Correlation is not causation

Alex Tutor

Morning, Sofia. Last week you asked whether raising the ad budget actually gets us more orders. Let's find out with real numbers instead of vibes.

You

Isn't that obvious? More money in, more orders out. Case closed, right?

Alex Tutor

Nothing in analytics is "obvious" until you've looked at the data. Let's pull ten weeks of ad spend and orders from ads.

Alex Tutor

Dataset · 10 rows

CatChow weekly ad spend and orders from ads

Ten weeks of weekly ad spend and the orders attributed to ads. Synthetic data made for this course; reused in the next lesson.

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

Each row is one week: how much we spent on ads, and how many orders that week's ads got credited with.

You

So I just... eyeball it and see if it goes up?

Alex Tutor

Ad spend vs orders from ads, 10 weeksEach dot is one week. Orders tend to be higher in weeks with more ad spend, but the dots don't sit on a perfect line.

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

That's a scatter plot: one dot per week, spend on the x-axis, orders on the y-axis. Look for a general drift, not a perfect line.

You

Okay, it clearly climbs up and to the right. So it IS a straight line — more spend always means more orders!

Alex Tutor

Close, but not quite. The dots drift upward, with real scatter around that drift. That drift is what we're about to measure precisely.

Alex Tutor

Question 1

Look at the scatter plot above. Which statement best describes the pattern?

Choose one answerA perfect straight line: every extra dollar buys exactly the same number of ordersA general upward drift with scatter around it: higher spend tends to come with more orders, but not exactlyNo pattern at all: spend and orders look unrelatedA downward drift: more spend comes with fewer orders

Right — a drift with scatter. Now we need a single number that captures both the direction and the strength of that drift.

You

How do you even calculate a number like that?

Alex Tutor

For each week, take how far spend is from its average, and how far orders are from their average — two deviations per week.

Multiply the two deviations for each week. If both are usually positive together (or both negative together), the product is positive.

Alex Tutor

\text{cov}(x,y) = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})

Covariance: the average product of deviations. Positive means the variables tend to move together; negative means they move oppositely.

You

What if some weeks go one way and some the other? Won't the pluses and minuses just cancel out?

Alex Tutor

If the relationship is real, one sign wins overall once you sum across all weeks. If spend and orders were unrelated, the sum would land near zero.

Alex Tutor

Question 2

If ad spend and orders from ads were completely unrelated, the sum of the deviation products across all weeks would land close to zero.

TrueFalse

Covariance works, but its units are a mess — dollars times orders. We fix that by dividing by both variables' spread.

Alex Tutor

r = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2 \sum (y_i-\bar{y})^2}}, \quad -1 \le r \le 1

Pearson correlation coefficient (r): covariance standardised to a fixed -1 to 1 scale, regardless of units.

r = 1 is a perfect positive line, r = -1 a perfect negative line, r = 0 no linear relationship at all.

You

So what counts as a "good" r in the real world?

Alex Tutor

Let's practice on a small example before touching the real dataset.

Alex Tutor

Question 3

Five CatChow product pages show 1 to 5 customer reviews. Their conversion rates (%) are: 1 review -> 9%, 2 -> 8%, 3 -> 10%, 4 -> 11%, 5 -> 12%. Compute the Pearson correlation coefficient r between reviews shown and conversion rate.

Your answer

Units: -1 to 1, e.g. 0.75

Nicely done — and yes, r = 0.9 is strong. Real business data is rarely this tidy, though.

You

So the ad-spend data won't be this clean?

Alex Tutor

Probably not — ten weeks include holidays, competitor moves, stock-outs. Let's compute r on the real dataset above and see.

Same formula, just ten weeks instead of five points. A spreadsheet's CORREL() function does the arithmetic for you.

You

On it.

Alex Tutor

Question 4

Using the dataset CatChow weekly ad spend and orders from ads, compute the Pearson correlation coefficient r between ad_spend_usd and orders_from_ads.

Your answer

Units: -1 to 1, e.g. 0.75

Strong positive relationship, confirmed with real numbers. But before you go tell the CFO "ads cause orders" — slow down.

You

Wait, why not? That r is huge!

Alex Tutor

A big r only tells you two things move together. It never tells you which one, if either, is causing the other.

Alex Tutor

Direct causation is one option: more spend really does buy more orders. But reverse causation is possible too — happy customers write more reviews; reviews don't make them happy.

Then there's a confounder: something else drives both. During flash sales the site slows down AND orders spike — the sale caused both, not one the other.

You

So how do I tell which one it is, just from the chart?

Alex Tutor

Usually you can't, from correlation alone. That's exactly why controlled experiments exist — you'll run one in module 3.

Alex Tutor

Question 5

Match each scenario to the explanation that fits it best.

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

CatChow raises prices; a controlled price test shows orders drop as a direct result
Subscribers with higher NPS also have longer tenure — but maybe long-time customers simply grow to like the product more, not the other way round
Site load time and refund requests both spike on the same days — both driven by a flash-sale traffic surge
The office cat's average nap length happens to track daily orders over one random month

Answers left to place

Good instincts. One more thing before you draft any recommendation.

You

There's more?!

Alex Tutor

Just a checklist: does a plausible confounder exist? Could the arrow point the other way? Is the sample big enough to rule out coincidence? Would an experiment settle it?

You

Okay, four questions before I trust any correlation. Got it.

Alex Tutor

Question 6

Before trusting a strong correlation between ad spend and orders, which checks are worth doing? Select all that apply.

Choose all that applyCheck whether an obvious confounder (like a seasonal sale) could explain bothConsider whether the causation could run in the opposite directionAssume causation, since r is above 0.7Run a controlled experiment (an A/B test) if the decision is costly enough to matter

Alex Tutor

Solid list. Now put it into words — imagine explaining this to someone who wasn't in this chat.

You

Do I need to write an essay?

Alex Tutor

No — a few honest sentences beat a long one that dodges the point.

Here's your first task.

Alex Tutor

Question 7Short answer · AI-checked task

A colleague says: "Orders and ad spend have r = 0.95, so more ad spend causes more orders. Let's triple the budget." What's wrong with this reasoning, and what would you check first? Answer in 2-4 sentences.

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Good instinct — not extrapolating is exactly the right worry. Now for the real recommendation.

You

No pressure, then.

Alex Tutor

You've got everything you need: the chart, the r, and four ways correlation can mislead. Use them.

Here's the scenario.

Alex Tutor

Question 8Case · AI-checked task

Write your recommendation to the CatChow leadership team about the ad-spend and orders relationship.

CatChow's marketing lead saw r = 0.95 between weekly ad spend and orders from ads (dataset CatChow weekly ad spend and orders from ads) and wants to triple next quarter's ad budget, arguing the correlation "proves" spend drives orders. Write a short recommendation: what the correlation does and doesn't tell you, at least one alternative explanation worth ruling out, and what you'd want to see before committing to a 3x budget increase.

Dataset for this task: CatChow weekly ad spend and orders from ads

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Solid work today, Sofia. Correlation tells us how strongly two things move together. Next time: regression, which tells us exactly how many orders each extra dollar buys.

Alex Tutor

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

Next lesson: Linear regression