Correlation Exercises in R: 20 Practice Problems

Explanation: cor(x, y) returns a single Pearson coefficient when both arguments are numeric vectors. The strong negative value near -0.87 says that heavier cars deliver markedly fewer miles per gallon, which matches physical intuition. If you pass cor(mtcars$wt, mtcars$mpg, method = "pearson") explicitly you get the same number; "pearson" is the default.

Explanation: The use argument controls NA handling. "complete.obs" casts out any row where either input is missing before computing the coefficient; "pairwise.complete.obs" is the matrix-mode equivalent that handles each pair independently. Without use, cor() returns NA whenever any value is missing. A common mistake is calling na.omit() on one column at a time, which misaligns the two vectors and yields a wrong number.

Explanation: Correlation is covariance rescaled by the product of the standard deviations: $r = \mathrm{cov}(x, y) / (s_x s_y)$. That standardization is exactly what makes correlation comparable across pairs and unit systems; covariance changes if you switch horsepower to kilowatts, but correlation does not. When you only care about strength and direction of a linear association, use correlation; covariance matters when you need the unstandardized magnitude (for example, portfolio variance from a covariance matrix).

Explanation: Pearson's formula is the centered cross-product divided by the geometric mean of the centered sums of squares. Re-deriving it once cements two ideas: the numerator is proportional to covariance, and the denominator equals $(n-1) s_x s_y$ if you divide by $n-1$ on both sides (those $n-1$ factors cancel). Real codebases never reimplement this, but writing it once helps you reason about why cor() is invariant under linear rescaling of either variable.

Explanation: Annotating the correlation directly in the title removes the back-and-forth between the chart and a separate stats table. Two small choices matter: se = FALSE keeps the ribbon off (it has nothing to do with $r$ and clutters the plot), and round(r, 2) matches the precision a reader can actually distinguish visually. For an in-panel annotation rather than a title, swap labs(title = ...) for annotate("text", x = 5, y = 30, label = paste("r =", r)).

Explanation: A pairs plot is the fastest way to spot non-linear, monotonic-but-curved, or grouped relationships before you ever compute a coefficient. The diagonal labels each column; the off-diagonal panels show every pairwise scatter. If you see a clear curve (for example hp vs mpg bending), a Pearson coefficient will understate the association and you should switch to Spearman or transform a variable. GGally::ggpairs() adds densities and correlations to the same grid if you prefer the ggplot2 look.

Explanation: hc.order = TRUE reorders rows and columns using hierarchical clustering on the correlation distance, so highly correlated blocks visually cluster together. That clustering is what turns a heatmap from "pretty colors" into a real diagnostic for finding latent factor structure. type = "lower" halves the visual load since the matrix is symmetric. For very wide matrices (20+ variables), turn off lab to avoid label collisions; for narrow ones, keeping the numbers is worth the extra ink.

Explanation: When cor() receives a data frame or matrix, it returns the full symmetric matrix of pairwise correlations. The diagonal is always 1 (every variable correlates perfectly with itself). Use round() purely for display; the underlying matrix keeps full precision. If any column is non-numeric, cor() errors; pre-filter with mtcars |> select(where(is.numeric)) in mixed-type frames.

Explanation: Zeroing the upper triangle and diagonal with NA is the cleanest trick to avoid double-counting symmetric pairs and self-correlations. as.table() then as.data.frame() melts the matrix into long form, which is much easier to filter and sort than a wide square matrix. The pattern generalizes to any pairwise-association screen: collinearity audits, gene-gene co-expression, asset-return clustering.

Explanation: When you pass two data frames to cor(x, y), it returns a rectangular matrix with rows from x and columns from y. That avoids computing within-group correlations you do not care about. The pattern is heavily used in canonical correlation setups, between-block PLS, and any screen where you want to see how one block of variables relates to another (for example, sensor channels vs. operating-condition tags).

Explanation: Pearson measures linear association, so any curvature drags the coefficient below 1 even when the relationship is perfectly monotonic. Spearman correlates the ranks of x and y, and because exponentiation preserves order, ranks agree exactly: Spearman = 1. Switching to Spearman is the fix whenever a scatter plot shows a clean curve rather than a straight line, or when one variable is on a log/exponential scale.

Explanation: Kendall's tau counts concordant minus discordant pairs and divides by the total number of pairs, so it has a direct probabilistic interpretation: if you pick two routines at random, $(1 + \tau)/2$ is the probability the two judges agree on their relative order. With small $n$ and ordinal data (rankings, Likert ratings), Kendall is more robust to ties and outliers than Spearman, and a tau of about 0.86 says the two judges agree on the relative order in roughly 93 percent of pair comparisons.

Explanation: Spearman and Pearson agree closely here because the relationship in iris is roughly linear; Kendall's tau is on a different scale (it counts pair concordances) so it is always closer to zero than Spearman for the same monotonic data, even when both indicate the same direction. Report Pearson when you have ruled out non-linearity and outliers, Spearman when the relationship is monotonic but not linear, and Kendall when ties are common or sample size is small.

Explanation: cor.test() returns an htest object with $estimate, $p.value, $conf.int, and $statistic. Pulling fields by name is more robust than parsing the printed output, especially when you are piping results into a downstream report or dashboard. The vanishingly small p-value here is unsurprising: with $n = 150$ and an estimate near 0.82, the test has overwhelming power. Always look at the estimate alongside the p-value, since a tiny p-value with a tiny estimate just means the sample was large.

Explanation: cor.test() builds the CI by applying Fisher's z transformation, computing the standard CI on the z scale, and back-transforming via $\tanh$. The CI shrinks fast as $n$ grows; a sample of 150 produces a band only about 0.11 wide. Report the CI rather than just the point estimate when you want to communicate uncertainty, especially for moderate sample sizes where a 0.30 correlation might come with a CI that brushes zero.

Explanation: Fisher's transformation $z = \tanh^{-1}(r)$ stabilizes the variance of the sampling distribution of $r$: on the $z$ scale the standard error is approximately $1/\sqrt{n-3}$, regardless of the true correlation. You build the symmetric CI in $z$, then back-transform with $\tanh$ to land in the $[-1, 1]$ scale. This is exactly what cor.test() does internally; running it by hand once cements why the CI is asymmetric in the original scale (wider on the side closer to 0).

Explanation: Partial correlation residualizes both wt and mpg against disp and then correlates the residuals. The raw Pearson $r$ between wt and mpg is about -0.87; the partial $r$ controlling for disp shrinks to about -0.59, which says a meaningful chunk of the apparent weight effect was really displacement working through weight. Equivalently, you could fit lm(wt ~ disp) and lm(mpg ~ disp), correlate the residuals, and reproduce the same number; ppcor::pcor.test() just adds the inference.

Explanation: The percentile bootstrap makes no distributional assumption: you resample the rows (not the columns) to keep $x$ and $y$ paired, recompute $r$ on each resample, and read off quantiles. When the Fisher's z CI and the bootstrap CI roughly agree, you can trust both; when they disagree, the bootstrap is the safer report. For correlations near $\pm 1$ where Fisher's z is most accurate, both will be tight; for small $n$ or heavy-tailed data, the bootstrap is the standard recommendation.

Explanation: Comparing two correlations from independent samples is a Fisher-z test: transform each $r$, divide the difference by $\sqrt{1/(n_1-3) + 1/(n_2-3)}$, and use a normal reference. The two subgroup correlations look numerically different (-0.70 vs -0.89), but the test p-value of about 0.16 says that gap is well within sampling noise given small group sizes (19 and 13). The general lesson: eyeballed differences between subgroup correlations need a test, especially when subgroups are small.

Explanation: Pearson and Spearman both miss this association entirely because the relationship is symmetric (large $|x|$ produces large $y$, regardless of sign). Always look at a scatter plot before declaring "no correlation"; near-zero $r$ often hides U-shapes, V-shapes, or other symmetric patterns. Once you spot the symmetry, correlating $y$ with $|x|$ (or transforming to $\log y$, $\sqrt{y}$, or fitting lm(y ~ poly(x, 2))) recovers the real signal. This is the single most common way correlation analysis goes wrong in practice.