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Direction, form and strength read off a plot; what the correlation $r$ measures, what it cannot see, and the share of the variation $r^2$ accounts for.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you can describe the relationship between two variables in words — direction, form, strength and any point standing apart — and then measure it with $r$ where the form is straight. You can say why $r$ has no units, why it does not matter which variable you call $x$, and why a strong curved relationship can give an $r$ near zero, which is the reason the plot is looked at before the number is quoted.
You can describe one variable — its shape, centre and spread. Two variables raise a new question: do they move together, and how tightly? A scatterplot is the picture of that question, and $r$ is one number that answers part of it.
Scatterplot: one point per case, with the explanatory variable across and the response variable up.
Direction: positive if the cloud rises to the right, negative if it falls.
Form: straight, curved, or nothing.
Strength: how closely the points hug that form.
Correlation $r$: a number between $-1$ and $1$ measuring the strength and direction of a linear relationship.
$r^2$: the share of the variation in $y$ that the line accounts for.
A scatterplot shows one point per case. Describe it in four words — direction, form, strength, and any unusual points — and then, if the form is straight, measure it.
The correlation $r$ lies between $-1$ and $1$. Its sign is the direction; its size is the strength. It is computed from standardised values, $\frac{x - \bar{x}}{s_x}$ and $\frac{y - \bar{y}}{s_y}$, which is the source of three facts worth holding on to: $r$ has no units, so changing pounds to euros leaves it alone; $r$ is symmetric, so it does not matter which variable you call $x$; and $r$ measures linear strength only, so a strong curve can give $r \approx 0$.
$r^2$ is the more useful number in a report: it is the share of the variation in $y$ that the line accounts for. An $r$ of $0.8$ sounds like most of the story and is $r^2 = 0.64$ — just under two thirds, with a third of the variation still unexplained.
One point far from the rest can dominate all of this. A point that is extreme in $x$ is influential: removing it swings the line and moves $r$ a long way. Always look at the plot before quoting the number.
Another way: picture
Picture the same cloud of points stretched vertically. Every point is twice as far up as before, the line is twice as steep — and the cloud hugs it exactly as closely as it did, so $r$ has not changed at all. Strength and steepness are different questions.
Another way: steps
"$r$ near zero means unrelated." It means no straight-line relationship. A perfect hill or valley can have $r$ near zero while every point sits exactly on a curve.
"A larger $r$ means a steeper line." The two are independent. $r$ says how tightly the points hug the line; the slope says how steep it is. Doubling every $y$ doubles the slope and leaves $r$ untouched.
"A strong correlation shows a cause." It shows that the two move together. Everything in the study-design lesson applies here, and applies most strongly when the correlation is impressive.
Negative: older cars cost less, so the cloud falls.
The sign.
$|{-0.9}| = 0.9$, so the points lie close to a straight line: strong.
The size.
$r^2 = 0.81$: age accounts for about $81\%$ of the variation in price, leaving $19\%$ to mileage, condition and everything else.
Square it to report it.
The plot rises and then falls: too much fertiliser harms the crop.
Look before quoting.
The rising half and the falling half cancel, so $r$ comes out near zero.
$r$ sees straight lines only.
Reporting 'no relationship' would be badly wrong; there is a strong one, and it is a curve.
Positive and moderate: the cloud rises but is fairly loose.
$r^2 = 0.36$, so study time accounts for about $36\%$ of the variation in marks — and $64\%$ is something else.
And since nobody assigned study hours, none of this shows that studying more causes a higher mark.
A scatterplot puts the daily temperature on the horizontal axis and the amount of heating gas a house burns on the vertical axis. What direction does the association have?
Two scatterplots have correlations $r = -0.8$ and $r = 0.2$. Which shows the stronger linear relationship?
Heights in inches and weights in pounds have correlation $r = 0.6$. The heights are converted to centimetres. What is $r$ now?
A scatterplot of a plant's height plotted against the amount of fertiliser, which helps at first and harms in large doses shows a hill that rises then falls, and the correlation is $r = 0.02$. What does that tell you?
A least-squares line has $r = 0.5$. What percentage of the variation in $y$ does it account for?
answer
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A scatterplot puts the daily temperature on the horizontal axis and the amount of heating gas a house burns on the vertical axis. What direction does the association have?
You can read a scatterplot and interpret $r$ and $r^2$. Without looking: what does an $r$ near zero not rule out, and what share of the variation does an $r$ of $0.8$ account for?
8. Your turn: hours of study against exam mark, $r = 0.6$, step 3