Regression — gradient interpretation and extrapolation/reliability

~50 min · WST01 · 4.2

WST01 · 4.2 · 50 min

Every one of the five examiner reports read for this unit flags the same failure, worded a different way each time. Computing $b = S_{xy}/S_{xx}$ is not where the marks go missing — candidates who can barely finish a $S_{xx}$ calculation still reliably reach the right line. What goes missing is the sentence after it: saying, in words, what one extra unit of $x$ actually buys you in $y$ , and knowing the difference between a prediction the data has already tested and one it is only guessing at.

Before you read on

Two or three questions on exactly what this lesson teaches. Being wrong here is fine — it's the fastest way to find out what to pay attention to next.

Scatter diagrams, and which variable explains which

A scatter diagram plots one variable against another to show whether a linear relationship between them is worth pursuing at all (spec 4.1) — it's the picture you look at before committing to fit a line, not decoration added afterwards. Drawing one won't be the direct focus of an exam question; reading and using one will be.

Spec item 4.2 names the two variables precisely, and the names carry real content, not just labels. The explanatory (independent) variable, conventionally $x$ , plotted on the horizontal axis, is the one you set, choose, or already know. The response (dependent) variable, conventionally $y$ , plotted on the vertical axis, is the one you're trying to predict or explain FROM the other. Advertising spend is set by a shop each week; the number of enquiries that follows is the outcome being investigated — spend explains, enquiries respond.

This distinction isn't symmetric, and treating it as symmetric is a real source of error further down the line. The least-squares line of $y$ on $x$ (minimising vertical distances, used whenever you're predicting $y$ from a given $x$ ) is a different line, in general, from the least-squares line of $x$ on $y$ (minimising horizontal distances instead). Which one a question wants is decided entirely by which variable it asks you to predict — so identifying explanatory and response correctly isn't a formality, it fixes which formula applies at all.

One dataset runs through this whole lesson. A shop records, for five separate weeks, how much it spends on advertising ( $x$ , in hundreds of dollars) and how many customer enquiries it receives that week ( $y$ ): $x$ : 2, 4, 6, 8, 10. $y$ : 19, 27, 35, 44, 53. Read the units on $x$ twice before doing anything else with this data — $x = 6$ means $600 was spent, not $6 — because that scale is exactly where the richest trap in this whole topic lives, and it's worth meeting it here, once, before it costs a mark later.

Fitting the line: what $S_{xx}$ and $S_{xy}$ are, and what the formula booklet already gives you

The least squares regression line of $y$ on $x$ has equation $\hat y = a + bx$ — the hat on $y$ signals a fitted or predicted value, not an actually-observed one, a distinction worth keeping visible, since a real recorded $y$ and a value read off the line are different claims. Both $a$ and $b$ are estimates, computed from two summary totals: $S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n}$ and $S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n}$ . Both of these — this computational form and an equivalent 'deviation' form written with $(x - \bar x)$ — are printed in full in the formula booklet's Statistics S1 section, so neither has to be memorised; what has to be practised is using them correctly.

From those two totals, $b = \frac{S_{xy}}{S_{xx}}$ — the gradient — and $a = \bar y - b\bar x$ — the intercept, using the actual means $\bar x$ and $\bar y$ , not the raw column totals $\sum x$ and $\sum y$ those means came from. Confirmed directly from a real regression question: examiners recorded intercept errors that came from substituting the raw totals straight into the intercept formula — the specific numbers named in that report were 273 and 93 — instead of dividing each by $n$ first to get the means the formula actually needs (Oct 2021, Q2(c)).

Working the running example: $n = 5$ , $\sum x = 30$ , $\sum y = 178$ , $\sum x^2 = 220$ , $\sum xy = 1238$ . So $S_{xx} = 220 - \frac{30^2}{5} = 220 - 180 = 40$ , and $S_{xy} = 1238 - \frac{30 \times 178}{5} = 1238 - 1068 = 170$ . Then $\bar x = 6$ , $\bar y = 35.6$ , so $b = \frac{170}{40} = 4.25$ and $a = 35.6 - 4.25 \times 6 = 35.6 - 25.5 = 10.1$ . The fitted line is $\hat y = 10.1 + 4.25x$ .

Two habits protect this whole calculation. First: never let $\bar x^2$ stand in for $\frac{(\sum x)^2}{n}$ inside $S_{xx}$ — they aren't the same number ( $\bar x^2 = 36$ here; $\frac{(\sum x)^2}{n} = 180$ ) — and this exact substitution is independently confirmed as *'the most common error made'* on a real $S_{xx}$ question (Oct 2021, Q2(b)). Second: once $b$ is evaluated, write both $a$ and $b$ as decimals to at least 3 significant figures and finish the job by writing the actual equation of the line — a real examiner report records candidates who correctly found $a$ and $b$ separately and then never wrote them back up as $\hat y = a + bx$ , losing the final mark for stopping one step early (Oct 2021, Q2(c)); a separate report states outright that *'fractions are not accepted in a final regression equation; the values of a and b are both estimates and so a fractional answer is not appropriate'* (Jun 2022, Q2(c)).

Mechanism

Why 'as x increases, y increases' is not an interpretation

A gradient interpretation question is not asking whether $x$ and $y$ move together — the scatter diagram and the sign of $b$ already answered that, and restating it earns nothing on its own; examiner reports say so independently, across several different series and contexts. What the question is checking is whether you can read $b$ as a genuine RATE: the amount $y$ changes, on average, for one whole extra unit of $x$ — a number with a size and a unit attached, not a description of direction. Two things then have to survive the translation from algebra into English, and both are independently confirmed as the specific things that go wrong. First, 'one extra unit of $x$ ' has to match how $x$ was actually defined in the question, not whatever feels like a natural unit — a real report puts the cost of getting this wrong in exact numbers: the main error 'was not recognising that a single rise in the number of employees led to a rise in the amount spent on paper of $156 and not $1.56' (Oct 2021, Q2(d)), a hundred-fold slip from reading the gradient's raw value as if $x$ had been measured in ones. Another report shows the identical failure from the opposite direction, where $x$ was compressed into millions rather than expanded into ones: 'only the most able candidates successfully managed to write that the GDP increases by 31.2 billion dollars for every 1 million increase in the population' (Jun 2022, Q2(d)) — most answers that series gave the direction of the relationship and stopped, never converting the raw coefficient into a sentence about what one real unit of $x$ buys. Second, the sentence has to be anchored in the actual variables and their real meaning, not a generic template: one report records exactly that failure — *'too many students referred to positive correlation or "as x increases then y increases"... some students mixed up the units (grams and °C)'* (Jan 2023, Q6(a)) — and another records a sentence that survives everything except getting turned around: *'a few lacked the context required, whilst others gave the interpretation the wrong way round'* (Jun 2024, Q4(d)). A credited answer names both variables in their own terms, states the correct SIZE of one real unit of $x$ , states the direction, and does all three in a single sentence — not four separate half-marks' worth of fragments.

Mechanism

Why extrapolation is a real danger, not a caution to recite

The least squares method has no way to know what happens outside the $x$ -values it was actually given — $S_{xx}$ and $S_{xy}$ are sums over the recorded data points and nothing else, so the fitted line is a statement about the pattern INSIDE that range, not a law of nature that happens to be expressed as a straight line. Predicting inside the recorded range (interpolation) borrows strength from data on both sides of the value being predicted; predicting outside it (extrapolation, in EITHER direction — below the smallest recorded $x$ counts exactly as much as above the largest) borrows nothing, because there's no data on that side to borrow from. Spec guidance states the principle directly: use the line to make predictions 'within the range of the explanatory variable' and stay alert to 'the dangers of extrapolation' (spec 4.2 guidance). Real relationships are very often only locally linear — advertising spend probably does buy roughly proportional enquiries over the range a shop has actually tried, but there's no reason to expect that to hold at ten times the recorded spend, where a market could saturate, a budget could hit diminishing returns, or the relationship could simply change shape, and this particular data has never been anywhere near that region to say one way or the other. This is exactly what the mark scheme rewards and what most answers miss: an examiner report on a real reliability question notes it 'was rare to see responses which accurately assessed the reliability of the estimate found... many stated the estimate was unreliable, but they were unable to refer to the correct variable or value that is not in the range... it was also no surprise to find responses saying that the estimate was reliable, even with correct working earlier in the question [that showed it wasn't]' (Jun 2022, Q2(e)). The credited form is specific, not a vague verdict: a separate report records that successful answers 'made reference to 90 being outside of the range and therefore unreliable' (Jan 2023, Q6(c)) — name the value, name the range, connect them, in that order.

Diagram — The fitted line, the data it was built from, and where it stops being trustworthy

x-axis: Advertising spend, in hundreds of dollars (x) · y-axis: Enquiries received that week (y)

ŷ = 10.1 + 4.25x, drawn across the recorded data (2 ≤ x ≤ 10): The least squares line, solid only where the data actually reaches: from (2, 18.6) to (10, 52.6). Every point on this stretch is a genuine interpolation — a prediction sitting between two weeks the shop actually recorded.
The same line, extended beyond the data: The identical straight line, continued past x = 10 with nothing new to support it. Drawn dashed deliberately: the algebra doesn't change at x = 10, but the evidence for it does — the picture should look less certain exactly where the argument is less certain.

(2, 19): Week 1: $200 spent, 19 enquiries received.
(4, 27): Week 2: $400 spent, 27 enquiries received.
(6, 35): Week 3: $600 spent, 35 enquiries received.
(8, 44): Week 4: $800 spent, 44 enquiries received.
(10, 53): Week 5: $1000 spent, 53 enquiries received — the highest spend actually recorded.
Mean point (x̄, ȳ) = (6, 35.6): Not a real data point — the average of the five weeks. The least squares line always passes through this exact point, because a = ȳ − bx̄ is built to guarantee it. A free check on any computed a and b, not a coincidence of this particular dataset.
Estimate at x = 20 ($2000 spend): Where the dashed line reaches y = 95.1. Mathematically well-defined, statistically unsupported — no week in the data went anywhere near $2000, so nothing here has actually been tested.

Common error: Drawing the fitted line at the same solid weight all the way out to an extrapolated prediction, so the picture itself implies the far-out estimate is exactly as trustworthy as one read from inside the plotted points.

Correct: Draw the line solid only across the x-values actually in the data (2 to 10 here) and dashed beyond it in either direction. The visual boundary should match the boundary in the argument: solid means 'the data says this', dashed means 'the model is guessing'.

Beyond the spec

Spec 4.2's own guidance says outright that derivations are 'not required' for this unit — a student can score full marks using $b = S_{xy}/S_{xx}$ and $a = \bar y - b\bar x$ exactly as the formula booklet hands them over, with no idea where either formula comes from. This is where they come from, and it explains, rather than just asserts, a fact used elsewhere in this lesson: why the fitted line always passes through $(\bar x, \bar y)$ .

'Least squares' names an actual optimisation: choose $a$ and $b$ to make $S = \sum (y_i - a - bx_i)^2$ — the total of the squared vertical gaps between each real data point and the line — as small as possible. Calculus finds that minimum by setting both partial derivatives to zero. $\frac{\partial S}{\partial a} = -2\sum(y_i - a - bx_i) = 0$ gives $\sum y_i - na - b\sum x_i = 0$ , so $a = \bar y - b\bar x$ — the exact intercept formula used throughout this lesson, arrived at without assuming it. Substitute $x = \bar x$ into $\hat y = a + bx$ and this identity forces $\hat y = (\bar y - b\bar x) + b\bar x = \bar y$ : the point $(\bar x, \bar y)$ satisfies the line's own equation for ANY value of $b$ , which is exactly why every least squares line, for every dataset, passes through its own mean point — not a property of this data, a property of the method itself. The second equation, $\frac{\partial S}{\partial b} = -2\sum x_i(y_i - a - bx_i) = 0$ , gives $\sum x_iy_i - a\sum x_i - b\sum x_i^2 = 0$ ; substituting the expression for $a$ just found and collecting every term containing $b$ onto one side eventually reduces to $b\left(\sum x_i^2 - \frac{(\sum x_i)^2}{n}\right) = \sum x_iy_i - \frac{(\sum x_i)(\sum y_i)}{n}$ — which is exactly $b \cdot S_{xx} = S_{xy}$ , so $b = S_{xy}/S_{xx}$ . Nobody chose that formula for its shape; it's the unique answer to 'which line makes the total squared error smallest', and the booklet simply saves every student the calculus needed to arrive at it.

Worked, in full

Two predictions from the same line — why one is trustworthy and the other isn't

01
Start from what the line was actually built from. The fitted line $\hat y = 10.1 + 4.25x$ came from five weeks with spend $x$ between 2 and 10 (i.e. $200 to $1000) — that closed interval, $2 \le x \le 10$ , is the entire evidence base behind the line. Nothing about the arithmetic that produced $a$ and $b$ has any way of knowing what happens for an $x$ outside it.
Earns: B1 — states the recorded range of the explanatory variable correctly, as the necessary first step before judging any prediction.
02
Predict the enquiries for a week where $700 is spent, so $x = 7$ : $\hat y = 10.1 + 4.25(7) = 10.1 + 29.75 = 39.85$ , about 40 enquiries. Check where $x = 7$ sits: it's between the recorded $x = 6$ and $x = 8$ , so it lies strictly inside $2 \le x \le 10$ .
Earns: M1 — correct substitution of x = 7 into the fitted line, and correct identification that this value sits inside the recorded range.
03
Predict the enquiries for a week where $1600 is spent, so $x = 16$ : $\hat y = 10.1 + 4.25(16) = 10.1 + 68 = 78.1$ , about 78 enquiries. Check where $x = 16$ sits: the highest spend ever recorded was $x = 10$ , so $x = 16$ is a long way past the top of the range — 6 units of $x$ , or $600, beyond the last week the shop ever actually tried.
Earns: M1 — correct substitution of x = 16 into the fitted line, and correct identification that this value sits well outside the recorded range, on the high side.
04
The two predictions used identical arithmetic — the same line, the same substitution method — so arithmetic correctness can't be what separates them. What separates them is where each $x$ -value sits relative to $2 \le x \le 10$ : $x = 7$ is interpolation, reading a value from inside data that surrounds it on both sides; $x = 16$ is extrapolation, reading a value the fitted pattern has never once been tested against.
Earns: B1 — correctly classifies x = 7 as interpolation and x = 16 as extrapolation, by explicit comparison against the stated range, not by intuition alone.
05
State the conclusion the way the mark scheme actually rewards it: name the value, name the range, connect them. The estimate of 39.85 (≈ 40) enquiries at $700 is reliable, since $700 lies inside the $200–$1000 range the line was fitted to. The estimate of 78.1 (≈ 78) enquiries at $1600 is not reliable, because $1600 lies well outside that range — there's no evidence in the data that the same linear relationship between spend and enquiries continues to hold once spending is pushed that far past anything the shop has actually tried.
Earns: B1 — the credited answer form specifically: naming the value that falls outside the range and stating that it is outside the range, not a generic 'it might not be accurate' comment with nothing attached to it.

Source — Examiner report, Jan 2023

"made reference to 90 being outside of the range and therefore unreliable"

Complete it yourself

Complete the chain — is a prediction at x = 0 (no advertising at all) reliable?

01
The line fitted to this data is $\hat y = 10.1 + 4.25x$ , built entirely from five weeks with recorded spend between $200 and $1000, i.e. $2 \le x \le 10$ .
02
The shop asks: what does the model predict for a week where it spends nothing at all on advertising, $x = 0$ ?

Marked, line by line

A shop records, for five separate weeks, the amount spent on advertising that week ( $x$ , in hundreds of dollars) and the number of customer enquiries it receives ( $y$ ): $x$ : 2, 4, 6, 8, 10; $y$ : 19, 27, 35, 44, 53. You are given $\sum x = 30$ , $\sum y = 178$ , $\sum x^2 = 220$ , $\sum xy = 1238$ . (a) Find $S_{xx}$ and $S_{xy}$ , and hence find the equation of the regression line of $y$ on $x$ in the form $y = a + bx$ , giving $a$ and $b$ to 3 significant figures where appropriate. (5) (b) Give an interpretation of the gradient of your regression line. (2) (c)(i) Use your regression line to estimate the number of enquiries received in a week when $2000 is spent on advertising. (ii) Comment on the reliability of this estimate. (3) — VERIDIAN-original question, inspired by the real structure of WST01 regression questions (a data table, a 'find Sxx/Sxy hence the line' part, a gradient-interpretation part, and a prediction-plus-reliability part all appear together in real series) but this dataset, these numbers and this exact question are not a reproduction of any past-paper question.

10 marks available

(a) — 5 marks

01
$S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n} = 220 - \frac{30^2}{5} = 220 - 180 = 40$
Independent mark for the correct value of Sxx, from the computational form given in full in the booklet. The most commonly recorded slip at this exact step is substituting x̄² (=36) for (Σx)²/n (=180) — outwardly similar, numerically nothing alike.
B1
02
$S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n} = 1238 - \frac{30 \times 178}{5} = 1238 - 1068 = 170$
Independent mark for the correct value of Sxy, computed separately from Sxx — an error in one does not force an error in the other, since the two are unrelated sums.
B1
03
$b = \frac{S_{xy}}{S_{xx}} = \frac{170}{40} = 4.25$
Method mark for correct use of b = Sxy/Sxx on their own two values above. Dependent on both marks above having been attempted, though not on their being correct — this method scores even against a wrong Sxx or Sxy, provided it's applied correctly to whatever was found.
M1
04
$a = \bar y - b\bar x = 35.6 - 4.25 \times 6 = 35.6 - 25.5 = 10.1$
Method mark for a = ȳ − bx̄ using the actual means (ȳ = 178/5 = 35.6, x̄ = 30/5 = 6), not the raw column totals 30 and 178 substituted directly — a real report records exactly that substitution error, naming the two raw totals involved (273 and 93) on the question it reviewed.
M1
05
$y = 10.1 + 4.25x$
Correct answer only: the actual equation of the line, stated explicitly with both values as decimals to 3 s.f. Unavailable if a and b are found correctly but never assembled into this equation, and unavailable if either is left as an unresolved fraction rather than converted — both are separately confirmed as real ways to lose this exact mark.
A1

(b) — 2 marks

101
For every extra $100 spent on advertising in a week (one whole unit of $x$ , since $x$ was measured in hundreds of dollars), the number of enquiries received increases, on average, by 4.25.
Independent mark for stating the rate of change at the CORRECT scale of x — reading the gradient against $1 instead of the $100 that one unit of x actually represents is the single most-repeated version of this error across the archive.
B1
102
This is an average/estimated effect from the fitted line, not a guarantee for any individual week — and the direction is an increase, matching the positive value of b.
Independent mark for the correct direction, stated as part of a genuine sentence about the two named variables, rather than a bare 'positive correlation' — restating the sign of b or the direction of correlation alone, with no rate attached, is recorded as scoring nothing on this type of question.
B1

201
$x = 20$ (since $2000 is 20 hundreds of dollars): $\hat y = 10.1 + 4.25(20) = 10.1 + 85 = 95.1$
Method mark for correctly converting $2000 into the x = 20 the line actually uses, then substituting.
M1
202
About 95 enquiries.
Accuracy mark for the correct estimate (awrt 95.1), dependent on the method above.
A1
203
Unreliable: $x = 20$ is well outside the recorded range $2 \le x \le 10$ ($200 to $1000), so this is extrapolation — there's no evidence the linear relationship between spend and enquiries continues to hold that far past the data the line was fitted to.
Independent mark for the credited answer form: naming the specific value (20, or $2000) and naming that it sits outside the specific recorded range, not a general 'estimates from regression lines are always a bit risky' comment.
B1

Named traps

sxx-computed-with-xbar-squared: Confirmed directly on a real $S_{xx}$ question: *'the most common error made was using x̄² in the calculation of Sxx'* (Oct 2021, Q2(b)). $\bar x^2$ and $\frac{(\sum x)^2}{n}$ are not the same number — for this lesson's own dataset they're 36 against 180 — and the two formulas that use them, $S_{xx} = \sum x^2 - \bar x^2$ (wrong) against $S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n}$ (correct, and the one printed in the booklet), look similar enough on the page that the substitution can slip past unnoticed.
gradient-fraction-inverted: Confirmed on a real regression question: gradient errors came from *'the fraction the wrong way up'* (Oct 2021, Q2(c)) — computing $S_{xx}/S_{xy}$ instead of $S_{xy}/S_{xx}$ . The fix is naming the formula out loud before substituting: $b$ is 'the sum that mixes x and y' over 'the sum that's just x', in that order, every time.
intercept-from-raw-totals-not-means: Confirmed on the same question, with the exact numbers the report names: intercept errors came from substituting the raw column totals — the report gives 273 and 93 — straight into $a = \bar y - b\bar x$ in place of the actual means those totals needed to be divided by $n$ first to produce (Oct 2021, Q2(c)). $\sum x$ and $\bar x$ differ by a factor of $n$ ; using one where the formula asks for the other doesn't produce a slightly-off answer, it produces one wrong by exactly that factor.
stops-after-a-and-b-never-states-the-line: Confirmed on the same question again: candidates who correctly found both $a$ and $b$ but never wrote them back up as the actual equation of the line lost the final mark of the part for stopping one step early (Oct 2021, Q2(c)). Finding the two numbers is not the same task as answering 'find the equation of the regression line' — the sentence $y = a + bx$ , with the values substituted in, is the actual deliverable.
fraction-left-in-final-line: Confirmed directly: *'it is also important to note that fractions are not accepted in a final regression equation; the values of a and b are both estimates and so a fractional answer is not appropriate'* (Jun 2022, Q2(c)). Once $b = S_{xy}/S_{xx}$ has actually been divided out, it stays a decimal (to at least 3 s.f.) for the rest of the question — reverting to the exact fraction it started as, e.g. $\frac{170}{40}$ , in the final line costs the mark even when the value is numerically correct.
gradient-interpreted-at-the-wrong-scale: The richest single trap in this topic, confirmed independently in two different subjects' worth of context. One report gives the cost in dollars: the main error 'was not recognising that a single rise in the number of employees led to a rise in the amount spent on paper of $156 and not $1.56' (Oct 2021, Q2(d)) — a hundred-fold misreading of the gradient's own scale. Another shows the same failure from the other direction: 'only the most able candidates successfully managed to write that the GDP increases by 31.2 billion dollars for every 1 million increase in the population' (Jun 2022, Q2(d)) — most answers gave the direction of change and stopped, never converting the raw coefficient into a sentence about what one real unit of the explanatory variable actually buys. Whatever $x$ is measured in — dollars, hundreds of dollars, millions of people — 'one unit of x' in the interpretation sentence has to mean that, not '1' read off the page with no units attached.
interpretation-gives-correlation-not-a-rate: Confirmed twice: *'too many students referred to positive correlation or "as x increases then y increases"... some students mixed up the units (grams and °C)'* (Jan 2023, Q6(a)); and separately, *'a few lacked the context required, whilst others gave the interpretation the wrong way round'* (Jun 2024, Q4(d)). Stating that $x$ and $y$ move together restates something the scatter diagram already showed, and earns nothing on its own — a full-credit sentence names both variables, states the correct-scale rate of change, and gets the direction right, all three, in one sentence.
reliability-judged-without-naming-the-range: Confirmed directly: *'it was rare to see responses which accurately assessed the reliability of the estimate found... many stated the estimate was unreliable, but they were unable to refer to the correct variable or value that is not in the range... it was also no surprise to find responses saying that the estimate was reliable, even with correct working earlier in the question [that showed it wasn't]'* (Jun 2022, Q2(e)). The credited form is specific: a separate report records that successful answers 'made reference to 90 being outside of the range and therefore unreliable' (Jan 2023, Q6(c)) — name the value, name the range, connect the two, or the comment doesn't score even when the underlying judgement (reliable / unreliable) happens to be right.

In your own words

In one sentence: why can two predictions made from the exact same regression line, using the exact same correct arithmetic, have completely different reliability?

Retrieval — with feedback on every choice

Question 1

3 marks

For a different dataset, $n = 6$ , $\sum x = 54$ , $\sum y = 96$ , $\sum x^2 = 522$ , $\sum xy = 918$ . What is the gradient $b$ of the regression line of $y$ on $x$ ?

Question 2

2 marks

A scientist fits the regression line $\hat t = 5.2 + 0.8h$ , where $h$ is hours of sunshine in a day and $t$ is the maximum temperature reached that day, in °C. Which is the best interpretation of the gradient?

Question 3

2 marks

A regression line is fitted to data recorded for $x$ between 10 and 50. Which of these predictions is NOT extrapolation?

Question 4

2 marks

A student correctly finds $S_{xx} = 25$ and $S_{xy} = 40$ for a dataset, and correctly finds $\bar x = 4$ , $\bar y = 11$ . Which of these is an acceptable way to state the final regression line?

Reference — not a study method, a lookup

Regression line of y on x: ŷ = a + bx. b = Sxy/Sxx, a = ȳ − bx̄. Sxx = Σx² − (Σx)²/n, Sxy = Σxy − ΣxΣy/n.
Never x̄² for (Σx)²/n inside Sxx. Never leave a or b as a fraction — decimals, 3 s.f., in the final line.
Gradient sentence: 'for each extra 1 unit of x, y changes by b units, on average.' Check what 1 unit of x actually is.
Interpolation = inside the recorded range = usually reliable. Extrapolation = outside it, either direction = name the value and the range.
The fitted line always passes through (x̄, ȳ) — a free check on any computed a, b.

Not affiliated with or endorsed by Pearson Edexcel. Every quotation and figure attributed to a mark scheme or examiner report in this lesson was independently verified against WST01-verified-facts.md, not carried over from prior course material. The advertising-spend/enquiries dataset that runs through this lesson, its regression line (ŷ = 10.1 + 4.25x, from Sxx = 40 and Sxy = 170), and every question built on it — prequestion, worked chain, chain drill, marked solution and MCQ alike — are VERIDIAN-original: no real WST01 question uses this scenario or these numbers, and every arithmetic result (including the deliberately wrong values on the common-wrong-path) was independently verified by script before being written into this file. The named traps (fraction left in the final line, Sxx computed with x̄² instead of (Σx)²/n, the intercept built from raw totals rather than means, the gradient's scale left unstated, extrapolation judged without naming the value and the range) and every direct quotation are transcribed from real Pearson examiner reports (Oct 2021, Jun 2022, Jan 2023, Jun 2024) as recorded in the research bank; the per-line mark allocations attached to the original questions are modelled on verified mark-scheme conventions (M/A/B marks, follow-through, cao) rather than transcribed from a real scheme, which for an original question does not exist.

Question 13 marks

For a different dataset, $n = 6$ , $\sum x = 54$ , $\sum y = 96$ , $\sum x^2 = 522$ , $\sum xy = 918$ . What is the gradient $b$ of the regression line of $y$ on $x$ ?

$b = 1.5$
Correct. $S_{xx} = 522 - \frac{54^2}{6} = 522 - 486 = 36$ ; $S_{xy} = 918 - \frac{54 \times 96}{6} = 918 - 864 = 54$ ; $b = \frac{S_{xy}}{S_{xx}} = \frac{54}{36} = 1.5$ .
B $b = 0.667$
This is $S_{xx}/S_{xy} = 36/54$ — the fraction the wrong way up, a real, named error on this exact step of a real regression question. The gradient is the sum that mixes x and y, over the sum that's just x, not the other way round.
C $b = 0.122$
This substitutes $\bar x^2 = 81$ for $\frac{(\sum x)^2}{n} = 486$ inside $S_{xx}$ , giving $S_{xx} = 522 - 81 = 441$ instead of $36$ — the exact substitution independently confirmed as 'the most common error' on a real Sxx question. The two quantities aren't close: $\frac{(\sum x)^2}{n}$ divides by $n$ once, correctly; $\bar x^2$ has already divided by $n$ once to form $\bar x$ and then divides by it again when squared, which is why it under-corrects so badly.
D $b = 25.5$
This uses $\sum xy = 918$ directly as if it were already $S_{xy}$ , skipping the correction term $\frac{(\sum x)(\sum y)}{n} = 864$ that has to be subtracted first. $\sum xy$ and $S_{xy}$ measure different things: one is a raw sum, the other has already had the part explained by the two means removed.

Traps tested: Gradient fraction inverted · Sxx computed with xbar squared · Sxy correction term omitted

Question 22 marks

For each extra hour of sunshine, the maximum temperature reached rises, on average, by 0.8°C
Correct. It names both variables, states the correct unit for one extra unit of h (one hour), states the correct unit for the change in t (°C), and gives the right direction — everything a credited interpretation needs in one sentence.
BThere is a strong positive correlation between hours of sunshine and temperature
This restates the direction the scatter diagram already showed and attaches no rate to it at all — examiner reports record this exact substitution ('positive correlation' in place of an actual rate) as scoring nothing on interpretation questions.
CFor each extra hour of sunshine, the temperature rises by 0.8 hours
The number is right and the unit is wrong: 0.8 is measured in °C (it's a change in t), not in hours (a unit of h). A real report on a question mixing exactly these two kinds of units — grams and °C — records candidates making this same category error.
DTemperature increases by 0.8°C in total, regardless of how many hours of sunshine there are
0.8 is a RATE — per additional hour — not a one-off fixed amount. Two extra hours of sunshine predicts roughly 1.6°C more, not another flat 0.8°C bolted on top of an already-flat 0.8°C.

Traps tested: Interpretation gives correlation not a rate · Units of y dropped or swapped · Gradient treated as a fixed total not a rate

Question 32 marks

A regression line is fitted to data recorded for $x$ between 10 and 50. Which of these predictions is NOT extrapolation?

x = 30
Correct. 30 sits strictly between the smallest (10) and largest (50) recorded values, so this prediction is supported by data on both sides — interpolation, not extrapolation.
Bx = 5
This IS extrapolation — 5 is below the smallest recorded value, 10. Leaving the range at the bottom carries exactly the same danger as leaving it at the top; the data never went there either way.
Cx = 60
This is extrapolation too — 60 is above the largest recorded value, 50, by a margin similar to how far 5 sits below 10. Distance from the range changes how risky an extrapolation is, but it doesn't change whether it counts as one at all.
Dx = 9
This is still extrapolation, even though 9 is only one unit outside the range. 'Inside the range' means between 10 and 50, with genuinely recorded data on both sides — being close to the boundary doesn't confer safety, because the data still stops exactly at 10, and 9 is on the wrong side of that stop.

Traps tested: Extrapolation recognised only above the range · Range boundary not checked · Near boundary extrapolation mistaken for safe

Question 42 marks

A student correctly finds $S_{xx} = 25$ and $S_{xy} = 40$ for a dataset, and correctly finds $\bar x = 4$ , $\bar y = 11$ . Which of these is an acceptable way to state the final regression line?

$y = 4.6 + 1.6x$
Correct. $b = \frac{40}{25} = 1.6$ ; $a = 11 - 1.6 \times 4 = 11 - 6.4 = 4.6$ . Both values are decimals, and they're assembled into the actual equation of the line.
B $y = 4.6 + \frac{8}{5}x$
Numerically identical to the correct line ( $\frac{8}{5} = 1.6$ ), and marked wrong anyway: a real report states plainly that fractions are not accepted in a final regression equation, since a and b are both estimates and a fractional answer isn't an appropriate way to state one.
C $a = 4.6$ , $b = 1.6$
Both values are right and the question was never actually answered — it asked for the equation of the line, and two separate numbers sitting on the page are not that equation. A real examiner report records exactly this: candidates who found a and b correctly but never wrote them back up as y = a + bx, losing the final mark for stopping one step early.
D $x = 4.6 + 1.6y$
This states x as a function of y, but the question asked for the regression line of y ON x. Writing the variables the wrong way round doesn't just relabel the same line — it's a genuinely different line in general, since y-on-x and x-on-y minimise distances in different directions.

Traps tested: Fraction left in final line · Stops after a and b never states the line · Regression direction reversed

Practice this for real

This site teaches the mechanism; the exam is sat on Pearson's own real questions. Go find and attempt these yourself — nothing here substitutes for actually sitting a timed paper.

Examiner report: Jan 2023 · Q6(c) — cited directly in this lesson

Pearson's official past-papers portal

Select International Advanced Level → Mathematics → any series, then look for WST01.

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Up next

Discrete random variables — the probability function and the discrete uniform distribution

A probability function isn't finished the moment every value looks reasonable on its own — it's finished when the whole list adds up to exactly 1. A real WST01 examiner report names the single most common way this goes wrong, and it isn't a miscalculation: it's forgetting that a value — often x=0 specifically — belongs in the domain at all. This lesson builds the probability function and the cumulative distribution function from the one fact that makes that self-check work (\sum p(x) = 1, because X is certain to take *some* value), then uses a genuinely rich real exam question — two four-sided dice, from the most recent series reviewed for this course — to show a discrete uniform distribution, a probability, a mean found "by symmetry," and a variance that has to be summed by hand all compose inside one real question.

50 min

Regression — gradient interpretation and extrapolation/reliability

Scatter diagrams, and which variable explains which

Fitting the line: what SxxS_{xx}Sxx​ and SxyS_{xy}Sxy​ are, and what the formula booklet already gives you

Why 'as x increases, y increases' is not an interpretation

Why extrapolation is a real danger, not a caution to recite

Two predictions from the same line — why one is trustworthy and the other isn't

Complete the chain — is a prediction at x = 0 (no advertising at all) reliable?

Deep dive

Practice this for real

Fitting the line: what $S_{xx}$ and $S_{xy}$ are, and what the formula booklet already gives you