What is the difference between correlation and regression in Statistics 1 for Edexcel International A Levels?

In Statistics 1 for Edexcel International A Levels, correlation measures the strength and direction of a linear relationship between two variables, typically represented by the correlation coefficient. Regression, on the other hand, involves predicting the value of a dependent variable based on the value of at least one independent variable, using a regression line or equation. While correlation quantifies the relationship, regression provides a model for prediction.

How do you interpret the correlation coefficient in the context of Edexcel International A Levels Mathematics?

In Edexcel International A Levels Mathematics, the correlation coefficient, denoted as 'r', ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 suggests no linear relationship. Values closer to 1 or -1 imply a stronger linear relationship, while values near 0 indicate a weaker relationship.

What are the assumptions of linear regression in the Statistics 1 syllabus for Edexcel International A Levels?

The assumptions of linear regression in the Statistics 1 syllabus for Edexcel International A Levels include linearity, independence, homoscedasticity, and normality. Linearity assumes a straight-line relationship between the variables. Independence implies that the residuals are independent. Homoscedasticity means that the residuals have constant variance, and normality assumes that the residuals are normally distributed.

How can outliers affect correlation and regression analysis in Edexcel International A Levels Statistics 1?

In Edexcel International A Levels Statistics 1, outliers can significantly affect both correlation and regression analysis. Outliers can distort the correlation coefficient, making it appear stronger or weaker than it actually is. In regression, outliers can skew the regression line, leading to inaccurate predictions. It's important to identify and assess the impact of outliers before drawing conclusions from the data.

Why is it important to distinguish between correlation and causation in Edexcel International A Levels Mathematics?

In Edexcel International A Levels Mathematics, distinguishing between correlation and causation is crucial because correlation does not imply causation. Two variables may be correlated without one causing the other. This distinction is important to avoid incorrect conclusions about the relationships between variables. Understanding this helps in making informed decisions based on statistical analysis.

Why is "Inferring causation from a high $r$" a common mistake in Correlation and regression?

Why it happens: Strong correlations feel like 'proof'. How to avoid it: Always answer 'no — correlation does not prove causation' and name a plausible confounding variable. Mark schemes routinely award marks for this exact phrasing. Source: WST01/01 examiner reports — annual

Why is "Predicting $y$ at $x$ far outside the data range without warning" a common mistake in Correlation and regression?

Why it happens: Mechanical substitution into the regression equation. How to avoid it: Check whether x lies in the observed range. If not, explicitly call the prediction 'extrapolation' and mark it as unreliable.

Why is "Claiming 'no relationship' when $r \approx 0$" a common mistake in Correlation and regression?

Why it happens: Forgetting that r only measures LINEAR relationships. How to avoid it: r 0 means no LINEAR relationship. A perfect parabola y = x^2 on [-1, 1] has r = 0. Plot the scatter diagram before drawing conclusions.

Why is "Confusing $y$ on $x$ with $x$ on $y$" a common mistake in Correlation and regression?

Why it happens: The two regression lines are different. Symmetry of r does not extend to b. How to avoid it: Read the question. The line of y on x has slope S_xy/S_xx and is used to predict y from x. The line of x on y has slope S_xy/S_yy.

Why is "Forgetting to back-transform the regression line after coding" a common mistake in Correlation and regression?

Why it happens: Confusion between PMCC (invariant) and slope/intercept (not invariant). How to avoid it: PMCC is invariant under positive linear coding. Slope and intercept must be back-transformed using the inverse coding. Show the substitution explicitly. Source: WST01/01 examiner reports — recurring

Why is "Saying 'strong positive correlation' without naming the variables" a common mistake in Correlation and regression?

Why it happens: Generic statistical language. How to avoid it: Use the actual variable names: 'As temperature increases, ice cream sales increase' — not just 'positive correlation'.

Edexcel International A Levels Mathematics (XMA01-YMA01)

Correlation and regression

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Short Notes - Correlation and regression

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Detailed Study Notes

Detailed notes on Statistics 1 for Edexcel International A Levels Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Correlation and Regression Study Notes — Edexcel IAL Mathematics S1 (WST01/01, 2018 spec — 2026 onwards)

Two related ideas for bivariate data $(x, y)$ : $r$ (PMCC) measures the strength and direction of LINEAR association; the regression line of $y$ on $x$ is the best-fit straight line for predicting $y$ from $x$ . Both rely on the booklet sums $S_{xx}$ , $S_{yy}$ , $S_{xy}$ . Discipline matters — interpretation in context, awareness of extrapolation, and the famous 'correlation $\neq$ causation' caveat.

At a glance

$S_{xx} = \sum x^{2} - (\sum x)^{2}/n$ and similarly for $S_{yy}$ , $S_{xy}$ .
PMCC $r = S_{xy}/\sqrt{S_{xx} S_{yy}}$ , with $-1 \le r \le 1$ .
$r = +1$ perfect positive line; $r = -1$ perfect negative; $r \approx 0$ no linear pattern.
Regression $y$ on $x$ : $b = S_{xy}/S_{xx}$ , $a = \bar{y} - b\bar{x}$ , line passes through $(\bar{x}, \bar{y})$ .
Coding: PMCC is invariant under positive linear coding; slope and intercept are not.
Interpolation inside data range — reliable. Extrapolation outside — unreliable.
Correlation $\neq$ causation — always mention confounding variables.

What you’ll learn

Mapped to the Pearson Edexcel International A Levels XMA01-YMA01 syllabus (2026 onwards).

S1 5.1 — Calculate and interpret the product-moment correlation coefficient $r$ .
S1 5.2 — Understand the limitations of $r$ (linearity, causation, outliers).
S1 5.3 — Calculate the equation of the regression line of $y$ on $x$ using least squares.
S1 5.4 — Use the regression line to estimate $y$ values within the data range and discuss the reliability of estimates outside the range.

Scatter diagrams and the PMCC

Always plot first; then $r = S_{xy}/\sqrt{S_{xx} S_{yy}}$ measures the LINEAR pattern.

Scatter diagram first. Before computing any number, plot the data. The picture tells you whether a linear model is even sensible. A clear curve, U-shape, or fan-out should make you cautious.

The PMCC.

$r = \dfrac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}, \qquad -1 \le r \le 1.$

Value of $r$	Meaning
$r = 1$	Perfect positive linear pattern
$r \approx 0.8$	Strong positive linear pattern
$r \approx 0.5$	Moderate positive
$r \approx 0$	No linear pattern
$r \approx -0.7$	Strong negative
$r = -1$	Perfect negative

Compute via the booklet sums. Plug summary statistics ( $\sum x$ , $\sum y$ , $\sum x^{2}$ , $\sum y^{2}$ , $\sum xy$ , $n$ ) into

$S_{xx} = \sum x^{2} - \dfrac{(\sum x)^{2}}{n}, \;\; S_{yy} = \sum y^{2} - \dfrac{(\sum y)^{2}}{n}, \;\; S_{xy} = \sum xy - \dfrac{\sum x \sum y}{n}.$

Then plug into the PMCC formula.

Worked example. $n = 10$ , $\sum x = 50$ , $\sum y = 70$ , $\sum x^{2} = 285$ , $\sum y^{2} = 540$ , $\sum xy = 386$ .

$S_{xx} = 285 - 250 = 35$ .
$S_{yy} = 540 - 490 = 50$ .
$S_{xy} = 386 - 350 = 36$ .
$r = 36/\sqrt{1750} \approx 0.861$ — strong positive linear correlation.

Least-squares regression line of y on x passes through the centroid (x̄, ȳ) and best fits the scatter pattern.

Always plot the scatter diagram first.
Use booklet sums $S_{xx}$ , $S_{yy}$ , $S_{xy}$ .
$r \in [-1, 1]$ ; close to $\pm 1$ = strong linear.
$r \approx 0$ means no LINEAR pattern (possibly non-linear).

Regression line of $y$ on $x$

Best-fit line $y = a + bx$ with $b = S_{xy}/S_{xx}$ and $a = \bar{y} - b\bar{x}$ .

Goal. Find the straight line that best summarises the linear relationship for predicting $y$ from $x$ .

Method (least squares). Choose $a, b$ to minimise

$\sum (y_i - (a + bx_i))^{2}.$

The minimisers (in the formula booklet) are

$b = \dfrac{S_{xy}}{S_{xx}}, \qquad a = \bar{y} - b\bar{x}.$

So the regression line is

$y = a + bx = \bar{y} + b(x - \bar{x}).$

It always passes through the mean point $(\bar{x}, \bar{y})$ .

Worked example. Same data as before: $\bar{x} = 5$ , $\bar{y} = 7$ , $S_{xy} = 36$ , $S_{xx} = 35$ .

$b = 36/35 \approx 1.029$ .
$a = 7 - 1.029 \times 5 \approx 1.857$ .
Line: $y \approx 1.86 + 1.03 x$ .

Slope interpretation. A one-unit increase in $x$ is associated with an estimated increase of $b$ units in $y$ . Phrase it in context: 'each extra hour studied is associated with an estimated $1.03$ -mark increase in test score'.

$y$ on $x$ vs $x$ on $y$ . These are DIFFERENT lines. $y$ on $x$ has slope $S_{xy}/S_{xx}$ and minimises vertical residuals; $x$ on $y$ has slope $S_{xy}/S_{yy}$ and minimises horizontal residuals. They coincide only when $r = \pm 1$ .

$b = S_{xy}/S_{xx}$ , $a = \bar{y} - b\bar{x}$ .
Line passes through $(\bar{x}, \bar{y})$ .
Slope: 'a unit change in $x$ is associated with $b$ units change in $y$ '.
$y$ on $x$ and $x$ on $y$ are different lines.

Using the regression line — interpolation and extrapolation

Predict $y$ for $x$ inside the data range; refuse (or qualify) for $x$ outside.

Interpolation. Substituting an $x$ within the observed range into the regression line gives an estimate of the corresponding $y$ . This is interpolation and is generally reliable, especially when $r$ is high.

Extrapolation. Substituting an $x$ outside the observed range gives an estimate that may be far from reality. The linear pattern observed in the data is not guaranteed to continue. Examples of failure modes:

Saturation: an effect levels off at high $x$ .
Sign change: the relationship reverses outside the range.
Non-linear regime: the true relationship is quadratic or exponential, which looks linear over a small window.

Worked example. Regression $y = 12 + 1.4x$ from data with $x \in [5, 25]$ , $r = 0.96$ .

$x = 18$ : $y = 37.2$ — within range, interpolation, reliable.
$x = 60$ : $y = 96$ — well outside range, extrapolation, unreliable.

Best-practice template for an exam answer.

'This estimate uses the regression line at $x = X$ . Since $X$ lies within / outside the data range $[a, b]$ , the estimate is reliable / unreliable. [If unreliable]: the linear model may not extend beyond the data, so the prediction at $X$ is an extrapolation and should be treated with caution.'

Interpolation = $x$ inside range = reliable.
Extrapolation = $x$ outside range = unreliable.
High $r$ does NOT excuse extrapolation.
Always cite the data range explicitly.

Coding and its effect on $r$ and the regression line

$r$ is invariant (under positive scaling); slope and intercept need back-transformation.

Coding simplifies arithmetic when raw data has large magnitudes or awkward offsets. For example $u = x - 100$ , $v = (y - 50)/2$ converts heights around $100$ cm and weights around $50$ kg to small whole numbers.

Effect on $r$ . For linear codings $u = \alpha x + \beta$ , $v = \gamma y + \delta$ with $\alpha, \gamma > 0$ :

$r_{uv} = r_{xy}.$

The PMCC is dimensionless and unaffected by shift or positive scaling. (A negative scale factor flips the sign of $r$ .)

Effect on the regression line. Slope and intercept DO change.

Method.

Find the regression line of $v$ on $u$ from the coded data.
Substitute the coding relations $u = \alpha x + \beta$ , $v = \gamma y + \delta$ .
Rearrange to express $y$ in terms of $x$ .

Worked example. $u = x - 100$ , $v = (y - 50)/2$ . Coded regression: $v = 1.5 + 0.4 u$ .

Substitute: $(y - 50)/2 = 1.5 + 0.4(x - 100)$ .
Multiply by $2$ : $y - 50 = 3 + 0.8(x - 100) = 3 + 0.8x - 80$ .
Rearrange: $y = -27 + 0.8 x$ .

So the regression line of $y$ on $x$ is $y = -27 + 0.8x$ . PMCC unchanged at whatever value it had on the coded data.

PMCC invariant under positive linear coding.
Slope and intercept must be back-transformed.
Substitute coding relations into the coded regression line.
Verify by checking the line passes through $(\bar{x}, \bar{y})$ .

Limitations and the correlation-causation trap

PMCC measures association, not cause. Outliers and non-linearity wreck it. Beware extrapolation.

Correlation does not imply causation. A high $r$ between two variables can arise from:

$X$ causes $Y$ (the interpretation we'd like).
$Y$ causes $X$ (reverse causation).
A confounding variable $Z$ causes both — neither $X$ nor $Y$ causes the other.
Pure coincidence in a small sample.

Famous example. Ice cream sales and drowning deaths are strongly correlated across summers. Neither causes the other; hot weather (confounder) causes both.

Outliers. A single extreme point can shift $r$ substantially. Always check the scatter diagram, identify outliers, and consider whether they are real measurements, recording errors, or come from a different population.

Non-linearity. $r \approx 0$ does not mean 'no relationship' — only 'no linear relationship'. Quadratic, exponential, sinusoidal relationships can have $r = 0$ over a symmetric range.

Extrapolation. Reliable predictions require interpolation within the observed range of $x$ .

Linearity assumption. The regression line is only sensible if the scatter looks roughly linear. Otherwise transform the variables (e.g. fit $\log y$ vs $x$ for exponential data) or use a different model.

Correlation $\neq$ causation: list confounders.
Outliers can dominate $r$ — check the scatter.
$r \approx 0$ may hide a non-linear pattern.
Avoid extrapolation; transform variables if a curve fits better.

Quick recap

$S_{xx}, S_{yy}, S_{xy}$ are the booklet sums.
PMCC: $r = S_{xy}/\sqrt{S_{xx} S_{yy}}$ — measures LINEAR correlation only.
Regression $y$ on $x$ : $b = S_{xy}/S_{xx}$ , $a = \bar{y} - b\bar{x}$ .
Line passes through $(\bar{x}, \bar{y})$ .
Interpolation reliable; extrapolation not.
PMCC invariant under positive linear coding; regression line is not.
Correlation $\neq$ causation; cite confounders.

Memorise this

Verbatim phrases and definitions Edexcel mark schemes credit.

$S_{xx}, S_{yy}, S_{xy}$ : $\sum x^{2} - (\sum x)^{2}/n$ etc.
PMCC: $r = S_{xy}/\sqrt{S_{xx} S_{yy}}$ .
Range: $-1 \le r \le 1$ .
Slope of regression $y$ on $x$ : $b = S_{xy}/S_{xx}$ .
Intercept: $a = \bar{y} - b\bar{x}$ .
Line passes through $(\bar{x}, \bar{y})$ .

How it’s examined

Correlation/regression is a Q7-Q9 fixture in WST01/01, worth $10$ - $14$ marks in total per paper. Typical tasks: compute $r$ from summary statistics with an interpretation ( $5$ - $6$ marks); find the regression line $y = a + bx$ and use it to estimate ( $5$ - $7$ marks); discuss reliability of an estimate as interpolation or extrapolation ( $2$ - $3$ marks); back-transform a coded regression line ( $4$ - $6$ marks); explain why a high $r$ does not imply causation ( $2$ - $4$ marks). Marks are routinely lost for generic interpretation without naming variables and for accepting extrapolation without comment.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WST01/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

Take this whole topic with you

Step-by-step worked examples — Correlation and regression

Step-by-step solutions to past-paper-style questions on correlation and regression, written exactly the way a tutor would explain them at the board.

1Product-moment correlation coefficient $r$ (5 marks, S1)
Extended• Adapted from WST01/01 January 2024 Q8• correlation, PMCC, r
Question
For a sample of $n = 10$ pairs $(x, y)$ : $\sum x = 50$ , $\sum y = 70$ , $\sum x^{2} = 285$ , $\sum y^{2} = 540$ , $\sum xy = 386$ . Calculate the product-moment correlation coefficient $r$ and interpret it.
Step-by-step solution
1. Step 1
  Compute the corrected sums.
  $S_{xx} = \sum x^{2} - \dfrac{(\sum x)^{2}}{n} = 285 - \dfrac{50^{2}}{10} = 285 - 250 = 35$
2. Step 2
  $S_{yy}$ .
  $S_{yy} = 540 - \dfrac{70^{2}}{10} = 540 - 490 = 50$
3. Step 3
  $S_{xy}$ .
  $S_{xy} = 386 - \dfrac{50 \times 70}{10} = 386 - 350 = 36$
4. Step 4
  Apply the PMCC formula (in the booklet):
  $r = \dfrac{S_{xy}}{\sqrt{S_{xx} S_{yy}}} = \dfrac{36}{\sqrt{35 \times 50}} = \dfrac{36}{\sqrt{1750}}$
5. Step 5
  Evaluate:
  $r = \dfrac{36}{41.83} \approx 0.861 \;(3\,\text{sf})$
6. Step 6
  Interpretation: $r \approx 0.861$ is close to $+1$ , indicating a strong positive linear correlation between $x$ and $y$ — as $x$ increases, $y$ tends to increase.
Answer
$S_{xx} = 35$ , $S_{yy} = 50$ , $S_{xy} = 36$ . $r \approx 0.861$ — strong positive linear correlation.
Examiner tip
M1 for $S_{xx}$ , $S_{yy}$ , $S_{xy}$ (one M each, max $2$ ). A1 for $36$ and another value. M1 for substituting into $r$ . A1 for $r \approx 0.861$ . B1 for the contextual interpretation. Mark schemes accept any wording that links direction + strength + the named variables.
2Regression line of $y$ on $x$ (6 marks, S1)
Extended• Adapted from WST01/01 June 2024 Q9• regression, y on x, prediction
Question
Using the same data ( $n = 10$ , $\bar{x} = 5$ , $\bar{y} = 7$ , $S_{xx} = 35$ , $S_{xy} = 36$ ), find the equation of the regression line of $y$ on $x$ in the form $y = a + bx$ . Use the line to estimate $y$ when $x = 4$ , and comment on the reliability of this estimate (the data has $x$ values between $1$ and $9$ ).
Step-by-step solution
1. Step 1
  Slope:
  $b = \dfrac{S_{xy}}{S_{xx}} = \dfrac{36}{35} \approx 1.029$
2. Step 2
  Intercept:
  $a = \bar{y} - b \bar{x} = 7 - 1.029 \times 5 \approx 7 - 5.143 = 1.857$
3. Step 3
  Equation:
  $y = 1.857 + 1.029 x \;(3\,\text{dp})$
4. Step 4
  Estimate at $x = 4$ :
  $y \approx 1.857 + 1.029 \times 4 = 1.857 + 4.114 = 5.97$
5. Step 5
  Reliability: $x = 4$ lies within the data range ( $1$ to $9$ ), so this is interpolation — relatively reliable. Extrapolating outside the range (e.g. $x = 20$ ) would be unsafe.
Answer
$y = 1.857 + 1.029 x$ . At $x = 4$ , $y \approx 5.97$ . Reliable (interpolation within data range $1$ - $9$ ).
Examiner tip
M1A1 for $b$ . M1A1 for $a$ . A1 for the equation. M1A1 for the estimate. B1 for the reliability comment with mention of interpolation. Examiner reports flag the omission of 'within range' as a recurring mark loss.
3Correlation does not imply causation (4 marks, S1)
Core• Adapted from WST01/01 June 2023 Q7• correlation, causation, interpretation
Question
A researcher finds $r = 0.92$ between the number of swimming pool drowning deaths and the per-capita consumption of ice cream in a country across $20$ summers. (a) Interpret this value. (b) Does this prove that eating ice cream causes drownings? Explain.
Step-by-step solution
1. Step 1
  $r = 0.92$ indicates strong positive linear correlation — more ice cream consumption is associated with more drownings across the summers in this data set.
2. Step 2
  No, this does not prove causation. Correlation only measures association, not cause.
3. Step 3
  A plausible alternative explanation is a confounding variable: hot weather increases both ice cream consumption and the number of people swimming (and so the number of drownings). The hot weather causes both — neither variable causes the other.
4. Step 4
  To establish causation we would need a controlled experiment in which ice cream consumption is varied while everything else (including weather, swimming behaviour) is held constant — generally infeasible here.
Answer
(a) Strong positive linear correlation between drownings and ice cream sales. (b) No — correlation is not causation. A confounding variable (hot weather) plausibly explains the association.
Examiner tip
B1 for interpreting the value. B1 for stating 'correlation $\neq$ causation'. B1 for naming a confounding variable (heat, weather, season). B1 for explaining how the confounder produces the correlation. Examiners reward sensible context-specific confounders.
4Effect of coding on $r$ and on the regression line (6 marks, S1)
Extended• Adapted from WST01/01 January 2023 Q8• coding, correlation, regression
Question
Variables $x$ and $y$ are coded as $u = x - 100$ and $v = (y - 50)/2$ . For the coded data: $r_{uv} = 0.87$ , regression of $v$ on $u$ is $v = 1.5 + 0.4 u$ . (a) State the value of $r_{xy}$ . (b) Find the regression line of $y$ on $x$ .
Step-by-step solution
1. Step 1
  PMCC is invariant under linear coding with positive scale factors. Since both $u$ and $v$ use linear codings $u = 1 \cdot x + (-100)$ and $v = (1/2)y + (-25)$ , both with positive scale factors, $r_{xy} = r_{uv} = 0.87$ .
2. Step 2
  Back-transform the regression line. From $v = 1.5 + 0.4 u$ : substitute $u = x - 100$ and $v = (y - 50)/2$ :
  $\dfrac{y - 50}{2} = 1.5 + 0.4(x - 100)$
3. Step 3
  Multiply by $2$ :
  $y - 50 = 3 + 0.8(x - 100) = 3 + 0.8x - 80$
4. Step 4
  Rearrange:
  $y = 50 - 80 + 3 + 0.8x = -27 + 0.8x$
5. Step 5
  So the regression line of $y$ on $x$ is $y = -27 + 0.8 x$ .
Answer
(a) $r_{xy} = 0.87$ (invariant under linear coding). (b) $y = -27 + 0.8 x$ .
Examiner tip
B1 for $r_{xy} = 0.87$ with reason. M1 for substituting coded values into the regression line. M1 for clearing the denominator. A1 for the slope $0.8$ . A1 for the intercept $-27$ . A1 for the equation. The reverse-coding requires care with signs — a frequent mark loser.
5Extrapolation and limits of linearity (5 marks, S1)
Extended• Adapted from WST01/01 January 2024 Q9• regression, extrapolation, limitations
Question
A regression line $y = 12 + 1.4x$ was fitted to data with $x$ in the range $5 \le x \le 25$ . The PMCC was $r = 0.96$ . (a) Use the line to estimate $y$ at $x = 18$ and at $x = 60$ . (b) Comment on the reliability of each estimate.
Step-by-step solution
1. Step 1
  At $x = 18$ : $y = 12 + 1.4 \times 18 = 12 + 25.2 = 37.2$ .
2. Step 2
  At $x = 60$ : $y = 12 + 1.4 \times 60 = 12 + 84 = 96$ .
3. Step 3
  Reliability of $x = 18$ : this lies within the data range $[5, 25]$ , so it is interpolation. The high PMCC ( $r = 0.96$ ) means the linear model fits well; the estimate is reliable.
4. Step 4
  Reliability of $x = 60$ : this lies far outside the data range. This is extrapolation, and we have no evidence that the linear relationship continues beyond $x = 25$ . The estimate $96$ is unreliable — there could be saturation, change of regime, or measurement issues at high $x$ .
Answer
(a) $y(18) = 37.2$ ; $y(60) = 96$ . (b) $y(18)$ is reliable (interpolation; high $r$ ). $y(60)$ is unreliable (extrapolation outside data range).
Examiner tip
M1A1 for each estimate. B1 for stating that $x = 18$ is within range = interpolation. B1 for stating $x = 60$ is outside range = extrapolation. B1 for the consequence (unreliable). Mark schemes do NOT credit 'unreliable because $r$ is not $1$ ' — the issue is data range, not perfection of fit.

Key Formulae — Correlation and regression

The formulae you need to memorise for correlation and regression on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Corrected sums $S_{xx}$ , $S_{yy}$ , $S_{xy}$
$S_{xx} = \sum x^{2} - \dfrac{(\sum x)^{2}}{n}, \;\; S_{yy} = \sum y^{2} - \dfrac{(\sum y)^{2}}{n}, \;\; S_{xy} = \sum xy - \dfrac{\sum x \sum y}{n}$
$n$
sample size
$x, y$
the two variables
When to use
First step in any correlation or regression calculation. Plug in the summary statistics, then use $r$ and $b$ formulas. ARE in the IAL formula booklet under S1.
Example
$\sum x = 50$ , $\sum x^{2} = 285$ , $n = 10$ : $S_{xx} = 285 - 250 = 35$ .
Product-moment correlation coefficient $r$
$r = \dfrac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}$
$r$
PMCC, $-1 \le r \le 1$
When to use
Measures strength + direction of LINEAR correlation. Close to $\pm 1$ = strong; near $0$ = weak (or non-linear). IS in the IAL formula booklet.
Example
$S_{xy} = 36$ , $S_{xx} = 35$ , $S_{yy} = 50$ : $r = 36/\sqrt{1750} \approx 0.861$ .
Regression line of $y$ on $x$
$y = a + bx, \quad b = \dfrac{S_{xy}}{S_{xx}}, \quad a = \bar{y} - b\bar{x}$
$b$
slope of regression line
$a$
intercept
$\bar{x}, \bar{y}$
sample means
When to use
Best-fit line minimising sum of squared $y$ -residuals. The line always passes through $(\bar{x}, \bar{y})$ . ARE in the IAL formula booklet.
Example
$S_{xy} = 36$ , $S_{xx} = 35$ , $\bar{x} = 5$ , $\bar{y} = 7$ : $b \approx 1.029$ , $a \approx 1.857$ , so $y \approx 1.86 + 1.03 x$ .
PMCC invariance under linear coding
$u = a_1 x + b_1, \;\; v = a_2 y + b_2 \;\; (a_1, a_2 > 0) \Rightarrow r_{uv} = r_{xy}$
$u, v$
coded variables
When to use
When data has been coded for arithmetic. Carry the $r$ from the coded variables directly — no recalculation needed. If a coding scale factor is negative, $r$ flips sign. NOT explicitly in the booklet — definitional consequence.
Example
$u = x - 100$ , $v = (y - 50)/2$ , $r_{uv} = 0.87$ : $r_{xy} = 0.87$ .

Key Definitions and Keywords — Correlation and regression

Definitions to memorise and the exact keywords mark schemes credit for correlation and regression answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Bivariate data
Examiner keyword
Paired observations $(x_i, y_i)$ on two variables. Displayed in a scatter diagram. The aim of correlation/regression is to describe how $y$ depends on $x$ .
Scatter diagram
Examiner keyword
Plot of $(x_i, y_i)$ points. Used to spot patterns: linear, non-linear, no pattern, clusters, outliers. Always plot before computing $r$ .
Product-moment correlation coefficient (PMCC)
Examiner keyword
Number $r \in [-1, 1]$ measuring the strength and direction of the LINEAR relationship. $r = 1$ perfect positive line; $r = -1$ perfect negative line; $r \approx 0$ no linear pattern (could still be a non-linear one).
Regression line of $y$ on $x$
Examiner keyword
Straight line $y = a + bx$ minimising $\sum(y_i - (a + bx_i))^{2}$ . Passes through $(\bar{x}, \bar{y})$ . Used to predict $y$ from $x$ — but only the line of $y$ on $x$ (not $x$ on $y$ , which is different).
Interpolation / extrapolation
Examiner keyword
Interpolation: predicting $y$ for an $x$ inside the data range — reasonably reliable. Extrapolation: predicting $y$ for an $x$ outside the data range — unreliable, since the linear model may not extend.
Causation vs correlation
Examiner keyword
Correlation = statistical association. Causation = one variable produces the change in the other. A non-zero $r$ does NOT prove causation — confounding variables (lurking) often explain the association.

Common Mistakes and Misconceptions — Correlation and regression

The traps other students keep falling into on correlation and regression questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Inferring causation from a high $r$
WST01/01 examiner reports — annual
Why it happens
Strong correlations feel like 'proof'.
How to avoid it
Always answer 'no — correlation does not prove causation' and name a plausible confounding variable. Mark schemes routinely award marks for this exact phrasing.
✕Predicting $y$ at $x$ far outside the data range without warning
Why it happens
Mechanical substitution into the regression equation.
How to avoid it
Check whether $x$ lies in the observed range. If not, explicitly call the prediction 'extrapolation' and mark it as unreliable.
✕Claiming 'no relationship' when $r \approx 0$
Why it happens
Forgetting that $r$ only measures LINEAR relationships.
How to avoid it
$r \approx 0$ means no LINEAR relationship. A perfect parabola $y = x^{2}$ on $[-1, 1]$ has $r = 0$ . Plot the scatter diagram before drawing conclusions.
✕Confusing $y$ on $x$ with $x$ on $y$
Why it happens
The two regression lines are different. Symmetry of $r$ does not extend to $b$ .
How to avoid it
Read the question. The line of $y$ on $x$ has slope $S_{xy}/S_{xx}$ and is used to predict $y$ from $x$ . The line of $x$ on $y$ has slope $S_{xy}/S_{yy}$ .
✕Forgetting to back-transform the regression line after coding
WST01/01 examiner reports — recurring
Why it happens
Confusion between PMCC (invariant) and slope/intercept (not invariant).
How to avoid it
PMCC is invariant under positive linear coding. Slope and intercept must be back-transformed using the inverse coding. Show the substitution explicitly.
✕Saying 'strong positive correlation' without naming the variables
Why it happens
Generic statistical language.
How to avoid it
Use the actual variable names: 'As temperature increases, ice cream sales increase' — not just 'positive correlation'.

Correlation and regression — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Correlation and regression

Short Notes - Correlation and regression

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Product-moment correlation coefficient rrr (5 marks, S1)

2Regression line of yyy on xxx (6 marks, S1)

3Correlation does not imply causation (4 marks, S1)

4Effect of coding on rrr and on the regression line (6 marks, S1)

5Extrapolation and limits of linearity (5 marks, S1)

Corrected sums SxxS_{xx}Sxx​, SyyS_{yy}Syy​, SxyS_{xy}Sxy​

Product-moment correlation coefficient rrr

Regression line of yyy on xxx

PMCC invariance under linear coding

Bivariate data

Scatter diagram

Product-moment correlation coefficient (PMCC)

Regression line of yyy on xxx

Interpolation / extrapolation

Causation vs correlation

✕Inferring causation from a high rrr

✕Predicting yyy at xxx far outside the data range without warning

✕Claiming 'no relationship' when r≈0r \approx 0r≈0

✕Confusing yyy on xxx with xxx on yyy

✕Forgetting to back-transform the regression line after coding

✕Saying 'strong positive correlation' without naming the variables

Correlation and regression — frequently asked questions

1Product-moment correlation coefficient $r$ (5 marks, S1)

2Regression line of $y$ on $x$ (6 marks, S1)

4Effect of coding on $r$ and on the regression line (6 marks, S1)

Corrected sums $S_{xx}$ , $S_{yy}$ , $S_{xy}$

Product-moment correlation coefficient $r$

Regression line of $y$ on $x$

Regression line of $y$ on $x$

✕Inferring causation from a high $r$

✕Predicting $y$ at $x$ far outside the data range without warning

✕Claiming 'no relationship' when $r \approx 0$

✕Confusing $y$ on $x$ with $x$ on $y$