Drawing the one line that lets you predict.
Last unit, correlation told us a relationship exists — social media and wellbeing move together. Nice. But it can't answer the question everyone actually asks: "If a student is on social media 6 hours a day, what's their wellbeing likely to be?" For that you need a line. Regression draws the single best line through your dots so you can plug in one number and predict the other. Same data, one step further — and it's the engine behind weather forecasts, insurance pricing, GPA predictors, and most of the "data science" you'll ever meet.
Correlation gives you a single number, r, that says how tightly two things move together and in which direction. Useful — but it stops there. Regression takes that same relationship and lays a straight line right through the middle of the scatterplot. That line is the whole payoff: once you have it, prediction is just plugging in a number and reading off what the line says. Correlation describes; regression predicts.
| Step | Question it answers | What you get |
|---|---|---|
| Correlation (last unit) | Is there a relationship, and how strong? | One number, r |
| Regression (today) | What's the best line, and what does it predict? | An equation, Ŷ = a + bX |
The two are deeply linked — regression is built directly out of the correlation and the standard deviations from Unit 6. They're the same relationship looked at two ways: correlation is the snapshot, regression is the formula you can actually use.
Picture the scatterplot of wellbeing vs. social media — 200 dots scattered in a cloud. You could draw a hundred different straight lines through that cloud. Which one is "best"? We need a rule, not a vibe.
The rule is this: for any line you draw, every dot sits some distance above or below it — that vertical gap is the line's error for that dot. The best line is the one that makes those errors, taken all together, as small as possible. Because we don't want positives and negatives to cancel out, we square each gap before adding them up — which is exactly why this is called the least-squares line.
The least-squares line is the single straight line that sits closest to all the dots at once — it makes the total of the squared vertical gaps between the dots and the line as small as it can possibly be. You don't compute it by hand; Sheets finds it for you. Just know what it is: the one best line through the data.
Here's the beautiful part — that whole line is captured by just two numbers. The equation is:
Ŷ = a + bXTwo numbers and you can predict. The slope b is the rate: "for every one step right on X, the line rises (or falls) by b." The intercept a is just the line's starting height — its value when X is zero. Get a and b from your data and the line is fully pinned down.
If b = 0.5, then every 1-unit increase in X raises the prediction by half a point. If b = −1.43, every 1-unit increase in X lowers the prediction by 1.43 points — the minus sign means the line slopes down. Read the sign for direction, read the size for steepness.
This is the moment regression earns its keep. Plug an X into the equation and do the arithmetic — that's the whole skill. For our data, predicting wellbeing (M) from social media hours (H), the line comes out to Ŷ = 28.80 − 1.43X. So what wellbeing would we predict for a student on social media 6 hours a day?
Ŷ = 28.80 − 1.43 × 6
= 28.80 − 8.58
= 20.2So we'd predict a wellbeing score of about 20 for that student. Notice we did not need that exact person in our data — we read the prediction straight off the line. That's the power: one equation predicts Y for any X you hand it.
The slope of −1.43 is the headline: each extra hour of daily social media predicts about 1.4 fewer wellbeing points. The minus sign means the line slopes down — more hours, lower predicted wellbeing. Whenever you meet a regression, find the slope and translate it into a plain-English sentence like that one.
No line threads through every dot perfectly. For each student, there's the wellbeing the line predicted (Ŷ) and the wellbeing they actually had (Y). The gap between them is the residual:
residual = actual Y − predicted ŶA residual of 0 means the dot landed exactly on the line — a perfect call. A positive residual means the person scored higher than the line predicted (dot above the line); a negative residual means they scored lower (dot below). Those vertical gaps are precisely what least-squares makes as small as possible — squaring them first so a big miss in either direction counts heavily and positives can't cancel negatives.
The best-fit line is the one with the smallest total squared residual. Square each gap, add them all up, and no other straight line gives a smaller sum. That single idea is the whole engine — the residuals are the errors, and the line is chosen to crush them.
r² (r-squared, the coefficient of determination) tells you what fraction of the variation in Y the line actually explains. The recipe is delightfully simple: square the correlation and you've got it. It's the bridge straight back to Unit 6.
Our wellbeing correlation was r = −.46, so r² = (−.46)² = .22 — about 22% of the differences in students' wellbeing are explained by their social media use. The other 78%? Sleep, money, friends, classes, genetics — everything else regression didn't measure. So the line is real and useful, but social media is one slice of a much bigger pie.
And when we predict loneliness from social media (our headline r = +.52), r² = (.52)² = .27: about 27% of the variance in loneliness is shared with social media use. A few anchors worth memorizing:
| r² | As a percent | What it means |
|---|---|---|
| 0 | 0% | The line explains nothing; X is useless for predicting Y. |
| .22 | 22% | Our wellbeing line — modest but real. |
| .50 | 50% | The line explains half the variation — quite strong. |
| 1.0 | 100% | Every dot is exactly on the line; perfect prediction. |
r² is always between 0 and 1 (0% to 100%) — it can never be negative and never exceed 1, because squaring a correlation (itself between −1 and +1) lands you in that range. Bigger r² means a tighter, more trustworthy line.
Let's walk a clean line end to end. Suppose a tutoring study fits Ŷ = 2.0 + 0.5X, predicting an exam grade improvement (Y, in points) from the number of tutoring sessions attended (X). Predict the improvement at X = 10 sessions, then say what each piece means in plain English. Predict each step before you reveal it.
Now the same thing on our dataset. Open the SJNY Student Life dataset and predict wellbeing (M) from social media hours (H) using three functions — type them in any empty cell:
=SLOPE(M2:M201, H2:H201) → b = −1.43 (the slope)
=INTERCEPT(M2:M201, H2:H201) → a = 28.80 (the intercept)
=RSQ(M2:M201, H2:H201) → r² = .22 (22% explained)It's (y-range, x-range) — the thing you're predicting comes FIRST. Here that's wellbeing (M), then social media (H). Flip them and your slope and intercept come out wrong.
A regression line is only trustworthy inside the range of data you actually measured. Step outside it and you get nonsense. Picture a toddler growing a few inches a year. Draw that line and stretch it forward 30 years — congratulations, you've predicted a 50-foot-tall adult. The line was real for toddlers; it falls apart everywhere else. Anytime you hear "if this trend continues…" run far into the future, that's the joke — the trend almost never continues.
Our wellbeing line was built from students with roughly 0 to 8 hours of social media. Predicting wellbeing for someone at 20 hours a day is extrapolation — outside the data, off the map. And remember: a great-fitting line still isn't proof that X causes Y. Regression predicts; it never proves cause.
Regression inherits correlation's assumptions and adds one of its own. Linearity — the line is only a fair summary if the cloud is actually line-shaped. Independent observations — one row per case. Watch for outliers — a single extreme point can grab the line and tilt it, especially at the edges of the X range. And the new one: even scatter along the line — the residuals should be about as puffy at low X as at high X (the ten-dollar word is homoscedasticity; the plain-English version is "even noise"). If the cloud fans out as X grows, the line still fits, but its predictions get much less trustworthy at the wide end. The scatterplot-first habit from Unit 6 screens every one of these.
You don't fit lines by hand — Sheets does it. Open the SJNY Student Life dataset. Four functions cover everything, and the order is always (y, x) — predicted variable first:
=SLOPE(M2:M201, H2:H201) → the slope b
=INTERCEPT(M2:M201, H2:H201) → the intercept a
=RSQ(M2:M201, H2:H201) → r² (variation explained)To let Sheets do the prediction for you — no arithmetic at all — use =FORECAST (newer Sheets call it =FORECAST.LINEAR) or its cousin =TREND:
=FORECAST(6, M2:M201, H2:H201) → 20.2 (predicted wellbeing at X = 6)
=FORECAST.LINEAR(6, M2:M201, H2:H201) → 20.2 (same thing, newer name)
=TREND(M2:M201, H2:H201, 6) → 20.2 (predicts Y for the new X = 6)Note that =FORECAST wants (new-X, y-range, x-range) — the X you're predicting for comes first, then the same (y, x) ranges. To graph the line: highlight your two columns, then Insert ▸ Chart, pick a scatter chart, and in the Chart editor turn on the Trendline (Customize ▸ Series) — and check the box to show the equation and r² right on the chart. There's your line of best fit, drawn for you.
Paste a predictor and an outcome, hit Run, and the calculator builds Ŷ = a + bX — then predicts Y for any X you choose. Hit Load class data to see a real result first.
Unit 7. Find the best-fit line (Ŷ = a + bX) and predict Y from any X.
Order in the functions is (y, x) — predicted variable first. The intercept can be meaningless: "predicted wellbeing at 0 hours" might be fine, but "predicted GPA at 0 study hours" may sit outside the data. Don't extrapolate — predict inside the range, not miles beyond it. And r² is a percent of variation explained, not the slope: r² = .22 means 22% explained; the slope says steepness and direction. A good-fitting line still isn't proof that X causes Y.
Ten quick true/false items with instant explanations. (Different from your Canvas quiz — real practice.)