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Grade 12 CAPS: bivariate data and scatter plots, the least-squares regression line, the correlation coefficient r, and interpolation vs. extrapolation — the final piece of the 3-year Statistics course, built on everything Grade 10 and 11 already taught.

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Grade 12 CAPS Mathematics

Statistics

How two-variable (bivariate) data relates, and how to summarise, quantify and use that relationship. Your calculator does the heavy arithmetic here — your job is knowing what to ask it for, and what the answer actually means.

Scope check:Official CAPS requirement

The 3-Year Statistics Journey — Complete

This is the final stop.

10
Build

Central tendency • grouped data • quartiles • box & whisker

11
Extend

Histograms • ogives • variance & standard deviation • outliers

12
Apply

Bivariate data • scatter plots • regression & correlation

Assumed knowledge
This page assumes Grade 10/11's mean, median, quartiles, standard deviation and skewness are already comfortable. Everything here is genuinely new: it's the first time you work with TWO variables at once instead of one.

What CAPS Actually Asks in Grade 12

One week, five connected skills, all about relating two variables.

  • 1

    Represent bivariate data as a scatter plot and judge by eye whether it looks linear, quadratic or exponential.

  • 2

    Use a calculator to find the equation of the least-squares regression line.

  • 3

    Use a calculator to find the correlation coefficient \(r\), and interpret it in terms of strength and direction.

  • 4

    Use the regression equation to interpolate and extrapolate, and discuss which is more reliable.

  • 5

    Revise symmetric and skewed data and be ready to discuss skewness — CAPS's own wording for this topic explicitly names it alongside interpolation and extrapolation.

By calculator, not by hand
Unlike Grade 11's manual variance calculation, CAPS Grade 12 expects the regression line and \(r\) to come from your calculator's regression mode. The exam tests whether you know what to DO with those values, not whether you can derive the formulas from scratch.
Where skewness actually shows up
In real NSC and provincial papers, the skewness part is almost always a short 1–2 mark sub-question — "describe/comment on the skewness of the data" — sitting inside the SAME multi-part Question 1 as your regression work, but attached to a DIFFERENT data set (a five-number summary, box plot or grouped table), not the \(x\)/\(y\) pairs you're regressing. It's a straight revision of the Grade 11 skill, just repackaged into a Grade 12 question.

Bivariate Data and Scatter Plots

Two measurements per subject, plotted as one point each.

What "bivariate" means

Everything in Grade 10 and 11 was UNIVARIATE — one variable per subject (just test scores, just heights). Bivariate data measures TWO variables on the SAME set of subjects — e.g. each learner's study hours (\(x\)) AND test score (\(y\)) together. Each learner contributes one point \((x,y)\).

Reading the shape

Before calculating anything, look at the overall pattern: does it rise roughly in a straight line (linear), curve up then down or down then up (quadratic), or start slow and rocket upward (exponential)?

Why the shape matters first
A regression LINE only makes sense if the data actually looks roughly linear. Fitting a straight line to clearly curved data would misrepresent the relationship — always look before you calculate.

Worked Example — Judging the Shape

Plotting the points yourself first, then classifying three already-drawn patterns.

Worked example — plotting a scatter plot from a table Level 1

Five seedlings' daily hours of sunlight (\(x\)) and height after two weeks in cm (\(y\)) are: \((2,8),\ (4,15),\ (6,19),\ (8,25),\ (10,30)\). Plot this as a scatter plot.

Show solution
  1. 1Draw and label the horizontal (\(x\)) and vertical (\(y\)) axes, scaled to cover the data's own range.
  2. 2For EACH pair, find the \(x\)-value on the horizontal axis, move straight up to the height given by the \(y\)-value, and mark a single point there — one point per \((x,y)\) pair, nothing joined.
  3. 3Plot all 5: \(\boxed{(2,8),(4,15),(6,19),(8,25),(10,30)}\)
Sunlight (hours) Height (cm)
Each of the 5 points plotted individually — no line joins them yet, that only comes later once you fit a regression line.
Worked example — classify each scatter plot Level 1–2

For each of the three scatter plots below, state whether the pattern looks linear, quadratic or exponential.

Panel A
Panel B
Panel C
Show solution
  1. 1Panel A: the points rise at a steady, roughly constant rate all the way across — \(\boxed{\text{linear}}\)
  2. 2Panel B: the points fall to a minimum in the middle, then rise again — a genuine U-shape — \(\boxed{\text{quadratic}}\)
  3. 3Panel C: the points stay low and nearly flat at first, then climb increasingly steeply — the classic "slow-then-rocket" shape — \(\boxed{\text{exponential}}\)

The Least-Squares Regression Line

The single straight line that best fits a linear-looking scatter plot.

The equation

\(\hat{y}=a+bx\) — \(a\) is the \(y\)-intercept, \(b\) is the gradient. The "hat" on \(\hat y\) signals this is a PREDICTED value, not an actual data point.

Finding it

Enter every \((x,y)\) pair into your calculator's statistics/regression mode, then read \(a\) and \(b\) directly off the screen — CAPS does not require deriving these values by hand.

"Least-squares" means...
Of every possible straight line, this is the ONE line that makes the total squared vertical distance from every data point to the line as small as possible — it is the best-fitting line in a precise, provable sense, not just a rough eyeball fit.

Worked Examples — Finding the Regression Line

Two data sets, using your calculator's regression mode.

Worked example — study hours vs. test score Level 2

Seven learners' weekly study hours (\(x\)) and test scores out of 100 (\(y\)) are: \((1,42),(2,48),(3,55),(4,60),(5,68),(6,74),(7,80)\). Determine the equation of the least-squares regression line.

Study hours (x) Score (y)
The red line is the least-squares regression line — the single best-fitting straight line through all 7 points.
Show solution
  1. 1Enter all 7 \((x,y)\) pairs into your calculator's linear regression (LR) mode.
  2. 2Read off \(a\approx35{,}43\) and \(b\approx6{,}39\).
  3. 3\(\boxed{\hat{y}=35{,}43+6{,}39x}\)
Worked example — a weaker relationship Level 3

A different data set gives \((10,50),(15,58),(20,52),(25,65),(30,60),(35,72),(40,68)\). Determine the regression line equation.

Show solution
  1. 1Enter all 7 pairs into regression mode.
  2. 2Read off \(a\approx44{,}64\) and \(b\approx0{,}64\).
  3. 3\(\boxed{\hat{y}=44{,}64+0{,}64x}\)
  4. 4Notice the much smaller gradient here — \(y\) still rises with \(x\), but far less steeply and less consistently than the first example. The correlation coefficient (next section) will confirm this is a weaker relationship.

What Your Calculator Is Actually Doing

The exam only ever requires the calculator method — but seeing the real formula once builds genuine trust in the shortcut.

The least-squares formulas

\(b=\dfrac{n\sum xy-\left(\sum x\right)\left(\sum y\right)}{n\sum x^2-\left(\sum x\right)^2}\)\(\qquad a=\bar{y}-b\bar{x}\)

Worked example — computing \(a\) and \(b\) by hand Level 3

Find the regression line for \((1,3),(2,5),(3,6),(4,8),(5,10)\) using the formulas above, then confirm it matches your calculator.

Show solution
  1. 1\(n=5\). \(\sum x=1+2+3+4+5=15\); \(\sum y=3+5+6+8+10=32\)
  2. 2\(\sum xy=(1)(3)+(2)(5)+(3)(6)+(4)(8)+(5)(10)=3+10+18+32+50=113\)
  3. 3\(\sum x^2=1+4+9+16+25=55\)
  4. 4\(b=\dfrac{5(113)-(15)(32)}{5(55)-(15)^2}=\dfrac{565-480}{275-225}=\dfrac{85}{50}=\boxed{1{,}7}\)
  5. 5\(\bar{x}=\dfrac{15}{5}=3\), \(\bar{y}=\dfrac{32}{5}=6{,}4\). \(a=6{,}4-(1{,}7)(3)=6{,}4-5{,}1=\boxed{1{,}3}\)
  6. 6\(\boxed{\hat{y}=1{,}3+1{,}7x}\) — enter the same 5 points into your calculator's regression mode and you'll get exactly these same values for \(a\) and \(b\).
Why "least squares"
Of every possible straight line through the data, this specific \(b\) and \(a\) combination is the ONE choice that makes the total squared vertical distance from every point to the line as small as it can possibly be — not a rough eyeball fit, but a mathematically provable best fit. Your calculator runs exactly this formula internally; it just does the arithmetic instantly.
Quick revision — standard deviation
You met this in Grade 11: \(\sigma^2=\dfrac{\sum(x-\bar{x})^2}{n}\) (the average squared distance from the mean), and \(\sigma=\sqrt{\sigma^2}\) (back to the original units). It's about to come back — \(r\) itself is built directly out of it.
Worked example — where \(r\) actually comes from Level 3

Using the study-hours data from earlier, \((1,42),(2,48),(3,55),(4,60),(5,68),(6,74),(7,80)\), with \(b\approx6{,}39\): find \(\sigma_x\) and \(\sigma_y\), then use them to compute \(r\).

Show solution
  1. 1\(\bar{x}=4\). Deviations: \(-3,-2,-1,0,1,2,3\); squared: \(9,4,1,0,1,4,9\), sum \(=28\). \(\sigma_x^2=\dfrac{28}{7}=4\), so \(\boxed{\sigma_x=2}\)
  2. 2\(\bar{y}=61\). Deviations: \(-19,-13,-6,-1,7,13,19\); squared: \(361,169,36,1,49,169,361\), sum \(=1146\). \(\sigma_y^2=\dfrac{1146}{7}\approx163{,}71\), so \(\boxed{\sigma_y\approx12{,}80}\)
  3. 3The correlation coefficient is actually the gradient RESCALED by how spread out \(x\) is compared to \(y\): \(r=b\times\dfrac{\sigma_x}{\sigma_y}\)
  4. 4\(r=6{,}39\times\dfrac{2}{12{,}80}\approx\boxed{0{,}999}\) — exactly the value your calculator's regression mode already gave for this data.
What this actually tells you
\(r\) isn't an arbitrary black-box number — it's the gradient \(b\) rescaled into a fixed \(-1\) to \(1\) range by comparing how spread out the \(x\)-values are to how spread out the \(y\)-values are. That rescaling is exactly why \(r\) can compare relationships measured in completely different units, while \(b\) alone can't.
Quick check: scatter plots & regression

A scatter plot's points start nearly flat and then climb increasingly steeply. Which shape does this suggest?

In the regression equation ŷ = a + bx, what does b represent?

The Correlation Coefficient \(r\)

One number that says how strong and which direction the linear relationship is.

Range

\(-1\le r\le1\), read directly off your calculator alongside \(a\) and \(b\).

Sign

Positive \(r\): as \(x\) increases, \(y\) tends to increase too. Negative \(r\): as \(x\) increases, \(y\) tends to decrease.

Size

\(|r|\) close to 1: a strong linear relationship — the points sit close to the regression line. \(|r|\) close to 0: a weak or no linear relationship — the points are scattered widely around any line.

\(|r|\) rangeStrengthExample
\(0\) to \(0{,}25\)Very weak\(r=0{,}05\) — almost no linear relationship
\(0{,}25\) to \(0{,}5\)Weak\(r=-0{,}40\) — weak negative
\(0{,}5\) to \(0{,}75\)Moderate\(r=0{,}55\) — moderate positive
\(0{,}75\) to \(0{,}9\)Strong\(r=-0{,}82\) — strong negative
\(0{,}9\) to \(1\)Very strong\(r=0{,}98\) — very strong positive
Correlation is NOT causation
A strong \(r\) only shows that two variables move together — it never proves one CAUSES the other. Classic example: ice-cream sales and shark attacks are strongly positively correlated, but ice cream doesn't attract sharks. Both simply rise in warm weather — a third factor (temperature) drives both. Always be cautious about claiming cause and effect from correlation alone.

Worked Examples — Finding and Interpreting \(r\)

A perfect-fit lead-in, then the two regression examples, plus a genuinely negative case.

Worked example — a perfectly linear set Level 1

Find \(r\) for: \((1,10),(2,20),(3,30),(4,40),(5,50)\).

Show solution
  1. 1Every point lies EXACTLY on the same straight line (\(y=10x\)), with zero scatter around it.
  2. 2Calculator regression mode confirms: \(\boxed{r=1}\) — a perfect positive linear relationship, the strongest value \(r\) can ever take.
\(r\approx0{,}997\): points sit tightly against the line.
\(r\approx0{,}79\): a real trend, but more scattered.
\(r\approx0{,}23\): almost no linear pattern.
\(r\approx-0{,}999\): very strong negative — as \(x\) rises, \(y\) falls almost perfectly.
\(r\approx-0{,}83\): strong negative, but with visibly more scatter than the panel on the left.
The visual meaning of \(r\)
This is exactly what \(r\) is measuring: how tightly the actual data points hug the regression line. It has nothing to do with how steep the line is — only how close the points sit to whatever line best fits them.
Worked example — the study-hours data Level 2

Find and interpret \(r\) for the study-hours/test-score data from the previous slide.

Show solution
  1. 1Your calculator's regression mode gives \(r\) alongside \(a\) and \(b\): \(\boxed{r\approx0{,}999}\)
  2. 2\(r\) is positive and extremely close to 1: \(\boxed{\text{a very strong positive linear relationship}}\) — more study hours are very consistently linked to a higher score.
Worked example — the weaker data Level 3

Find and interpret \(r\) for the second (weaker) data set from the previous slide.

Show solution
  1. 1\(\boxed{r\approx0{,}853}\)
  2. 2\(r\) is positive but noticeably further from 1: \(\boxed{\text{a moderately strong positive linear relationship}}\) — the trend is real, but individual points sit further from the line than in the first example.
Worked example — a negative relationship Level 3

A runner's speed (km/h, \(x\)) and time to finish a fixed race (minutes, \(y\)) are: \((5,95),(10,80),(15,68),(20,55),(25,42),(30,30)\). Find and interpret \(r\).

Show solution
  1. 1Regression mode gives \(a\approx106{,}87\), \(b\approx-2{,}58\), \(\boxed{r\approx-0{,}9995}\)
  2. 2\(r\) is negative and extremely close to \(-1\): \(\boxed{\text{a very strong negative linear relationship}}\) — as speed increases, finishing time decreases almost perfectly consistently, which matches common sense here.
Quick check: correlation coefficient

A data set has r = -0,92. What does this tell you?

Which value of r shows the WEAKEST linear relationship?

Interpolation vs. Extrapolation

Using the regression equation to predict — and knowing when to trust the prediction.

Interpolation

Predicting a \(y\)-value for an \(x\) INSIDE the original data's range. Generally reliable, since the pattern is directly supported by real observed data on both sides.

Extrapolation

Predicting for an \(x\) OUTSIDE the original range. Less reliable — the relationship might not continue the same way beyond the data you actually collected.

A concrete danger
A regression line fitted to a child's height (ages 2 to 10) would predict absurd, ever-increasing heights if extrapolated to age 40 — the real relationship clearly changes outside the observed range. Always ask whether the pattern is likely to keep holding before trusting an extrapolated prediction.

Worked Example — Predicting With the Regression Equation

A direct substitution first, then the full interpolation-vs-extrapolation reasoning.

Worked example — a direct prediction Level 1

A regression line \(\hat{y}=20+5x\) was fitted to data from \(x=2\) to \(x=10\). Predict \(\hat y\) when \(x=6\).

Show solution
  1. 1Substitute \(x=6\) directly into the equation: \(\hat{y}=20+5(6)=20+30=\boxed{50}\)
Worked example — interpolation and extrapolation Level 3

Using \(\hat{y}=35{,}43+6{,}39x\) (fitted to data from \(x=1\) to \(x=7\)): (a) predict the score for 4,5 study hours. (b) predict the score for 15 study hours. (c) which prediction is more reliable, and why?

Show solution
  1. 1(a) \(x=4{,}5\) is INSIDE the original range \([1,7]\) — this is interpolation. \(\hat{y}=35{,}43+6{,}39(4{,}5)=35{,}43+28{,}76\approx\boxed{64{,}2}\)
  2. 2(b) \(x=15\) is far OUTSIDE the original range \([1,7]\) — this is extrapolation. \(\hat{y}=35{,}43+6{,}39(15)=35{,}43+95{,}85\approx\boxed{131{,}3}\)
  3. 3(c) The answer to (b), over 131%, is already impossible for a test score out of 100 — a clear sign the extrapolated prediction has broken down. \(\boxed{\text{The interpolated prediction (a) is far more reliable}}\), since it stays within the range the relationship was actually observed to hold.
original data range x=4,5 interpolation x=15 extrapolation
\(x=4{,}5\) sits inside the shaded original range \([1,7]\); \(x=15\) sits far beyond it — the visual gap is exactly why the second prediction cannot be trusted the same way.
Exam tip
Always check whether the given \(x\)-value falls inside or outside the original data's range BEFORE substituting — this determines whether you are interpolating or extrapolating, and CAPS regularly asks you to say so explicitly.

Integrated Worked Example

A full multi-part Question 1, exactly the shape a real NSC paper uses — a short skewness part on ONE data set, then the regression work on a completely different one.

Worked example — combining every skill Level 4

The recovery times (in days) of a group of patients after a minor procedure are summarised by the five-number summary: minimum 2, \(Q_1=4\), median 5, \(Q_3=9\), maximum 20. (a) Describe the skewness of this data set. Separately, a tutor records 6 learners' number of practice quizzes completed (\(x\)) and their exam mark (\(y\)): \((2,45),(4,52),(6,58),(8,63),(10,71),(12,75)\). (b) Describe the shape of the data. (c) Determine the regression equation. (d) Determine \(r\) and interpret it. (e) Predict the mark for a learner who completes 9 quizzes, and comment on the reliability of this prediction.

Show solution
  1. 1(a) Median to \(Q_1\) spans \(5-4=1\); median to \(Q_3\) spans \(9-5=4\) — the median sits much closer to \(Q_1\). The lower whisker spans \(4-2=2\); the upper whisker spans \(20-9=11\) — more than 5 times as long. Both checks agree: \(\boxed{\text{the data is positively (right) skewed}}\) — most patients recover quickly, but a small number take far longer, stretching the upper whisker out.
  2. 2(b) Plotting the 6 quiz/mark points shows a steady, roughly constant rise: \(\boxed{\text{the data looks linear}}\)
  3. 3(c) Calculator regression mode: \(a\approx39{,}47\), \(b\approx3{,}03\). \(\boxed{\hat{y}=39{,}47+3{,}03x}\)
  4. 4(d) \(\boxed{r\approx0{,}997}\) — positive and extremely close to 1: a very strong positive linear relationship between quizzes completed and exam mark.
  5. 5(e) \(x=9\) sits INSIDE the original range \([2,12]\), so this is interpolation. \(\hat{y}=39{,}47+3{,}03(9)=39{,}47+27{,}27\approx\boxed{66{,}7}\). Since this is interpolation within a very strong linear relationship, \(\boxed{\text{the prediction is highly reliable}}\).
2 4 5 9 20 min Q1 median Q3 max
Part (a)'s recovery-time data, drawn as a box plot — the tiny box against the long right whisker is exactly what "positively skewed" looks like.
Quick check: interpolation & extrapolation

A regression line is fitted to data with x ranging from 10 to 50. Predicting y at x = 30 is an example of...

Why is extrapolation generally considered less reliable than interpolation?

Grade 12 Mastery Sprint

Work first. Open one answer only when your own line of working is complete.

01 — Reading b

\(\hat{y}=12{,}5+2{,}1x\). Answer: gradient \(b=2{,}1\); for every 1-unit increase in \(x\), \(\hat y\) increases by 2,1.

02 — Interpreting r

\(r=-0{,}12\). Answer: very weak, almost no linear relationship (the negative sign barely matters when \(|r|\) is this small).

03 — Interpolation check

Data range \(x\in[5,25]\), predict at \(x=18\). Answer: interpolation (18 is inside [5,25]).

04 — Extrapolation check

Data range \(x\in[5,25]\), predict at \(x=40\). Answer: extrapolation (40 is outside [5,25]) — less reliable.

Exam Strategy

Judge a scatter plot's shape (linear, quadratic or exponential) before doing any calculation.
Use your calculator's regression mode to find \(a\), \(b\) and \(r\) directly — don't try to derive them by hand.
State \(r\)'s interpretation in words: strength (close to 0 or close to \(\pm1\)) AND direction (positive or negative).
Before substituting into \(\hat y=a+bx\), check whether \(x\) is inside or outside the original data range.
Always finish with a plain-language comment on reliability when asked to predict.
Avoid this

Fitting a straight-line regression to obviously curved (quadratic/exponential) data.

Reporting only the number for \(r\) without saying what it MEANS.

Treating an extrapolated prediction as equally trustworthy as an interpolated one.

Do this

Look at the scatter plot's shape first, every time.

Always pair \(r\)'s value with both a strength word and a direction word.

State explicitly whether a given prediction is interpolation or extrapolation.

More explanation and exercises:Siyavula Grade 12 Statistics
Summary complete

The 3-year Statistics course is done.

Use the Mastery Bank to build fluency. Then take the test without notes and use the result to choose the exact slide to revisit.

Next step 1

Mastery Bank

18 questions by skill, from reading a regression equation through to a full multi-part bivariate investigation. Answers reveal only after your attempt.

Next step 2

Test Your Knowledge

A short exam-style self-check. Mark it, then revisit the exact slide that matches a missed skill.

1 / 19

Learn It in Short Videos

Four free, independent videos — not made by Equation Station SA.

Scatter Plots

Plotting bivariate data and reading its shape.

Khan Academy · Constructing a scatter plot

The Regression Line

What "least-squares" actually means, visually.

Eugene O'Loughlin · How To... Perform Simple Linear Regression by Hand

Correlation Coefficient

Building intuition for what different r values look like.

The Organic Chemistry Tutor · Correlation Coefficient

Interpolation vs. Extrapolation

When a prediction can be trusted, and when it can't.

Khan Academy · Example estimating from regression line

Practise in the right order

The core teaching is above. These are the next steps, not a replacement for it.

01
Built-in practice
Statistics Mastery Bank

18 questions by skill, with concise reveal answers and methods.

Start after the slides
Open Mastery Bank
02
Built-in check
Test Your Knowledge

Use the short exam-style self-check when you want a fast confidence check.

Then target one weak skill
Take the Test
CAPS
Free textbook chapter
Siyavula: Grade 12 Statistics

Use its own worked examples for extra explanation and exercises.

Free • CAPS aligned
Open Siyavula
DBE
Official free books
DBE Grade 12 Textbooks

Official state-owned learner books and teacher support for Grade 12 Mathematics.

Official • free access
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Related Topics

This topic began in Grade 10 and Grade 11.

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Statistics (Grade 10)

Central tendency, quartiles, five-number summary

Study this →
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Statistics (Grade 11)

Histograms, ogives, variance and standard deviation

Study this →
?

Probability

Counting principle and contingency tables

Study this →
Student Questions

Frequently Asked Questions

Short answers for the checks learners make while preparing for the Grade 12 CAPS exam.

What does CAPS require for Grade 12 Statistics?

Represent bivariate (two-variable) data as a scatter plot and judge whether it looks linear, quadratic or exponential; use a calculator to find the least-squares regression line and the correlation coefficient r; interpret r in terms of the strength and direction of the relationship; and use the regression equation to interpolate and extrapolate, discussing the reliability of each; and revise symmetric and skewed data, ready to describe skewness from a five-number summary or box plot.

Do I calculate the regression line by hand?

No — CAPS Grade 12 expects you to use your calculator's regression (linear regression / LR) mode to find the equation and r directly. You do need to know what the values mean and how to use them, not just how to compute them from scratch.

What is the difference between interpolation and extrapolation?

Interpolation means predicting a y-value for an x-value INSIDE the range of the original data — this is generally reliable. Extrapolation means predicting outside that range, which is far less reliable since the pattern may not continue.

How do I interpret the correlation coefficient r?

r is always between -1 and 1. Values close to 1 or -1 show a strong linear relationship (positive or negative); values close to 0 show a weak or no linear relationship. The sign tells you the direction: positive means y increases as x increases, negative means y decreases as x increases.

Where should I practise next?

Finish the interactive slides, open the Mastery Bank, then take the short self-test without notes.