STAT5002 Chap.4 Scatter Plots and the Correlation Coefficient
Scatter Plots and the Correlation Coefficient
This chapter is about pairs of measurements taken on the same individuals, such as height and weight, or hours of practice and a test score. The first step with bivariate data is always a scatter plot, which shows the direction of any pattern, how tightly the points cluster, and whether the pattern is a line at all.
The plot is then summarised by five numbers: the mean and SD of each variable, and the correlation coefficient r. You will learn r as the average of the products of the two variables in standard units. Splitting the plot into four quadrants at the point of averages explains the sign: points to the top right and bottom left add positive products, the other two quadrants add negative ones.
The value always lies between −1 and 1, and the course formula gives the same r whether you divide by n with population SDs or by n − 1 with sample SDs. In R the whole calculation is cor(x, y), and because the formula treats the two variables alike, cor(x, y) and cor(y, x) always agree. Getting the sign right is mostly a matter of checking it against the slope you can see.
The chapter then compares what changes r and what does not. Shifting a variable, multiplying it by a positive number and swapping x with y all leave r unchanged, while a negative multiplier flips its sign.
Finally it collects the warnings that come up in exam questions: r = 0.8 does not mean 80% of points are near the line, one outlier can move r a long way, a curved relationship can be missed entirely, correlations of group averages are inflated, and association on its own never proves causation because a confounding variable may drive both.
What this chapter covers
- 01
Bivariate data and the scatter plot
- 02
Five summaries: two means, two SDs and r
- 03
Correlation as the average product of standard units
- 04
Quadrants at the point of averages and the sign of r
- 05
Interpreting the size and sign of r
- 06
Shift, scale and swap invariance of r
- 07
Negative scaling and the sign flip
- 08
Common misreadings of an r value
- 09
Outliers and non-linear patterns that r misses
- 10
Ecological correlation and group means
- 11
Association, causation and confounding variables
Worked example · free
Training distance and 5 km race time
- +1Summaries, dividing by n: mean x = 26, SD x = 10.677; mean y = 27, SD y = 3.033.
- +1Standard units: x gives −1.50, −0.56, −0.09, 0.84, 1.31; y gives 1.32, 0.66, 0.00, −0.33, −1.65.
- +1Products of matching pairs: −1.976, −0.371, 0.000, −0.278, −2.161. Their sum is −4.786.
- +1Average: r = −4.786/5 = −0.957, the same value cor(x, y) returns.
Key terms
- Bivariate data
- Data made of pairs of values (x, y) recorded on the same individuals, usually displayed in a scatter plot.
- Correlation coefficient
- A number r between −1 and 1 that measures the direction and strength of the straight-line association between two variables, computed in R with cor(x, y).
- Point of averages
- The point (mean of x, mean of y) at the centre of a scatter plot, where lines split the plot into four quadrants.
- Standard unit
- A value's distance from its variable's mean in SDs; the correlation coefficient is the average product of the x and y standard units.
- Ecological correlation
- A correlation computed from group means rather than individuals, which tends to be much stronger than the individual-level value.
- Confounding variable
- A third variable linked to both variables being studied, which can create or hide an association and mislead about cause and effect.
Scatter Plots and the Correlation Coefficient FAQ
Does a correlation of 0 mean there is no relationship between x and y?
No. A value of 0 only rules out a linear association. Points that follow a symmetric arch or U shape can have r equal to zero while y is completely determined by x, which is why the scatter plot must always come first.
What happens to r if I change the units of one variable?
Nothing, as long as the change is a shift or a multiplication by a positive number, such as Celsius to Fahrenheit or hours to minutes. Standard units are unchanged by these operations, so the products and their average stay the same.
Is r = 0.8 twice as strong as r = 0.4?
Not in any precise sense. The larger value signals a stronger linear association, but it does not mean the points are twice as tightly packed, and it does not mean 80% of points lie near the line.
Why can one outlier change the correlation so much?
Each point contributes the product of its two standard units. A point far from both means, in an opposite-sign quadrant, produces one very large product that can outweigh many ordinary points and pull r sharply towards zero or past it.
Can two data sets with the same r look completely different?
Yes. Data sets can share the same two means, the same two SDs and the same r while one is a straight-line cloud, another a smooth curve and another a line with one stray point. Only the scatter plot separates them, so describe the plot alongside the number.
Exam move
Make a small data set of five or six pairs and compute r by hand through standard units, then confirm it with cor() in R. Repeat once dividing by n with population SDs and once with sample SDs and n − 1 so you see the answer does not change.
Next, practise invariance questions without a calculator: for each transformation (adding a constant, multiplying by a positive or a negative number, swapping x and y), write whether r stays the same, flips sign or changes in size. Sketch one example for each warning: an outlier, an arch with r near zero, a rising curve with r near 1, and group means versus individuals.
Then explain in one sentence, using quadrants, why each picture gives the r it does. For every causal claim you meet in a question, name a plausible confounding variable before writing your answer.
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