Correlation

2.1 Core concepts

• Correlation analysis examines whether two variables are related, whether they move together, the direction of movement and the strength of the relationship (NCERT §1, pp. 74–75). It looks at pairs of observations like price-quantity, height-weight, income-consumption.
• Types of underlying relationships: cause-and-effect (low rainfall causing low agricultural productivity), pure coincidence (arrival of migratory birds and birth rates), and spurious — driven by a hidden third variable (ice-cream sales and drowning deaths, both driven by temperature) (NCERT §2, p. 75).
• Correlation measures covariation, not causation; the presence of correlation only means that when one variable changes, the other changes in the same or opposite direction in a definite way (NCERT §2, p. 75).
• Positive vs negative correlation: positive — variables move in the same direction (income and consumption; ice-cream sale and temperature); negative — variables move in opposite directions (price of apples and demand for apples) (NCERT §2, p. 76).
• For simplicity, correlation is assumed to be linear — i.e., relative movement can be represented by a straight line on a graph (NCERT §2, p. 76). Non-linear relations are real but not measured by Pearson's r.
• Three techniques are used: scatter diagrams, Karl Pearson's coefficient of correlation, and Spearman's rank correlation (NCERT §3, p. 76).
• A scatter diagram visually presents the nature of association without giving any specific numerical value; closeness and direction of plotted points indicate strength and type of correlation (NCERT §3, p. 76). Figures 6.1–6.5 illustrate positive, negative, no, perfect positive and perfect negative correlation; Figures 6.6–6.7 show non-linear relations.
• Karl Pearson's coefficient (also called product moment correlation coefficient) gives a precise numerical value of the degree of linear relationship between X and Y; it must NOT be used when the relation is non-linear (NCERT §3, p. 77).
• Formulas for r (NCERT §3, p. 79):
• r = Σxy ÷ (N · σx · σy) — using covariance and standard deviations.
• r = Σ(X − X̄)(Y − Ȳ) ÷ √[Σ(X − X̄)² · Σ(Y − Ȳ)²] — direct deviation form.
• Actual-values form using ΣXY, ΣX², ΣY².
• Properties of r (NCERT §3, pp. 79–80):
• r has no unit; it is a pure number.
• Negative r indicates inverse relation; positive r indicates same-direction movement.
• r lies between −1 and +1; a value outside this range indicates calculation error.
• r is unaffected by change of origin and change of scale (basis of step-deviation method).
• r = 0 means no linear relation, but non-linear relation may still exist (zero correlation ≠ independence).
• r = ±1 indicates perfect linear correlation; values near ±1 are "high"; values near 0 are "weak".
• Step-deviation method: transform variables as U = (X − A) ÷ B and V = (Y − C) ÷ D, where A, C are assumed means and B, D are common factors of the same sign; then rUV = rXY (NCERT §3, pp. 80, 82–83). This makes computation easier when raw values are large.
• Spearman's rank correlation was developed by C.E. Spearman. It is used when variables cannot be precisely measured (beauty, honesty), when only ranks are available, when relations are non-linear in direction-defined ways, or when data contain extreme values (NCERT §3, pp. 83–84).
• Spearman's formula: rₛ = 1 − [6 ΣD² ÷ (n³ − n)], where D is the difference in ranks and n the number of observations (NCERT §3, p. 84).
• Correction for ties: when ranks are repeated, a correction factor (m³ − m) ÷ 12 is added for each tied group inside the bracket of the formula's numerator (NCERT §3, p. 86).
• Properties of rₛ: it lies between −1 and +1; generally rₛ ≤ r because some information is lost when individual values are replaced by ranks; when first differences are constant, r and rₛ are identical (NCERT §3, p. 84).
• Conclusion: the scatter diagram is the only one of the three tools not confined to linear relations; Pearson and Spearman both measure linear relationship; none implies causation (NCERT §4, p. 87).
• Why study correlation: everyday questions — does demand really fall when price rises? does smoking really raise the risk of cancer? — require a tool to measure pairwise variation rather than single-variable summary (NCERT §1, pp. 74–75).
• Two-variable framing: correlation always involves paired observations (Xᵢ, Yᵢ) on the same unit (the same household, the same year, the same firm). Without pairing the data lose meaning — a common CUET trap is to give two unrelated lists and ask whether r can be computed (it cannot) (NCERT §1, p. 75).
• Spurious correlation example (NCERT): number of storks counted in a Danish village and number of human births in the same village rose together for years — pure coincidence reflecting common demographic trends. NCERT uses such examples to warn against reading causation into correlation (NCERT §2, p. 75).
• Negative-correlation classic examples: price and quantity demanded (Law of Demand), study hours and exam errors, alcohol consumption and motor coordination — each shows X↑ ⇒ Y↓ in a roughly linear fashion (NCERT §2, p. 76).
• Positive-correlation classic examples: height and weight of children, household income and expenditure, advertisement and sales, temperature and ice-cream demand — each shows X↑ ⇒ Y↑ (NCERT §2, p. 76).
• Why linearity is assumed: the algebra of Pearson's r (variance, covariance, square root) implicitly fits a best straight line through the scatter; a curved relation would give a misleadingly low r. NCERT cautions that one must first sketch the scatter before applying r (NCERT §2, p. 76; §3, p. 77).
• Three reasons to prefer rₛ: (i) attributes cannot be measured numerically (beauty, honesty) — only ranked; (ii) extreme values would distort Pearson's r — but ranking caps the influence of any single observation at ±1; (iii) only ranks are reported in the data (e.g., contest standings) and original scores are unavailable (NCERT §3, pp. 83–84).
• Why rₛ ≤ r generally: converting raw numbers to ranks throws away magnitude information; what remains is only ordinal information. Pearson's r exploits magnitudes, so for well-behaved cardinal data Pearson's r captures slightly more information and is at least as large as Spearman's rₛ (NCERT §3, p. 84).
• Step-deviation legality: r is unchanged by U = (X − A)/B, V = (Y − C)/D so long as B and D are of the same sign. If B and D are of opposite signs, the sign of r flips — students often miss this subtlety (NCERT §3, p. 80).
• r interpretation bands (informal): |r| < 0.3 — weak; 0.3 ≤ |r| < 0.7 — moderate; |r| ≥ 0.7 — strong; |r| = 1 — perfect. NCERT does not codify these cut-offs but the bands are widely used in CUET context items.
• Covariance and units: covariance Σ(X − X̄)(Y − Ȳ)/N carries units (e.g., kg·cm for weight-height) — that is exactly why dividing by σx · σy in Pearson's formula makes r dimensionless. Without the standardisation, covariance values across different unit systems cannot be compared (NCERT §3, p. 79).
• Why r² is useful (extension): r² (the coefficient of determination, implicit in NCERT) gives the proportion of total variation in Y explained by linear movement in X. An r of 0.8 means r² = 0.64, i.e., 64% of Y's variation is linearly accounted for by X — a more interpretable number than r itself.
• Tied-rank logic: when m observations tie at, say, ranks 7, 8, 9, the average rank 8 is assigned to all three. The correction factor (m³ − m)/12 added inside Spearman's bracket — once for each tied group — compensates for the deflated ΣD² that results from artificial ties (NCERT §3, p. 86).
• Perfect-positive vs perfect-negative diagrams: in Fig. 6.4 every point lies on an upward line at slope > 0 (r = +1); in Fig. 6.5 every point lies on a downward line at slope < 0 (r = −1). The numerical value of the slope is not the same as r — slope depends on units, r does not. CUET sometimes tests this slope-vs-r distinction.
• Non-linear examples that defeat r: a U-shaped relation (e.g., income vs age) or an inverted-U relation (e.g., productivity vs hours of sleep) can have r ≈ 0 even though the variables are strongly related — illustrating why r = 0 ≠ independence (NCERT §3, p. 80; Fig. 6.6, 6.7).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 6.1 — Positive Correlation: points scattered around an upward-rising line (p. 78).
• Fig. 6.2 — Negative Correlation: points scattered around a downward-sloping line (p. 78).
• Fig. 6.3 — No Correlation: no rising or falling pattern; random scatter (p. 78).
• Fig. 6.4 — Perfect Positive Correlation: all points lie ON an upward line (p. 78).
• Fig. 6.5 — Perfect Negative Correlation: all points lie ON a downward line (p. 78).
• Fig. 6.6 — Positive non-linear relation & Fig. 6.7 — Negative non-linear relation: curved patterns; Pearson's r should NOT be used here (p. 78).
• Table 6.1: worked example computing r = 0.644 between years of schooling of farmers and annual yield per acre (p. 81).
• Table 6.3: step-deviation example yielding r = 0.98 between price index and money supply (p. 83).
• Example 5: worked Spearman calculation with repeated ranks (Y = 50 at ranks 9, 10, 11 averaged to 10; (m³ − m)/12 correction applied), giving rₛ = 0.30 (pp. 86–87).
• Correlation decision flow: data type (cardinal vs ordinal/attribute) → if cardinal and linear use Pearson's r; if ordinal or non-linear monotonic use Spearman's rₛ; always sketch a scatter diagram first.
• Worked Pearson's r (small example): take 5 pairs — (X, Y) = (1, 2), (2, 4), (3, 5), (4, 4), (5, 5). Means X̄ = 3, Ȳ = 4. Deviations x = X − X̄: −2, −1, 0, 1, 2; y = Y − Ȳ: −2, 0, 1, 0, 1. xy products: 4, 0, 0, 0, 2 → Σxy = 6. x²: 4, 1, 0, 1, 4 → Σx² = 10. y²: 4, 0, 1, 0, 1 → Σy² = 6. r = 6 / √(10 × 6) = 6/√60 = 6/7.746 ≈ 0.775. Interpretation: strong positive linear correlation between X and Y.
• Worked Spearman's rₛ (no ties): ranks of two judges for 5 contestants — Judge1: 1, 2, 3, 4, 5; Judge2: 2, 1, 4, 3, 5. D = R1 − R2: −1, 1, −1, 1, 0; D²: 1, 1, 1, 1, 0; ΣD² = 4. rₛ = 1 − [6 × 4 / (5³ − 5)] = 1 − [24/120] = 1 − 0.2 = 0.8. Strong agreement between the two judges.
• Worked Spearman with ties: Y-values 50, 60, 50, 70, 50, 80 → ranks: three 50s tie at ranks 1, 2, 3, averaged to (1+2+3)/3 = 2; so awarded ranks are 2, 4, 2, 5, 2, 6. Correction factor for one tied group of size m=3 is (3³ − 3)/12 = (27 − 3)/12 = 2. The factor 2 is added inside Spearman's bracket numerator before dividing — illustrating mechanically how repeated-rank cases differ from the no-tie case (NCERT §3, p. 86 logic).
• Scatter diagram reading drill: in a 10-point scatter that slopes upward and clusters tightly around an imaginary line, r is high positive (e.g., 0.9); if the same 10 points are scattered widely with a faint upward tendency, r is low positive (e.g., 0.3); if they form a random cloud with no slope, r ≈ 0. The visual feel of "tightness" is the qualitative analogue of |r|, and "tilt" is the analogue of sign(r) (NCERT §3, p. 78 figures).

2.5 Key formulas

2.4 Common confusions / NTA trap points

• Correlation vs causation: r measures covariation only — high r does not prove cause-and-effect.
• Zero correlation is NOT independence: r = 0 means no LINEAR relation, but a non-linear relation may still exist.
• Unit of r: r is a pure number — has no unit (not kg/feet or %).
• Range of r: strictly −1 ≤ r ≤ 1. A value outside this range means calculation error.
• Pearson vs Spearman applicability: Pearson's r is valid only for linear relations between precisely-measured variables; for qualitative attributes (honesty, beauty) or extreme values, use Spearman's.
• Scatter diagram is the only tool that works for any (including non-linear) relationship — both r and rₛ measure only linear relationships.
• Repeated ranks need a correction factor (m³ − m)/12 for each tied group.
• r is unaffected by change of origin and scale — basis of the step-deviation method.
• Generally rₛ ≤ r because rank reduction loses information.
• Sign of r matches the sign of Cov(X, Y) — both denominators are positive.
• Perfect correlation (±1) means all points on a line, not "near" a line.
• Spearman's formula uses n³ − n in the denominator, not n² or n + 1.