📉 Scatter Diagrams & Pearson Correlation
Does cutting speed affect tool wear? Does temperature affect yield strength? Scatter diagrams show the relationship at a glance, and Pearson's coefficient turns that impression into a precise number.
2Topics
8Worked Examples
4Eng. Cases
12Practice Qs
📍Scatter DiagramsVisual relationships
🔢Pearson's rMeasuring linear correlation
⚙️ApplicationsReal engineering
🎮Quick-Fire GameTest your speed
📝PracticeGraded questions
🎯What you will learn
Plot and interpret scatter diagrams to spot patterns between two variables; calculate Pearson's product-moment correlation coefficient r; and understand what r does and doesn't tell you about a relationship.
💡 Correlation is not causation. Two variables can move together perfectly (high |r|) without one causing the other — both might be driven by a third factor entirely.
🧭How to use this resource
1. Read the rules banner at the top of each tab
2. Experiment with the interactive tool
3. Check the key vocabulary panel
4. Study the visual diagram, then the worked examples
📍
Scatter Diagrams
Plotting paired data reveals relationships that a table of numbers hides.
🔧Scatter Diagram PlotterEnter paired x,y data to see the plot and pattern
Enter matching x and y values, then press Plot to see the scatter diagram and pattern description…
📖 Key Vocabulary
Bivariate data — data collected in pairs, one value of each of two variables per item.
Positive correlation — as x increases, y tends to increase too (upward pattern).
Negative correlation — as x increases, y tends to decrease (downward pattern).
No correlation — points scattered with no clear pattern; x and y appear unrelated.
👁️ Visual aid: the overall trend of the points — not any single point — tells you whether the correlation is positive, negative, or absent.
✏️Worked Example 1 — Identifying a positive correlationType: Pattern ID
Question Data: hours studied (1,2,3,4,5) vs test score (52,58,64,70,78). Describe the correlation.
Observe As hours increase, score consistently increases
Answer Strong positive correlation — more study time is associated with higher scores.
📉Worked Example 2 — Identifying a negative correlationType: Pattern ID
Question Data: cutting speed (m/min) vs tool life (hours): (50,120),(70,90),(90,60),(110,35),(130,20). Describe the correlation.
Observe As cutting speed increases, tool life consistently decreases
Answer Strong negative correlation — higher cutting speeds wear tools out faster.
🎲Worked Example 3 — Recognising no correlationType: Pattern ID
Question Data: employee shoe size vs typing speed shows points scattered randomly with no visible trend. What does this tell us?
Observe No consistent upward or downward pattern
Answer No correlation — the two variables appear unrelated, which makes physical sense here.
⚙️Worked Example 4 — Engineering context (choosing variables)Type: Engineering
Question An engineer wants to study how furnace temperature affects the hardness of a heat-treated steel sample. Which variable goes on which axis, and why?
Reasoning Temperature is set/controlled by the engineer (independent); hardness is the resulting measurement (dependent)
Answer Temperature on the x-axis, hardness on the y-axis — the standard convention of "cause" on x, "effect" on y.
🧠 Common slip: assuming a visible pattern proves causation. A scatter diagram only shows association — a controlled experiment or engineering theory is needed to establish cause and effect.
🔢
Pearson's Correlation Coefficient
A single number between −1 and +1 that measures the strength and direction of a linear relationship.
🔧Pearson's r CalculatorEnter matching x,y data
Enter matching x and y values, then press Calculate to see Sxx, Syy, Sxy and Pearson's r worked out…
📖 Key Vocabulary
Pearson's r — the product-moment correlation coefficient, measuring linear association strength.
Sxx, Syy — the sum of squared deviations of x and y from their means (like a variance without dividing by n).
Sxy — the sum of the products of x- and y-deviations; positive if they tend to move together.
Coefficient of determination (r²) — the proportion of variation in y "explained" by x.
👁️ Visual aid: r is always trapped between −1 and +1 — the closer to either end, the stronger the straight-line relationship.
✏️Worked Example 1 — Calculating r step by stepType: Full calculation
Question Find r for x: 1,2,3,4,5 and y: 2,4,5,4,5.
Means
x̄=3, ȳ=4Sxy, Sxx, Syy
Sxy=6, Sxx=10, Syy=6 (from Σ(x−x̄)(y−ȳ) etc.)Answer
r = 6/√(10×6) = 6/√60 = 6/7.746 ≈ 0.775 — a strong positive correlation.🔍Worked Example 2 — Interpreting the strength of rType: Interpretation
Question Two datasets give r=0.92 and r=0.31 respectively. Compare their linear relationships.
Interpret r=0.92 is very close to 1: a very strong positive linear relationship. r=0.31 is much closer to 0: a weak positive relationship at best.
Answer The first dataset's x reliably predicts y along a line; the second's does not — much of y's variation is due to other factors.
📐Worked Example 3 — Coefficient of determinationType: r²
Question If r = 0.8, find and interpret r².
Square
r² = 0.8² = 0.64Answer 64% of the variation in y can be statistically explained by x; the remaining 36% is due to other factors or random variation.
⚙️Worked Example 4 — Engineering context (validating a sensor)Type: Engineering
Question A new low-cost pressure sensor is compared against a calibrated reference across 10 readings, giving r=0.995. What does this tell an engineer?
Interpret r extremely close to +1 means the new sensor's readings track the reference almost perfectly in a straight-line sense.
Answer The sensor is highly linearly consistent with the reference — strong evidence it could be calibrated and used as a cheaper substitute, pending a check of the actual scale/offset (regression, covered next).
🧠 Common slip: forgetting that Pearson's r only detects linear relationships. A perfect curved (non-linear) relationship can still give a low r, even though the variables are strongly related.
⚙️
Engineering Applications
Four live scenarios — each with its own calculator. Enter your own numbers.
🔧 Tool Wear vs Cutting Speed
Machining data typically shows strong negative correlation between cutting speed and tool life.
r close to −1 confirms the expected physical relationship
📟 Try it — Correlation Calculator
Enter both datasets, then press Calculate…
🌡️ Temperature vs Material Hardness
Heat-treatment studies routinely correlate furnace temperature with resulting hardness (HRC).
Strength of r guides how predictable the process is
📟 Try it — Correlation Calculator
Enter both datasets, then press Calculate…
📡 Sensor Calibration Check
Comparing a low-cost sensor's readings to a reference instrument, r close to 1 supports using the cheaper sensor.
r ≈ 1 → sensor tracks the reference linearly
📟 Try it — Correlation Calculator
Enter both sets of readings, then press Calculate…
💰 Production Volume vs Unit Cost
Cost engineers check correlation between batch size and per-unit cost to quantify economies of scale.
Negative r supports the case for larger batch runs
📟 Try it — Correlation Calculator
Enter both datasets, then press Calculate…
🎮
Quick-Fire Correlation Game
12 questions on scatter diagrams and Pearson's r. How many can you get?
⚡ Correlation Challenge
Score: 0Question: 1/12Streak: 0🔥
Press Start to begin!
📝
Practice Questions
Graded from warm-up to challenge. Attempt fully before revealing the working.
Question 1 🥉2 marks
State which variable normally goes on the x-axis of a scatter diagram.
The independent variable (the one believed to influence the other)
Question 2 🥉2 marks
State the range of possible values for Pearson's r.
−1 ≤ r ≤ +1
Question 3 🥉2 marks
r = −0.9 indicates what type and strength of correlation?
Strong negative correlation
Question 4 🥉2 marks
If r = 0.5, find r² and state what it represents.
r² = 0.25 → 25% of y's variation is explained by x
Question 5 🥈4 marks
Find Sxx, Syy for x: 1,3,5 (mean=3) and y: 2,6,10 (mean=6).
Deviations x: −2,0,2 → Sxx=4+0+4=8
Deviations y: −4,0,4 → Syy=16+0+16=32; Sxx=8
Deviations y: −4,0,4 → Syy=16+0+16=32; Sxx=8
Question 6 🥈4 marks
Using Sxx=8, Syy=32, and Sxy=16 (from Q5's data), find Pearson's r.
r = 16/√(8×32) = 16/√256 = 16/16 = 1.0 (perfect positive)
Question 7 🥈3 marks
A scatter diagram shows points randomly spread with no visible trend. What r value would you expect?
r close to 0
Question 8 🥈4 marks
Explain why a strong r value alone cannot prove that x causes y.
Correlation only measures association, not causation — a third (confounding) variable could be driving both x and y, or the relationship could be coincidental.
Question 9 🥇5 marks
Calculate Pearson's r for x: 2,4,6,8 (mean=5) and y: 3,5,6,10 (mean=6).
Dev x: −3,−1,1,3; Dev y: −3,−1,0,4
Sxy = 9+1+0+12 = 22
Sxx = 9+1+1+9 = 20
Syy = 9+1+0+16 = 26
r = 22/√(20×26) = 22/√520 = 22/22.80
r ≈ 0.965
Sxy = 9+1+0+12 = 22
Sxx = 9+1+1+9 = 20
Syy = 9+1+0+16 = 26
r = 22/√(20×26) = 22/√520 = 22/22.80
r ≈ 0.965
Question 10 🥇5 marks
Two sensors are compared: r=0.999 for Sensor A vs reference, r=0.72 for Sensor B vs reference. Which sensor should be trusted more, and why?
Sensor A — its near-perfect r (0.999) shows an almost exact linear match to the reference, while Sensor B's weaker r (0.72) suggests meaningful deviation or noise.
Question 11 🥇5 marks
A study of 5 furnace batches gives r = 0.88 between temperature and hardness. Find r² and explain what the remaining percentage might represent physically.
r² = 0.88² = 0.7744 → 77.4% explained by temperature
The remaining ≈22.6% could be due to alloy composition variation, cooling rate differences, or measurement error.
The remaining ≈22.6% could be due to alloy composition variation, cooling rate differences, or measurement error.
Question 12 🥇5 marks
Explain, using the concept of Sxy, why a dataset where high x values pair with low y values (and vice versa) gives a negative r.
When x is above its mean, y tends to be below its mean (and vice versa)
So (x−x̄) and (y−ȳ) tend to have opposite signs, making most products in Sxy negative
A negative Sxy directly gives a negative r
So (x−x̄) and (y−ȳ) tend to have opposite signs, making most products in Sxy negative
A negative Sxy directly gives a negative r