:29:04,640
time you can tell the strength sometimes it's very

358
00:29:04,640 --> 00:29:07,960
clear that the relation is strong if the points

359
00:29:07,960 --> 00:29:11,480
are very close straight line that means the

360
00:29:11,480 --> 00:29:15,940
relation is strong now the other one the last one

361
00:29:15,940 --> 00:29:23,850
here As X increases, Y stays at the same value.

362
00:29:23,970 --> 00:29:29,450
For example, if Y is 20 and X is 1. X is 1, Y is

363
00:29:29,450 --> 00:29:33,870
20. X increases to 2, for example. Y is still 20.

364
00:29:34,650 --> 00:29:37,230
So that means there is no relationship between X

365
00:29:37,230 --> 00:29:41,830
and Y. It's a constant. Y equals a constant value.

366
00:29:42,690 --> 00:29:50,490
Whatever X is, Y will have constant value. So that

367
00:29:50,490 --> 00:29:54,790
means there is no relationship between X and Y.

368
00:29:56,490 --> 00:30:01,850
Let's see how can we compute the correlation

369
00:30:01,850 --> 00:30:07,530
between two variables. For example, suppose we

370
00:30:07,530 --> 00:30:12,150
have data for father's height and son's height.

371
00:30:13,370 --> 00:30:16,510
And we are interested to see if father's height

372
00:30:16,510 --> 00:30:21,730
affects his son's height. So we have data for 10

373
00:30:21,730 --> 00:30:28,610
observations here. Father number one, his height

374
00:30:28,610 --> 00:30:38,570
is 64 inches. And you know that inch equals 2

375
00:30:38,570 --> 00:30:39,230
.5.

376
00:30:43,520 --> 00:30:52,920
So X is 64, Sun's height is 65. X is 68, Sun's

377
00:30:52,920 --> 00:30:58,820
height is 67 and so on. Sometimes, if the dataset

378
00:30:58,820 --> 00:31:02,600
is small enough, as in this example, we have just

379
00:31:02,600 --> 00:31:08,640
10 observations, you can tell the direction. I

380
00:31:08,640 --> 00:31:12,060
mean, you can say, yes, for this specific example,

381
00:31:12,580 --> 00:31:15,280
there exists positive relationship between x and

382
00:31:15,280 --> 00:31:20,820
y. But if the data set is large, it's very hard to

383
00:31:20,820 --> 00:31:22,620
figure out if the relation is positive or

384
00:31:22,620 --> 00:31:26,400
negative. So we have to find or to compute the

385
00:31:26,400 --> 00:31:29,700
coefficient of correlation in order to see there

386
00:31:29,700 --> 00:31:32,940
exists positive, negative, strong, weak, or

387
00:31:32,940 --> 00:31:37,820
moderate. but again you can tell from this simple

388
00:31:37,820 --> 00:31:40,280
example yes there is a positive relationship

389
00:31:40,280 --> 00:31:44,660
because just if you pick random numbers here for

390
00:31:44,660 --> 00:31:49,240
example 64 father's height his son's height 65 if

391
00:31:49,240 --> 00:31:54,600
we move up here to 70 for father's height his

392
00:31:54,600 --> 00:32:00,160
son's height is going to be 72 so as father

393
00:32:00,160 --> 00:32:05,020
heights increases Also, son's height increases.

394
00:32:06,320 --> 00:32:11,700
For example, 77, father's height. His son's height

395
00:32:11,700 --> 00:32:15,160
is 76. So that means there exists positive

396
00:32:15,160 --> 00:32:19,740
relationship. Make sense? But again, for large

397
00:32:19,740 --> 00:32:20,780
data, you cannot tell that.

398
00:32:31,710 --> 00:32:36,090
If, again, by using this data, small data, you can

399
00:32:36,090 --> 00:32:40,730
determine just the length, the strength, I'm

400
00:32:40,730 --> 00:32:43,490
sorry, the strength of a relationship or the

401
00:32:43,490 --> 00:32:47,590
direction of the relationship. Just pick any

402
00:32:47,590 --> 00:32:51,030
number at random. For example, if we pick this

403
00:32:51,030 --> 00:32:51,290
number.

404
00:32:55,050 --> 00:33:00,180
Father's height is 68, his son's height is 70. Now

405
00:33:00,180 --> 00:33:02,180
suppose we pick another number that is greater

406
00:33:02,180 --> 00:33:05,840
than 68, then let's see what will happen. For

407
00:33:05,840 --> 00:33:11,060
father's height 70, his son's height increases up

408
00:33:11,060 --> 00:33:17,160
to 72. Similarly, 72 father's height, his son's

409
00:33:17,160 --> 00:33:22,060
height 75. So that means X increases, Y also

410
00:33:22,060 --> 00:33:25,740
increases. So that means there exists both of

411
00:33:25,740 --> 00:33:32,570
them. For sure it is hard to tell this direction

412
00:33:32,570 --> 00:33:36,130
if the data is large. Because maybe you will find

413
00:33:36,130 --> 00:33:40,250
as X increases for one point, Y maybe decreases

414
00:33:40,250 --> 00:33:43,610
for that point. So it depends on the data you

415
00:33:43,610 --> 00:33:49,010
have. Anyway, let's see how can we compute R. I

416
00:33:49,010 --> 00:33:53,770
will use Excel to show how can we do these

417
00:33:53,770 --> 00:33:54,550
calculations.

418
00:34:02,110 --> 00:34:06,530
The screen is clear. But give me the data of X and

419
00:34:06,530 --> 00:34:06,750
Y.

420
00:34:10,710 --> 00:34:14,310
X is 64. 68.

421
00:34:18,910 --> 00:34:26,830
68. 78. There is one 68. 78. 74.

422
00:34:31,120 --> 00:34:37,600
Seventy-four. Seventy-five. Seventy-six.

423
00:34:38,360 --> 00:34:42,240
Seventy-seven. Seventy-five. So that's the values

424
00:34:42,240 --> 00:34:49,440
of X, Y values. Seventy. Seventy-five. Seventy

425
00:34:49,440 --> 00:34:49,800
-seven.

426
00:35:17,230 --> 00:35:23,730
So first we have to compute it. x times y so

427
00:35:23,730 --> 00:35:28,270
that's as x times

428
00:35:28,270 --> 00:35:38,230
the value of y so 46 times 65 equals 4160 x

429
00:35:38,230 --> 00:35:46,050
squared so this value squared for y squared 65

430
00:35:48,660 --> 00:35:53,700
Square and we have to do this one for the rest of

431
00:35:53,700 --> 00:36:03,160
the data So

432
00:36:03,160 --> 00:36:07,600
that's the sum of XY sum X squared and Y squared

433
00:36:07,600 --> 00:36:13,480
now the summation So

434
00:36:13,480 --> 00:36:17,180
that's the sum of X and Y

435
00:36:20,380 --> 00:36:27,040
We have to compute the mean of x and y. So that is

436
00:36:27,040 --> 00:36:31,380
this sum divided by n, where n is 10 in this case.

437
00:36:34,600 --> 00:36:36,100
So this is the first step.

438
00:36:41,820 --> 00:36:48,900
Let's see how can we compute R. R, we have sum of

439
00:36:48,900 --> 00:37:00,780
x, y. minus n is 10 times x bar times y bar. This

440
00:37:00,780 --> 00:37:03,100
is the first quantity. The other one is square

441
00:37:03,100 --> 00:37:08,580
root of sum

442
00:37:08,580 --> 00:37:15,420
x squared minus n x bar squared.

443
00:37:18,830 --> 00:37:24,770
times some y squared minus n times y bar squared.

444
00:37:28,930 --> 00:37:34,090
And we have to find the square root of this value.

445
00:37:34,210 --> 00:37:40,810
So square root, that will give this result. So now

446
00:37:40,810 --> 00:37:46,990
R equals this value divided by

447
00:37:49,670 --> 00:37:54,890
155 and round always to two decimal places will

448
00:37:54,890 --> 00:38:05,590
give 87 so r is 87 so first step we have x and y

449
00:38:05,590 --> 00:38:12,470
compute xy x squared y squared sum of these all of

450
00:38:12,470 --> 00:38:18,100
these then x bar y bar values are given Then just

451
00:38:18,100 --> 00:38:20,820
use the formula you have, we'll get R to be at

452
00:38:20,820 --> 00:38:31,540
seven. So in this case, if we just go back to

453
00:38:31,540 --> 00:38:33,400
the slide we have here.

454
00:38:36,440 --> 00:38:41,380
As we mentioned, father's height is the

455
00:38:41,380 --> 00:38:45,640
explanatory variable. Son's height is the response

456
00:38:45,640 --> 00:38:46,060
variable.

457
00:38:49,190 --> 00:38:52,810
And that simple calculation gives summation of xi,

458
00:38:54,050 --> 00:38:57,810
summation of yi, summation x squared, y squared,

459
00:38:57,970 --> 00:39:02,690
and some xy. And finally, we'll get that result,

460
00:39:02,850 --> 00:39:07,850
87%. Now, the sign is positive. That means there

461
00:39:07,850 --> 00:39:13,960
exists positive. And 0.87 is close to 1. That

462
00:39:13,960 --> 00:39:17,320
means there exists strong positive relationship

463
00:39:17,320 --> 00:39:22,480
between father's and son's height. I think the

464
00:39:22,480 --> 00:39:25,060
calculation is straightforward.

465
00:39:27,280 --> 00:39:33,280
Now, for this example, the data are given in

466
00:39:33,280 --> 00:39:37,460
inches. I mean father's and son's height in inch.

467
00:39:38,730 --> 00:39:41,050
Suppose we want to convert from inch to

468
00:39:41,050 --> 00:39:44,750
centimeter, so we have to multiply by 2. Do you

469
00:39:44,750 --> 00:39:52,050
think in this case, R will change? So if we add or

470
00:39:52,050 --> 00:39:59,910
multiply or divide, R will not change? I mean, if

471
00:39:59,910 --> 00:40:06,880
we have X values, And we divide or multiply X, I

472
00:40:06,880 --> 00:40:09,460
mean each value of X, by a number, by a fixed

473
00:40:09,460 --> 00:40:12,600
value. For example, suppose here we multiplied

474
00:40:12,600 --> 00:40:19,460
each value by 2.5 for X. Also multiply Y by the

475
00:40:19,460 --> 00:40:24,520
same value, 2.5. Y will be the same. In addition

476
00:40:24,520 --> 00:40:28,920
to that, if we multiply X by 2.5, for example, and

477
00:40:28,920 --> 00:40:34,960
Y by 5, also R will not change. But you have to be

478
00:40:34,960 --> 00:40:39,400
careful. We multiply each value of x by the same

479
00:40:39,400 --> 00:40:45,700
number. And each value of y by the same number,

480
00:40:45,820 --> 00:40:49,640
that number may be different from x. So I mean

481
00:40:49,640 --> 00:40:56,540
multiply x by 2.5 and y by minus 1 or plus 2 or

482
00:40:56,540 --> 00:41:01,000
whatever you have. But if it's negative, then

483
00:41:01,000 --> 00:41:05,640
we'll get negative answer. I mean if Y is

484
00:41:05,640 --> 00:41:08,060
positive, for example, and we multiply each value

485
00:41:08,060 --> 00:41:13,000
Y by minus one, that will give negative sign. But

486
00:41:13,000 --> 00:41:17,640
here I meant if we multiply this value by positive

487
00:41:17,640 --> 00:41:21,320
sign, plus two, plus three, and let's see how can

488
00:41:21,320 --> 00:41:22,540
we do that by Excel.

489
00:41:26,320 --> 00:41:31,480
Now this is the data we have. I just make copy.

490
00:41:37,730 --> 00:41:45,190
I will multiply each value X by 2.5. And I will do

491
00:41:45,190 --> 00:41:49,590
the same for Y

492
00:41:49,590 --> 00:41:57,190
value. I will replace this data by the new one.

493
00:41:58,070 --> 00:42:00,410
For sure the calculations will, the computations

494
00:42:00,410 --> 00:42:09,740
here will change, but R will stay the same. So

495
00:42:09,740 --> 00:42:14,620
here we multiply each x by 2.5 and the same for y.

496
00:42:15,540 --> 00:42:19,400
The calculations here are different. We have

497
00:42:19,400 --> 00:42:22,960
different sum, different sum of x, sum of y and so

498
00:42:22,960 --> 00:42:31,040
on, but are the same. Let's see if we multiply

499
00:42:31,040 --> 00:42:38,880
just x by 2.5 and y the same.

500
00:42:41,840 --> 00:42:49,360
So we multiplied x by 2.5 and we keep it make

501
00:42:49,360 --> 00:42:57,840
sense? Now let's see how outliers will affect the

502
00:42:57,840 --> 00:43:03,260
value of R. Let's say if we change one point in

503
00:43:03,260 --> 00:43:08,480
the data set support. I just changed 64.

504
00:43:13,750 --> 00:43:24,350
for example if just by typo and just enter 6 so it

505
00:43:24,350 --> 00:43:33,510
was 87 it becomes 0.7 so there is a big difference

506
00:43:33,510 --> 00:43:38,670
between 0.87 and 0.7 and just we change one value

507
00:43:38,670 --> 00:43:45,920
now suppose the other one is zero 82. The other is

508
00:43:45,920 --> 00:43:48,260
2, for example. 1.

509
00:43:53,380 --> 00:43:59,200
I just changed half of the data. Now R was 87, it

510
00:43:59,200 --> 00:44:02,920
becomes 59. That means these outliers, these

511
00:44:02,920 --> 00:44:06,180
values for sure are outliers and they fit the

512
00:44:06,180 --> 00:44:07,060
correlation coefficient.

513
00:44:11,110 --> 00:44:14,970
Now let's see if we just change this 1 to be 200.

514
00:44:15,870 --> 00:44:20,430
It will go from 50 to up to 63. That means any

515
00:44:20,430 --> 00:44:26,010
changes in x or y values will change the y. But if

516
00:44:26,010 --> 00:44:30,070
we add or multiply all the values by a constant, r

517
00:44:30,070 --> 00:44:31,170
will stay the same.

518
00:44:35,250 --> 00:44:43,590
Any questions? That's the end of chapter 3. I will

519
00:44:43,590 --> 00:44:48,990
move quickly to the practice problems we have. And

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00:44:48,990 --> 00:44:55,270
we posted the practice in the course website.