Chapter Overview
🧭Your visual roadmap through the chapter
Association Between Variables
Contingency tables, scatter plots, positive/negative/no association
Correlation & the Correlation Coefficient
Strength & direction of a relationship; Spearman's rank correlation
Linear Regression Analysis
The line of best fit; forecasting with ŷ=a+bx
Establishing whether two variables are associated, measuring the strength of that association (correlation), and modelling the functional relationship between them for prediction (regression).
Businesses use these tools to understand relationships (e.g. cost vs output, advertising vs sales) and to forecast future values.
Construct and read scatter plots; calculate the correlation coefficient and Spearman's rank correlation; derive and use a regression line to forecast.
Key Definitions
📘Source-based terminology, exam-ready
Contingency Table
Displays two categorical variables in rows and columns, showing the frequency of observations at each intersection. Also called cross-tabulation.
Key word: categorical, cross-tab
Scatter Plot
A graphical display for two quantitative variables, using the x-axis for the independent variable and y-axis for the dependent variable.
Key word: x independent, y dependent
Correlation
A statistical relationship or association between two variables; changes in one are associated with changes in the other.
Key word: association
Correlation Coefficient (r)
A numerical value measuring the degree, strength and direction of association between two quantitative variables; ranges from −1 to +1.
Key word: −1 to +1
Coefficient of Determination (r²)
The square of the correlation coefficient; shows how much of the variation in y is explained by variation in x. Ranges 0 to +1.
Key word: % variation explained
Spearman's Rank Correlation
A correlation coefficient calculated when variables are based on rank/order rather than magnitude.
Key word: ranked data
Linear Regression / Line of Best Fit
A statistical technique for calculating the line that minimizes the sum of squared deviations (residuals) between the line and actual data points.
Key word: least squares
Residual
The vertical distance between an actual data point and the regression line.
Key word: actual − predicted
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Correlation
MEANING
Measures the strength and direction of an association — it does NOT prove causation.
PURPOSE
Determine whether, and how strongly, two variables move together.
APPLICATION
Testing whether two quantitative variables (or two rankings) are related.
CONCEPT → Regression
MEANING
Provides a functional (algebraic) relationship between a dependent and an independent variable.
PURPOSE
Predict/forecast the dependent variable from a given value of the independent variable.
APPLICATION
Estimating fixed/variable costs, forecasting sales, trend analysis.
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 Interpreting r
r = +1: perfect positive; r = −1: perfect negative; r = 0: no association
Closer to ±1 = stronger linear association; closer to 0 = weaker
r between +0.90 and +1 generally indicates good linear correlation
RULE 4 Coefficient of determination (r²)
r²=0: independent variable explains NONE of the variation in y
r²=1: independent variable explains ALL of the variation in y
Between 0 and 1: that % of variation in y is explained by x
RULE 2 Properties of r
Unit-less (a ratio); unaffected by the units the variables are measured in
Symmetric: rxy = ryx
Always lies between −1 and +1
Is the geometric mean of the two regression coefficients
RULE 5 Calculating the regression line
b must be calculated first, since a's formula depends on b
The regression line always passes through the point (x̄, ȳ)
b = r × (Sy÷Sx) links the regression coefficient to r
RULE 6 Limitations to remember
Scatter plots: only show 2 variables at once; few/atypical points mislead; can't quantify strength precisely
Regression: poor fit for non-linear data; distorted by outliers; unsafe to extrapolate beyond the observed data range
RULE 3 Correlation ≠ causation
A strong correlation does not mean the independent variable causes the dependent variable to change
Other factors may influence the relationship
Formula Zone — Correlation
🧮Every formula supported by the source text
CORRELATION COEFFICIENT (METHOD 1 — RAW SUMS)
x,y=paired data values, n=number of pairs. Most direct to compute from raw data.
CORRELATION COEFFICIENT (METHOD 2 & 3 — DEVIATIONS)
Σ(x−x̄)(y−ȳ)÷n is the covariance (sxy). Equivalent to r=sxy÷(n·sxsy) using standard deviations.
COEFFICIENT OF DETERMINATION
Simply square the value of r. Always between 0 and 1.
SPEARMAN'S RANK CORRELATION COEFFICIENT
d=difference between ranks in a pair, n=number of pairs. The "6" is a fixed constant, not related to n.
Formula Zone — Linear Regression
🧮Every formula supported by the source text
REGRESSION LINE
ŷ=predicted value of y, a=intercept, b=slope (regression coefficient). Calculate b BEFORE a.
SLOPE (B) — TWO EQUIVALENT FORMS
Use whichever form matches the data already tabulated (raw sums, or deviations from the mean).
INTERCEPT (A)
Requires b to already be calculated. The line always passes through (x̄, ȳ).
LINK BETWEEN B AND R
Sx, Sy are the standard deviations of x and y.
Calculation Roadmap
🗺️General path for any correlation/regression question
1· Read Question
Identify the paired data and what's required
2· Identify Method
Correlation (r), rank correlation, or regression?
3· Tabulate
Build the x, y, xy, x², y² (or deviation) columns
4· Substitute
Put the column totals into the formula
5· Calculate & Interpret
Solve, interpret strength/direction or forecast
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Constructing a Scatter Plot
- 1Identify the independent (x) and dependent (y) variable
- 2Draw the axes with appropriate scales
- 3Plot each (x, y) pair as a point
- 4Observe the pattern: positive, negative, or no association
METHOD C · Spearman's Rank Correlation
- 1Rank each variable's values (if not already ranked)
- 2Find d = difference between the two ranks for each pair
- 3Square each d and sum: Σd²
- 4Substitute into r=1−[6Σd²÷n(n²−1)]
METHOD B · Correlation Coefficient (Raw Sums)
- 1Tabulate x, y, xy, x², y² for every pair
- 2Sum each column: Σx, Σy, Σxy, Σx², Σy²
- 3Substitute into r = [nΣxy−ΣxΣy]÷√[...]
- 4Interpret the sign (direction) and size (strength) of r
METHOD D · Regression Line & Forecasting
- 1Tabulate the required sums or deviations (as for correlation)
- 2Calculate b FIRST using the slope formula
- 3Calculate a using a=ȳ−bx̄ (or the sums version)
- 4Write ŷ=a+bx, then substitute any x to forecast ŷ
Different Cases / Scenarios
🔀Recognise the data, pick the right tool
IF both variables are categorical
→ USE a contingency table
IF both variables are quantitative
→ USE a scatter plot and/or the correlation coefficient
IF data is based on rank/order, not magnitude
→ USE Spearman's rank correlation
IF you need to predict/forecast a value
→ USE the regression line (ŷ=a+bx), not correlation alone
IF r is close to +1 or −1
→ Strong linear association exists
IF r is close to 0
→ Little to no linear association
IF asked what % of variation in y is explained by x
→ USE r² (coefficient of determination)
IF forecasting a value OUTSIDE the range of the original data
→ Treat with caution — extrapolation may not be valid
Comparison Tables
⚖️Don't confuse these!
| Correlation | Regression | |
|---|---|---|
| What it measures | Degree/strength of association between two variables | A functional relationship, for predicting one variable from another |
| Provides causation? | No | Identifies dependent/independent roles, but still not proof of causation |
| Output | A single value, r (−1 to +1) | An equation, ŷ=a+bx |
| Pearson's r (Correlation Coefficient) | Spearman's Rank Correlation | |
|---|---|---|
| Data type | Actual quantitative values | Ranks/order (e.g. preference, quality) |
| Formula basis | Raw values or deviations from the mean | Differences between ranks (d) |
Reading Scatter Plots & Regression Lines
📌How to interpret the source's graphs
Reading a scatter plot
Points trending upward (left to right) = positive association. Points trending downward = negative association. Points scattered with no visible pattern = no association. Isolated points far from the rest may be outliers.
Reading a regression line graph
The line of best fit runs through the data cloud, minimizing the total squared vertical distance (residuals) from each point to the line. The line always passes through the point (x̄, ȳ).
Worked Calculation Logic
✍️Method students must reproduce in the exam
REGRESSION LINE: GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
A company's monthly output/cost data gives: n=6, Σx=41.8, Σy=268.3, Σxy=1,885.91, Σx²=295.88. Find the regression line and forecast cost at 10,000 units (x=10).
GIVEN: n=6, Σx=41.8, Σy=268.3, Σxy=1,885.91, Σx²=295.88
FORMULA: b=[nΣxy−ΣxΣy]÷[nΣx²−(Σx)²] then a=(Σy/n)−b(Σx/n)
SUBSTITUTION: b=[6(1,885.91)−(41.8)(268.3)]÷[6(295.88)−41.8²] = 100.52÷28.04 = 3.585
CALCULATION: a = 44.72 − 3.585(6.97) = 44.72−24.98 = 19.74 → ŷ=19.74+3.585x
FINAL ANSWER: At x=10 (10,000 units), ŷ=19.74+3.585(10)=Rs. 55.6 million
Exam Question Patterns
📝Correlation coefficient — calculate and interpret r
Coefficient of determination — interpret r² as % variation explained
Spearman's rank correlation — ranked/ordinal data
Regression line derivation — find a and b, write ŷ=a+bx
Forecasting — substitute a given x to predict y
Conceptual — correlation vs causation, limitations of each tool
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Correlation implies causation — a strong r never proves one variable causes the other.
- ✗ Wrong correlation formula — using Pearson's r on ranked data instead of Spearman's.
- ✗ Calculating a before b — a's formula depends on b; always find b first.
- ✗ r vs r² confusion — forgetting to square r, or re-squaring an already-squared r².
- ✗ Extrapolating too far — forecasting well beyond the observed x-range is unreliable.
Quick Decision Tree
🌳Chapter 10 in 60 Seconds
⚡r: [nΣxy−ΣxΣy]/√[...] r² = r × r (% variation explained)
Spearman's: 1−[6Σd²/n(n²−1)]
Regression: ŷ=a+bx b first, then a=ȳ−bx̄ b = r×(Sy/Sx) Line passes through (x̄, ȳ)
Trap: correlation ≠ causation
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