Linear regression

Alex Tutor

Yesterday we found ad spend and orders correlate strongly, r = 0.95. Today the CFO wants an actual number: how many extra orders does one more dollar buy?

You

Can't we just use r for that?

Alex Tutor

r only tells you how tightly the dots hug a line, not how steep that line is. For an actual number, we need linear regression.

Alex Tutor

Dataset · 10 rows

CatChow weekly ad spend and orders from ads

The same ten weeks of ad spend and orders from the correlation lesson, now used to fit a line.

Download CSV

Columns
week: Week number
ad_spend_usd: Weekly ad spend, USD
orders_from_ads: Orders attributed to ads that week

Data

week	ad_spend_usd	orders_from_ads
1	200	20
2	300	23
3	350	30
4	400	28
5	450	36

Show 5 more rows

6	500	33
7	550	42
8	600	38
9	650	48
10	700	44

Same ten weeks as yesterday. This time we'll fit an actual line through the dots, not just measure how straight they look.

Alex Tutor

Ad spend vs orders from ads, 10 weeksThe same scatter plot from the correlation lesson. Today's goal: the straight line that fits it best.

Chart data

Ad spend, USD	Orders from ads
200	20
300	23
350	30
400	28
450	36

Show 5 more rows

500	33
550	42
600	38
650	48
700	44

You

So we just draw a line through the middle of the dots?

Alex Tutor

That's the idea — but "through the middle" needs a precise rule, or every analyst would draw a different line.

Alex Tutor

Question 1

What does linear regression give you that correlation alone does not?

Choose one answerA precise equation to predict one variable from the otherProof that one variable causes the otherA number from -1 to 1 showing how tightly the points hug a lineNothing — it's the same information as r

Here's the rule: pick the line that minimises the total squared vertical gap between each dot and the line.

You

Squared again? Didn't we already do that for standard deviation?

Alex Tutor

Same trick: squaring stops positive and negative gaps from cancelling, and punishes big misses harder than small ones.

Alex Tutor

b_1 = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2}, \qquad b_0 = \bar{y} - b_1\bar{x}

The least-squares slope and intercept: the same numerator and denominator you computed for r, reused for the line's steepness.

Each dot's actual value minus its predicted value is called a residual. Add up all the residuals, and they always sum to exactly zero.

You

Wait, always? Even for a terrible line?

Alex Tutor

As long as the line includes an intercept and was fit by least squares — yes. It's the same reason deviations from a mean sum to zero.

Alex Tutor

Question 2

The residuals of a least-squares regression line (fit with an intercept) always sum to exactly zero.

TrueFalse

Good. Now let's actually fit the line on our ad-spend data.

You

Same ten weeks from the dataset above?

Alex Tutor

Exactly. A spreadsheet's SLOPE() and INTERCEPT() functions do this arithmetic, or you can grind through the formula by hand.

You

Let's see if spend really buys that many orders.

Alex Tutor

Question 3

Using the dataset CatChow weekly ad spend and orders from ads, compute the slope b1 of the least-squares line predicting orders_from_ads from ad_spend_usd. Give your answer as orders per $100 of extra weekly spend.

Your answer

Units: orders per 100 USD of ad spend, e.g. 4.0

5.39 extra orders per $100 — not bad. But a slope alone doesn't answer "how many orders next week": it needs the intercept too.

You

Right, the starting point when spend is zero?

Alex Tutor

Exactly —

b_0 = \bar y - b_1 \bar x \approx 34.2 - 0.0539 \times 470 \approx 8.85

orders with zero ad spend. Some organic orders happen even without ads.

You

So now we can predict any week?

Alex Tutor

Question 4

Using $b_0 \approx 8.85$ and $b_1 \approx 0.0539$ , predict orders_from_ads for a week with $550 of ad spend.

Your answer

Units: orders, e.g. 30.0

38.5 predicted, 42 actual for that week — a residual of about 3.5. Not bad for a straight line through messy real data.

You

Is there a way to say how good the whole line is, not just one week?

Alex Tutor

Yes — the coefficient of determination, R². It's the share of the variation in orders that the line explains, from 0 to 1.

Alex Tutor

R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}

R²: one minus the share of variance left over as residuals. R² = 1 is a perfect fit; R² = 0 means the line explains nothing.

You

For our data, is that just r squared?

Alex Tutor

For a single predictor, yes exactly:

R^2 = r^2 \approx 0.95^2 \approx 0.90

. About 90% of the week-to-week variation in orders lines up with ad spend.

Alex Tutor

Question 5

Match each regression term to what it means.

Tap an answer, then tap the row it belongs to. You can also drag.

How many more orders each extra $100 of spend predicts
The predicted orders when ad spend is zero
The gap between a week's actual orders and the line's prediction for it
The share of week-to-week variation in orders that the line explains

Answers left to place

One more thing before you start forecasting next quarter with this line.

You

Uh oh.

Alex Tutor

We only tested spend from $200 to $700. The line might not hold outside that interval — predicting for $5,000 is a guess, not a measurement.

And R² = 0.90 is still not 1 — a residual of a few orders each week is normal. Don't expect the line to nail every single week.

Alex Tutor

Question 6

Which of these are real risks when using this regression line to plan next quarter's ad budget? Select all that apply.

Choose all that applyPredicting orders for an ad spend far outside $200-$700 (extrapolation)Treating R² = 0.90 as if it were 1.0, expecting every single week to match exactlyAssuming the slope proves higher spend causes higher orders, rather than just describing this dataUsing the line at all, since its R² is below 1

Alex Tutor

Good catches, all three.

You

Go team.

Alex Tutor

One more diagnostic skill, then we wrap up with the memo.

Look at this residual pattern.

Alex Tutor

Question 7Short answer · AI-checked task

You plot the residuals against ad spend and they aren't scattered randomly: small negative residuals at low spend, positive around the middle, negative again at high spend — a curve. What might this pattern mean, and what would you consider doing next? Answer in 2-4 sentences.

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Right idea — a systematic pattern like that means look for curvature, not just noise. Now, the real memo.

You

For the CFO, no pressure.

Alex Tutor

You have the line, the R², and the extrapolation warning. Put it together.

Here's the scenario.

Alex Tutor

Question 8Case · AI-checked task

Write your recommendation about using this line to plan next quarter's ad budget.

Using the fitted line ( $b_0 \approx 8.85$ , $b_1 \approx 0.0539$ , $R^2 \approx 0.90$ ) from the dataset CatChow weekly ad spend and orders from ads, a colleague plugged in a planned $900 weekly ad spend and got a prediction of about 57 orders, then asked to lock in next quarter's budget around that level. Write a short recommendation covering: what the line and its R² tell you, why $900 deserves extra caution, and what you'd want to see before committing.

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Great work, Sofia. You can now fit a line and read what it does and doesn't promise. Next: what happens when more than one thing drives orders at once — multiple regression.

Alex Tutor

That’s the lesson. You answered every task — nicely done.

Next lesson: Multiple regression