🔗 Correlation

1. The big idea — two questions at once

A correlation describes how two quantitative variables move together. It compresses a whole cloud of data points into a single number — the correlation coefficient, written r — that answers two separate questions at the same time:

The version you'll use almost always is Pearson's r (full name: the Pearson product-moment correlation coefficient). It's built to detect straight-line patterns — a steady "as one goes up, the other tends to go up (or down)." Keep that word straight-line in mind: it's the source of r's biggest blind spot, which we'll meet later.

2. The coefficient r — direction (the sign)

r is a number that always lands between −1 and +1. It can never be 1.4, never −2 — if your arithmetic ever lands outside that range, you made a mistake. The sign tells you the direction of the relationship.

• Positive (+): the two go up together. More study hours, higher GPA. As x rises, y tends to rise.
• Negative (−): as one goes up, the other goes down. More sleep, less stress. As x rises, y tends to fall.
• Zero (≈ 0): no straight-line link at all. Knowing x tells you nothing about y.

3. The coefficient r — strength (the size)

The size of r — its distance from zero, ignoring the sign for a moment — tells you how tight the relationship is. Close to ±1, the dots hug a straight line: strong and dependable, so knowing one score lets you predict the other well. Close to 0, the dots are a scattered mess: weak or no relationship, and one score tells you almost nothing about the other.

There's no law of nature that sets the cutoffs, but psychologists lean on a rough convention (often credited to Cohen) for talking about effect size. Treat it as a guide, not gospel — and read it on the absolute value |r|, since direction doesn't change strength:

So r = +.52 is a strong, positive relationship. r = −.46 is a strong, negative one — the minus is just direction. And r = −.02 is basically nothing: knowing x leaves you no better off at guessing y. In psychology, where humans are messy and many causes pile up, even a "moderate" r of .3 is a genuinely useful finding — don't dismiss it for not being .9.

4. Scatterplots — seeing the relationship

A scatterplot is the picture behind r. It puts one dot per person, and each dot is that person's two scores: x across the bottom, y up the side. The shape of the cloud is the correlation — r is simply that cloud boiled down to one number.

• Positive → the cloud climbs uphill ↗ from left to right.
• Negative → the cloud slides downhill ↘ from left to right.
• None → a shapeless blob with no tilt; you can't draw an honest trend line through it.

The tighter the dots cling to an imaginary line, the larger |r|; the more they spray out, the closer to 0. Here are three clouds, each with the same dozen people, to calibrate your eye:

5. Worked example — estimating r by eye on OUR data

You won't always have a calculator handy — and on exam day you'll want to sanity-check whatever number Sheets hands you. So let's reason our way to an estimate from the picture alone. Open the SJNY Student Life dataset. We want to know: does social media go with loneliness? That's social_media_hours (column H) and loneliness (column L). Plot the cloud, then read it step by step.

A few more pairs from our dataset, already computed for you — cover the right column and try to predict each from its description before peeking:

6. The most important caveat — correlation ≠ causation

Here is the single most important — and most ignored — lesson in this whole course: a correlation cannot prove that one thing causes the other. Two variables can ride together for reasons that have nothing to do with one steering the other. There are two classic escape hatches, and a correlation alone can't rule either out.

The third-variable problem. Across a summer, ice-cream sales and drownings rise and fall together — a strong positive correlation. Does ice cream drown people? Of course not. A hidden third variable — hot weather — drives both. More heat means more ice cream and more swimming. The two are linked, but neither causes the other; a lurking variable pulls both strings at once.

The directionality problem. Even when two things are causally linked, a correlation can't tell you which way the arrow points. If phone time and loneliness travel together, maybe phones make people lonely — or maybe lonely people reach for the phone. Same r either way; the number is mute on direction.

Now turn the lesson on our own headline finding. We found r = +.52 between phone time and loneliness (a pattern echoed in real research; Primack et al., 2017). But ask three honest questions:

• Does the phone cause loneliness?
• Do lonely people reach for the phone more? (directionality)
• Does something else — anxiety, or being new on a commuter campus — drive both? (third variable)

7. Two traps that fool r — restricted range and outliers

Even when you remember not to claim causation, r itself can mislead you about the strength of a relationship. Two situations do most of the damage.

Restriction of range. Pearson's r measures how scores vary together, so it needs real variation to work with. If your sample has been squeezed into a narrow slice — only top students, only people who already use their phones a lot — the correlation can collapse toward zero even when a strong relationship exists in the full population. The classic case: SAT scores barely predict college grades at a highly selective school, because everyone admitted already scored high. Chop off the low end and the uphill tilt flattens out. So a small r can mean "no relationship" — or it can mean "I only looked at a narrow band." Always ask whether your sample covers the full range.

Outliers. Because r is built from means and squared distances, a single extreme point can drag it around dramatically — especially in a small sample. One person who is sky-high on x and y can manufacture a positive correlation out of an otherwise shapeless blob; one oddball in the corner can wipe out a real one. This is exactly why you plot the data first: the scatterplot reveals a lone dot living far from the pack that the bare number would never confess. Never report an r you haven't looked at.

And one structural limit worth repeating: r only catches straight-line patterns. A strongly curved relationship — say, arousal and performance, which rises then falls (an upside-down U) — can post an r near 0 even though the two are tightly linked. Near-zero r means "no line," not "no relationship of any kind." Once again: plot first.

🐮 Is meat male?

Correlation lets us test surprising ideas. Researchers have long suspected a link between traditional masculinity and meat-eating — that red meat carries a "manly" cultural charge (Rozin et al., 2012). Our dataset lets you check it: masculinity against meat_commitment gives r = +.35, a moderate positive relationship. Higher masculinity scores go with stronger commitment to eating meat. It's a real pattern in the data — a moderate-strength uphill cloud — though, as Section 6 warns, a correlation alone can't tell us why. Does endorsing masculine norms push someone toward meat, does a meat-heavy upbringing shape how masculine someone feels, or does some third thread — region, family, peer culture — weave through both? r = +.35 flags the pattern as worth studying; it doesn't close the case.

💻 In Google Sheets

You won't compute r by hand — Sheets does it with one function. Give it two equal-length columns and it returns Pearson's r:

=CORREL(H2:H201, L2:L201)            → r between social media and loneliness  (+.52)
=CORREL(I2:I201, M2:M201)            → r between exercise and wellbeing

The two ranges must be the same length and lined up row-for-row, so that row 2 is the same person in both columns. (=PEARSON does the identical job, if you ever see it.) To see the cloud behind the number — which you should, every time — highlight the two columns, then Insert ▸ Chart, and pick Scatter in the Chart editor. Looking at the picture is how you catch the outliers and curves that r quietly hides.

Key terms