A scatter diagram shows a clear relationship between two variables. What two things must you say when you describe the correlation?

Whether it is positive or negative , and whether it is strong or weak . Positive correlation means both variables increase together, and negative means one increases while the other decreases. Strong correlation means the points lie close to a straight line, and perfect linear correlation would put every one of them exactly on it.

True or False? If two variables show strong correlation, a change in one of them must be causing the change in the other.

False. Correlation does not imply causation. A causal relationship is one where a change in the first variable genuinely does cause the change in the second, and correlation on its own is not evidence that there is one. Temperature and ice cream sales at a park plausibly are causal, but global temperatures and the number of monkeys kept as pets in the UK could correlate with no connection between them whatsoever.

How is an outlier spotted on a scatter diagram, and how does that differ from finding one in a single set of data?

It is a point that does not fit the pattern made by the rest of the points, judged against the trend rather than by any formula. That is unlike the quartile and standard-deviation tests used on a single variable, because a point here is unusual in the combination of its two values rather than in either value on its own.

International ASMathsEdexcelStatistics 1FlashcardsData Presentation & InterpretationCorrelation & Regression

Correlation & Regression (Edexcel International AS Maths: Statistics 1): Flashcards

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FrontScatter Diagrams
In a scatter diagram, which variable goes on the $x$ -axis, and what is it called?
BackScatter Diagrams
The explanatory variable, also called the independent variable, which is the one that can be controlled during the data collection.
The variable that is measured or discovered as a result goes on the $y$ -axis, and is called the response or dependent variable.
For question packs completed against exam score, the number of packs is what the student chooses, so that is the explanatory variable.
BackScatter Diagrams
False.
Correlation does not imply causation. A causal relationship is one where a change in the first variable genuinely does cause the change in the second, and correlation on its own is not evidence that there is one.
Temperature and ice cream sales at a park plausibly are causal, but global temperatures and the number of monkeys kept as pets in the UK could correlate with no connection between them whatsoever.

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Cards in this collection (20)

In a scatter diagram, which variable goes on the $x$ -axis, and what is it called?
The explanatory variable, also called the independent variable, which is the one that can be controlled during the data collection.
The variable that is measured or discovered as a result goes on the $y$ -axis, and is called the response or dependent variable.
For question packs completed against exam score, the number of packs is what the student chooses, so that is the explanatory variable.
A scatter diagram shows a clear relationship between two variables. What two things must you say when you describe the correlation?
Whether it is positive or negative, and whether it is strong or weak.
Positive correlation means both variables increase together, and negative means one increases while the other decreases.
Strong correlation means the points lie close to a straight line, and perfect linear correlation would put every one of them exactly on it.
True or False?
If two variables show strong correlation, a change in one of them must be causing the change in the other.
False.
Correlation does not imply causation. A causal relationship is one where a change in the first variable genuinely does cause the change in the second, and correlation on its own is not evidence that there is one.
Temperature and ice cream sales at a park plausibly are causal, but global temperatures and the number of monkeys kept as pets in the UK could correlate with no connection between them whatsoever.
How is an outlier spotted on a scatter diagram, and how does that differ from finding one in a single set of data?
It is a point that does not fit the pattern made by the rest of the points, judged against the trend rather than by any formula.
That is unlike the quartile and standard-deviation tests used on a single variable, because a point here is unusual in the combination of its two values rather than in either value on its own.
Define the least squares regression line.
It is the line of best fit that makes the sum of the squares of the gaps between the line and the data points as small as possible.
Measuring those gaps vertically gives the regression line of $y$ on $x$ , and measuring them horizontally gives the regression line of $x$ on $y$ .
The $y$ on $x$ line is the one used at this level, and it is written $y = a + b x$ .
A regression line of $y$ on $x$ comes out as $y = 18 + 1.3 x$ . What do the 18 and the 1.3 each tell you?
The 1.3 is the gradient, so $y$ rises by 1.3 for every increase of one unit in $x$ .
The 18 is the $y$ -intercept, which is the value $y$ takes when $x$ is zero.
Because the gradient is positive, the data has positive correlation, and a regression line can tell you that even when there is no scatter diagram to look at.
A regression line was worked out from data in which $x$ ran from 10 to 60. Why should it not be used to predict $y$ when $x$ is 80?
Because 80 lies outside the range of the data the line came from, making the prediction an extrapolation, which is much less reliable.
Nothing in the data says the linear relationship carries on past the point where it was actually observed, and it may well not.
Predicting from a value inside the range is called interpolation and is the reliable case, and a larger original sample makes it more reliable still.
Define the product moment correlation coefficient.
The product moment correlation coefficient, written $r$ , puts a numerical value on the linear correlation of bivariate data.
It can take any value in the range $- 1 \leq r \leq 1$ .
The word linear matters: $r$ measures how closely the data lies to a straight line, and says nothing about any other kind of relationship between the variables.
What do the values $r = 0$ , $r = 1$ and $r = - 0.9$ each tell you about a set of bivariate data?
$r = 0$ means there is no linear correlation at all.
$r = 1$ means perfect positive correlation, with every point lying exactly on one straight line of positive gradient.
$r = - 0.9$ means strong negative correlation: negative because $r$ is negative, and strong because it is close to $- 1$ .
True or False?
The steeper the gradient of the regression line, the stronger the correlation.
False.
The two are separate things: $r$ measures how tightly the points cluster around the line, while the gradient measures how steeply that line rises.
A set of points lying exactly on a very gently sloping line has $r = 1$ , which is as strong as correlation can possibly get.
Which quantities do you need in order to calculate $r$ , and how much of that is given to you?
You need the three summary statistics $S_{x x}$ , $S_{y y}$ and $S_{x y}$ , and then
$r = \frac{S_{x y}}{\sqrt{S_{x x} S_{y y}}}$
All four of those formulae are given in the formula booklet, so the work is in getting $Σ x$ , $Σ y$ , $Σ x^{2}$ , $Σ y^{2}$ and $Σ x y$ out of the data and substituting them.
$S_{x y}$ is the one that connects the two variables, while the other two each involve a single variable only.
How are the gradient and the intercept of the regression line $y = a + b x$ calculated, and does the order matter?
The gradient is $b = \frac{S_{x y}}{S_{x x}}$ and the intercept is $a = \bar{y} - b \bar{x}$ , both given in the formula booklet.
The order does matter: $b$ has to be found first, because the formula for $a$ has $b$ in it.
True or False?
The point $(\bar{x}, \bar{y})$ always lies on the regression line of $y$ on $x$ .
True.
Whatever the data, the regression line passes through the point whose coordinates are the two means.
That is exactly where the formula $a = \bar{y} - b \bar{x}$ comes from: it is $y = a + b x$ rearranged for $a$ , applied at that point.
A question asks for the regression line of $d$ on $p$ . Which of the two variables plays the part of $y$ in $y = a + b x$ ?
$d$ does: in the phrase the regression line of $d$ on $p$ , the variable named first is the dependent one and stands in for $y$ , and the one named second is the independent one and stands in for $x$ .
So the equation comes out as $d = a + b p$ , with $b = \frac{S_{p d}}{S_{p p}}$ .
Taking them the wrong way round changes which summary statistic sits in the denominator, and gives a different line altogether.
Why is $r$ usually calculated before the equation of a regression line is worked out?
Because a regression line assumes a linear model, and $r$ is what tells you whether one fits: a value close to $1$ or $- 1$ suggests that it does.
Fitting a straight line to data whose $r$ is near zero produces an equation that describes nothing real.
The formulae will hand you numbers either way, which is why that check has to be made deliberately rather than assumed.
True or False?
Coding a set of bivariate data changes the value of the product moment correlation coefficient.
False.
Coding leaves $r$ completely unchanged, so $r_{X Y} = r_{x y}$ .
The multipliers do change the summary statistics, giving $S_{X X} = p^{2} S_{x x}$ and $S_{X Y} = p m S_{x y}$ , but they cancel out again as soon as those are substituted into the formula for $r$ .
Data is coded using $X = p x + q$ and $Y = m y + n$ . What happens to the summary statistics?
Each one is multiplied by the coding multipliers only:
$S_{X X} = p^{2} S_{x x}, S_{Y Y} = m^{2} S_{y y}, S_{X Y} = p m S_{x y}$
The added constants $q$ and $n$ have no effect at all, because these statistics are built from deviations from the mean and sliding every value along leaves those deviations exactly as they were.
$S_{x x}$ picks up a factor of $p$ twice because both of its $x$ s are coded, while $S_{x y}$ picks up one $p$ and one $m$ .
Salaries $s$ and lengths of service $l$ are coded, and the correlation coefficient between the coded variables is 0.739. What is the correlation coefficient between $s$ and $l$ ?
0.739, because coding does not change the product moment correlation coefficient, so the coded value is already the answer and there is nothing to convert back.
The contrast with the regression line in the same question is what catches people out, since that one does have to be converted.
Coding leaves $r$ unchanged. Why does it not leave the equation of the regression line unchanged?
Because $r$ is a ratio in which the multipliers cancel, whereas the gradient of the regression line keeps one: $B = \frac{S_{X Y}}{S_{X X}} = \frac{p m}{p^{2}} \times \frac{S_{x y}}{S_{x x}}$ , which still carries a factor of $\frac{m}{p}$ .
The line is also an equation in the coded variables themselves, so the added constants $q$ and $n$ shift it as well.
A regression line found from coded data therefore has to be converted back before it describes the original data.
Data is coded with $X = p x + q$ and $Y = m y + n$ , and the regression line of $Y$ on $X$ is found to be $Y = A + B X$ . How do you get the regression line of $y$ on $x$ ?
Substitute the coding into it, replacing $X$ and $Y$ by what they actually stand for:
$m y + n = A + B (p x + q)$
Then rearrange that into the form $y = a + b x$ , by expanding the bracket, subtracting $n$ and dividing through by $m$ .
There is no formula to remember here, which is rather the point: write the coding down and do the algebra.

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