[Auto-generated transcript. Edits may have been applied for clarity.]
Okay. What is a correlation. Um, it really is a measure of the relationship between two continuous or numeric variables.

So by numeric or continuous we mean variables that can be or that are measured on the ordinal or the rational scale.

So examples are height and weight. You can put a number of that.

We can measure somebody's weight. And that number means something in relation to other numbers on that scale.

Um, so does height relate to weight or is stamina absence related to WBA score?

Or does the number of cats that you own relate to how violent you are?

Now, some of these examples may sound a little silly, but all of these questions can be tested using correlation analysis.

So in other words, if something happens to x, what happens to y?

That is what you get an answer to. If you do a correlation analysis. Okay.

The scatterplot. So we assess the relationship between two variables visually via a scatter plot or a scatter gram.

You can see what that looks like on the slide here.

So each single point in this graph indicates the two values for one individual on the two variables of interest.

So this participant here, for instance, at the end of this arrow,

um scores about 23 on the variable on the X-axis and about 60 on the, on the variable on the y axis.

Now the first step in the correlation analysis is constructing and interpreting a scatter plot.

Now, two aspects of the relationship between our two variables of interest are reflected in the scatter plot.

So the first one is the strength of the relationship, and the second one is the direction of the relationship.

So here we have six different scatter plots.

And so the strength of the correlation can be visually interpreted from the

scatter plot by looking at the proximity of the scores to the line of best fit.

So how closely these points are to this line,

and the direction of the correlation can be visually interpreted from the scatter plot by looking at the direction of the line.

Does it go up as in here, or does it go down as in here?

So let's start maybe with the example here at the bottom right of this slide.

So here the the scores are very close to the line of best fit which suggests it is a strong correlation.

And the line starts kind of in the top left and goes down to the bottom right, which suggests that the correlation is negative.

So another example. Um, here the graph at the top right shows scores that fall further away from the line of best fit.

So there is more scatter around this line.

But overall you can still see that it that there is a is a positive relationship because it's kind of starts at the top,

the bottom left and it goes to the top right. So if x increases so does y.

So that's why we call it a positive relationship.

Now the graph in the top right hand middle shows scores that fall even further away from the line of best fit,

which is indicative of a weaker relationship. And here, um, the line goes down again indicative of a negative relationship.

So weak negative relationship would look like this.

Now finally, the graph in the top row here on the left shows a situation where there is a null correlation.

It's it's basically just a cloud of dots without a clear direction.

So the use of of scatter plots is a really good example of the importance of visualising your data.

So graphs are not just some end product or a pretty addition to your report.

They allow us to familiarise ourselves with the data and identify the distribution of the data and the initial relationships.

Um, and they also actually allow us to identify outliers.

And we'll talk a bit more about that next week. Okay.

The statistic that we use to quantify a linear relationship between two variables is called Pearson's product moment correlation coefficients,

or Pearson's are. So this is a number that ranges from -1 to 1.

So it can fall anywhere between -1 and 1.

And the same two aspects that are visible in the scatter plot are also reflected in the correlation coefficient.

So the value of the correlation coefficient, kind of ignoring the plus minus sign, reflects the strength of the relationship.

So the closer to 1 or -1, the stronger the correlation.

The sign reflects the direction of the relationship.

So a positive coefficient reflects a positive correlation and a negative coefficient reflects a negative correlation.

Now a coefficient close to zero reflects a null correlation.

Now let's look at some extreme examples of different types of correlation.

Here we have a positive correlation. So when x increases y also increases.

So the line starts at the bottom left of the plot and goes up to the top right corner.

So this suggests that a higher score on one on x.

So in this case height um is associated with a higher score of weight on y axis.

This is actually a perfect positive correlation because all the dots fall exactly on the line of best fit.

And that's why R is one. This doesn't usually happen in real life.

And we'll look at some more realistic, uh, scatter plot.

Later on in the lab. Another type of correlation is the negative correlation.

So a negative correlation is one x increases and y decreases.

So the line starts in the top left corner of the plot and goes down to the bottom right.

This suggests that a higher score on the variable on the x axis is associated with a lower score on the variable on the y axis.

So here you would say a seminar since increases w ba score decreases.

So seminar absence is negatively correlated with WBA score.

And here again we have a perfect negative correlation with an R of minus one.

So the perfect positive and negative correlations are really the two extremes.

Now when the correlation is null So R is zero or close to zero.

An increase in the variable on the x axis. Doesn't really have.

It's not associated with a consistent change in the variable on the y axis.

So as the number of cats owned increases, the level of violence doesn't change in a kind of systematic way.

Now here we have those scatter plots again. And this time you can also see the correlation coefficients that characterise these relationships.

So the strong negative correlation at the bottom right here has an R of minus 0.99.

And the data in the graph in the top row on the right here has an R of 0.5.

And this one and r -0.3.

So notice negative sign which indicates that the correlation is negative.

Now you might have heard of the term effect size, and effect size indicates how relevant an effect is.

So Pearson's r is an effect size in itself.

And there is a rule of thumb which is useful when describing your correlation or the results of your correlation analysis.

So the rule of thumb suggests that correlations bigger than 0.5 are referred to as as large or strong um.

Correlations between 0.3 and 0.5 are referred to as medium or moderate, and correlations between 0.1 and 0.3 are referred to as a small or weak.

Now, as is often the case with rules of thumb, there are different opinions about how useful they are.

But I think when you're starting out in psychology as you are, it is a useful way of describing correlations.

And so these indications, uh, these rules of bumps obviously apply to both the negative and positive correlation coefficients.

Now we've looked at the scatter plot and at the statistic Pearson's. Ah.

Let's now have a look at how you actually derive this correlation coefficient.

So if you think back to statistics last term psych 1121 um, you were taught how to derive a measure of variance.

Um, so of how much variation in scores there is around a mean in a sample?

Um, to do so, you use this formula. Um, so you have the score of a participant X.

From that, you subtract the mean score on that variable across the sample, and you square that number.

You do that for every participant in the sample. You add up all those totals and you divide by um, that's the sigma sign.

And then you divide by the number of participants in the sample.

And. So that is how you compute the variance of the sample.

And related to variance is something called covariance which we need for calculating the correlation coefficient.

The covariance is the extent to which two variables vary together.

So instead of multiplying the scores by by itself as you do here, by squaring it, you multiply it by the score on the y variable.

So we'll multiply it with the other variable which gives us kind of this this formula.

So let's have a look at that. So to compute the covariance, you take a participant score on variable x.

Subtract the mean of x across the sample and you do the same for y.

So you have the stats participant score for y, and you subtract the mean for y across the sample.

You multiply these two numbers and you do that for each participant.

And then you add up all those numbers. The sigma sign here.

And then you divide the outcome by n the number of participants.

That's n minus one. Okay. Let's look at an example.

So here we have some data. Um so here we have data for 15 participants here in these rows um are visible.

So and we have data on four different variables. So here in the columns the first column we have job performance scores.

The second column we have IQ scores. In the third column we have motivation scores.

And in the fourth column we have scores for social support.

So participant one scores 85 on job performance, 109 on IQ, 89 on motivation and 73 on social support.

And down here we have calculated earlier and the means and standard deviations for two of these variables.

So if you want to compute the covariance between job performance and motivation you would do the following.

We have the formula here at the top. Um so for participant one.

Participant one scores 85 um job performance.

Our first variable from that we subtract the mean across the sample of job performance, which is 78.

We do the same for Y. So participant one scores 89 on motivation.

And from that we subtract the mean on motivation across the sample.

That is 67. So if we do the same thing we end up with a number of 154 for participant one.

And you do you don't do this for each participant, separate exhibit two, etc.,

etc., etc. then you add up all of those totals and you divide by n minus one.

Now this is how you compute covariance by hand.

Of course, we wouldn't usually do it by hand, but it is helpful to kind of understand the principle.

Just to keep in mind, unlike variance that um covariance may have a positive or a negative value because we're not squaring basically.

so it can have a positive value and or a negative value, suggesting the direction of the variance.

So that kind of maps onto the positive and negative correlation coefficients that we looked at earlier.

So our example of covariance for job performance and motivation we get a number of 69.24.

And from this from this covariance we'll derive Pearson's r.

Now you might ask why do we need Pearson's r.

Isn't the covariance enough? Now the thing is that the size of the covariance is affected by the size of variances of the variables in the sample.

So depending on how big that is, um, the covariance could be really big or really small.

And that makes it very difficult to compare covariances across different measures or across different samples.

So the correlation coefficient Pearson's are improved that situation by dividing by the standard deviations.

So the standard deviation, not the thing that you might have remember from last term.

So you could say that Pearson's r is a standardised version of the covariance.

So down here you can see how it works.

So to calculate r Pearson's r we divide the covariance by the standard deviation of x multiplied by the standard deviation for y.

Now if we do that here we have our covariance. Here we have our standard deviations that we've calculated earlier.

And then we end up with an R of 0.63. Now as I mentioned previously, Pearson's r can range from -1 to 1 making it really easy to interpret.

So as I said, the covariance could have any value depending on the variances of the sample that are always is between -1 and 1.

So the important bits to remember from this part are that the covariance tells you to what extent one variable changes,

when the other variable changes, and the correlation coefficient is derived from the covariance by standardising it.

So you end up with a number that falls between -1 and 1.

the proportion of variance in one variable that can be accounted for by the other variable.

So to calculate the coefficient of determination or r squared is what it's also called.

You simply square the correlation coefficient. So because it's squared r squared is always positive.

So it doesn't matter whether a number is positive or negative.

If you square it it's always becomes positive. So r squared or the coefficient of determination is always a positive number.

So in our example we had an R of 0.63. If we square that we get an R squared.

Or coefficient of determination of 0.4. So this tells us that 40% of the variance in job performance is accounted for by variance in motivation,

giving us more information to interpret that relationship.

Okay, we're almost there. Why does correlation not infer causation.

So correlation only tells you whether there is a relationship between two variables.

It doesn't tell you anything about the direction of that relationship,

in the sense that you can't conclude, um, that if X goes up, it causes Y to go up.

It is associated it might be associated with y going up in case of a positive correlation,

but we don't know whether one thing causes the other or the other way around.

The only way to infer causation is to do an experiment in which you manipulate

an independent variable and measure what that does to your dependent variable.

So only in that case, um, can you say that this manipulation caused any variation observed in the dependent variable.

So for example, let's say we want to understand how children learn to read something that I'm interested in.

So in a group of children,

we observed a significant positive correlation between how much time they spend reading every day and their reading proficiency.

So from that correlation, we can tell that there is a relationship,

but we can't tell whether spending more time reading causes reading proficiency to go up

or whether children with higher reading proficiency choose to spend more time reading.

It could be either if we do an experiment.

We can figure this out, right? So imagine we we have a measure of how accurately children read a list of words.

And this is our measure of reading proficiency. That is our dependent variable.

We then divide the sample into two groups. One group continues with normal classroom reading instruction,

and the other group receives an intervention in which they read a book at their reading level together with uh,

with an older, with an older and older child, um, who reads fluently.

So after six weeks, we measure the children's reading proficiency again.

And if the children in the group who received the intervention improved more than

the group of children who received the kind of standard classroom instruction.

Then can we conclude that spending more time reading causes reading proficiency to go up?

Okay, so let's look at another example of kind of a significant positive correlation and why we cannot infer causation.

So here we have a scatter plot with the number of ice creams sold on the x axis here and the number of shark attacks on the y axis here.

So this would be a relatively strong positive correlation as the line goes up and

the individual dots are relatively close to that line and to that line of best fit.

So that suggests that shark attacks are related to ice cream sales.

Now, does that mean that eating ice cream causes you to be attacked by a shark?

No. And in this scenario, it is much more likely that there is a third factor or a third variable that affects both measures.

So hot weather results in more people swimming in the sea.

Which leads to, uh, which makes it more likely that more people are attacked by a shark.

Hot weather also leads to people eating more ice cream.

So there's this third factor, or weather, that influences both shark attacks and ice cream sold.

So the bottom line is you need to be careful in how you interpret it,

a significant correlation and really think about, um, what what a significant correlation means.