Explanatory and Response Variables, Correlation (2.1)

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“This video. We will be looking at explanatory variables response variables. And correlation in the the first video set. We talked about how we can use histogram stem plots and plots to describe one variable and we also used back to back stem plots and side by side box plots to help us compare two different populations with respect to the same variable these were good for describing one variable.

But what about two variables well we saw that we can use a time plot to show this. If there is a relationship between these two variables. One can be called the response variable and the other can be called the explanatory variable. The response variable measures.

The outcome of a study and the explanatory variable explains the outcome of a study in this example the age of the tree is the explanatory variable because as the tree gets older the taller it will be so we can see that the age of the tree explains its height this also means that the height of the tree is the response variable another way to show the relationship between two variables is by using a scatter plot. Unlike a time plot a scatter plot does not need to show time on the x. Axis. A scatter plot shows the values of two quantitative variables.

That were measured from the same population of individuals. This is a typical scatter plot you can think of each dot as being an individual so if we looked at this individual. We can see that this person studied for 14 hours. And got a test score of 62 and this individual study for almost 6 hours and got a test score of 30 you might have noticed that the explanatory variable is always plotted on the x axis and the response.

Variable is always plotted on the y axis for this reason. The explanatory variable is denoted as x. And the response variable is denoted as y. You can also think of the explanatory variable as being the independent variable and the response variable being the dependent variable note.

That it is possible to not have an explanatory variable and a response variable for example. The number of points scored during a football game..

Versus. The number of points scored during a basketball game. These are two unrelated events and one variable does not precisely explain the other if there are no explanatory or response variables. Then it doesn t matter where you plot each variable on the graph.

This is commonly seen when trying to compare two unrelated variables or events before we talk about correlation i d like to point out that when determining correlation explanatory and response variables are not necessary now correlation is denoted as r and it tells you about the direction and strength of a linear relationship shared between two quantitative variables correlation can be expressed using scatter plots so let s talk about how direction and strength is measured by correlation will talk about direction first correlation tells us about the direction or slope of a set of data. So it tells us if a data set has an upward slope or a downward slope. If we have an upward slope. We can say that r is positive if we have a downward slope.

Then r is negative. If we have an upward slope. And the data points. Follow a perfect straight line.

Then r is equal to positive one and this is called a perfect positive correlation in contrast. If we have a downward slope. And the data points. Follow a perfect straight line.

Then r is equal to negative one and this is called a perfect negative correlation in both of these cases. We have a perfect linear relationship. Correlation measures. The strength of this linear relationship.

We can actually notice a pattern about how correlation measures strength. We saw that r has values between positive 1 and negative..

1. And we saw that the strength of the linear relationship increased as arca hosts a positive 1 or negative. 1. So when r is equal to 0.

This means that there is no correlation in other words. There is no linear relationship whatsoever. We can see that as r gets closer and closer to positive. 1.

The linear relationship gets stronger and as r gets closer and closer to negative 1. The linear relationship also gets stronger so how do you calculate correlation correlation can be calculated using this formula. It seems like a complicated formula. But it s easier than it looks so suppose a teacher wants to determine the correlation between the number of hours spent studying and test scores to do this he would first have to gather some data.

I will assign the number of hours spent studying as x. And i will assign the test scores to be y. Remember that correlation doesn t care about explanatory or response variables. So it didn t matter how i assign these variables.

Because i would end up with the same value of r. When determining correlation. It s a good idea to make a table to help you with your calculations. This table corresponds to the formula specifically this part of the formula.

So the first step is to calculate the means for the x values and the y values. Which you should already know how to do then we will subtract each x..

Value from. X bar so we will. Have 13. Minus.

125 which is equal to. 05 for the. Second row we will have 15. Minus.

125 which is equal to 25. We will do this for each x value for the y values we would have 53 minus 68. Which is equal to negative 15. We are basically doing the same process for each y value for the next step.

We will have to multiply each of these terms. Together so we will have. 05 times. Negative.

15 and for the second. Row we will have 25 times. 1. And so on the next step is to add up all these products together and that gives us 821.

We can now plug this value into the formula and since we added. 6 products and will be equal to 6 next we need to calculate the standard deviations for each variable and you should already know how to do this at this point..

We have all the ingredients we need for the formula. So we can plug in sx and sy into the formula and when we simplify this we get our value of. R which is equal to 06. O2.

So if we plotted our data. It would look like this we determine that r is equal to positive point six and this makes sense. Because each data value seems to follow an upward direction. Be careful when trying to interpret correlation by just looking at the scatter plot.

When comparing these graphs we might think that the scatter plot on the right has a higher r value. Because the data points are closer together. And it looks like it visually displays a stronger linear relationship. Unless r is equal to plus or minus.

One it s really hard to determine the value of r. Just by using our eyes. Both of these graphs are actually the same and they have the same value of r. They were just made using different scales.

And this is why graphs can deceive us using the notion that numbers do not lie is applicable when ” ..

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