Big picture
Economists often ask how two variables move together: Does demand fall as price rises? Do families with higher incomes spend more on education? Correlation measures the direction and strength of the relationship between two variables.
They also need to measure change over time in a group of related variables: Have prices of essential goods risen since last year, and by how much? Index numbers answer this by expressing changes as percentages of a base year value (taken as 100). India's inflation rate, the stock market's Sensex and industrial growth figures are all index numbers.
Learning goals
Explain correlation and its types.
Interpret a scatter diagram.
Calculate Karl Pearson's coefficient of correlation and state its properties.
Calculate Spearman's rank correlation coefficient.
Explain index numbers and the problems in constructing them.
Construct simple and weighted index numbers (Laspeyres and Paasche).
Explain the CPI, WPI, IIP and Sensex, and use index numbers to measure inflation and real wages.
Core ideas
1. Correlation and its types
Correlation studies how two variables move in relation to each other.
Type | Meaning | Example |
|---|---|---|
Positive correlation | Both variables move in the same direction | Income and expenditure; height and weight |
Negative correlation | Variables move in opposite directions | Price and quantity demanded |
No correlation | No relationship | Shoe size and marks in economics |
Linear | The ratio of change is constant; points lie near a straight line | |
Non-linear (curvilinear) | The ratio of change is not constant |
Note. Correlation does not prove causation. Two variables may move together by chance, or because a third factor affects both. Sales of ice cream and sales of umbrellas may both fall in winter without one causing the other.
2. Scatter diagram
A scatter diagram plots pairs of values (X, Y) as points on a graph.
Points rising from bottom-left to top-right: positive correlation.
Points falling from top-left to bottom-right: negative correlation.
Points close to a straight line: high correlation; widely spread: low correlation.
All points exactly on a rising line: perfect positive correlation (r = +1); on a falling line: perfect negative (r = −1).
No pattern: no correlation.
The scatter diagram shows direction and roughly the strength, but not an exact numerical value.
3. Karl Pearson's coefficient of correlation
r = Σxy ÷ √(Σx² × Σy²)
where x = X − X̄ and y = Y − Ȳ (deviations from the means).
Worked example.
X | Y | x = X − 3 | y = Y − 4 | xy | x² | y² |
|---|---|---|---|---|---|---|
1 | 2 | −2 | −2 | 4 | 4 | 4 |
2 | 4 | −1 | 0 | 0 | 1 | 0 |
3 | 5 | 0 | 1 | 0 | 0 | 1 |
4 | 4 | 1 | 0 | 0 | 1 | 0 |
5 | 5 | 2 | 1 | 2 | 4 | 1 |
ΣX = 15 | ΣY = 20 | Σxy = 6 | Σx² = 10 | Σy² = 6 |
X̄ = 3, Ȳ = 4. r = 6 ÷ √(10 × 6) = 6 ÷ √60 = 6 ÷ 7.746 = 0.77: a fairly high positive correlation.
Properties of r
It lies between −1 and +1.
It has no unit; it is a pure number.
It is not affected by a change of origin or scale: adding a constant to all values, or multiplying them by a positive constant, doesn't change r.
r = 0 means no linear relationship (there may still be a non-linear one).
Interpreting r
Value of r | Meaning |
|---|---|
+1 | Perfect positive correlation |
0.7 to 0.99 | High positive correlation |
0.3 to 0.69 | Moderate positive correlation |
0 to 0.29 | Low positive correlation |
0 | No linear correlation |
Negative values | The same scale, with a negative relationship |
4. Spearman's rank correlation
Used when data are ranks (for qualitative attributes such as beauty, honesty or skill), or when exact values are not reliable.
rs = 1 − (6ΣD²) ÷ (n(n² − 1))
where D is the difference between the two ranks of each item and n the number of items.
Worked example. Two judges rank five singers.
Singer | Rank by judge 1 | Rank by judge 2 | D | D² |
|---|---|---|---|---|
A | 1 | 2 | −1 | 1 |
B | 2 | 1 | 1 | 1 |
C | 3 | 4 | −1 | 1 |
D | 4 | 3 | 1 | 1 |
E | 5 | 5 | 0 | 0 |
ΣD² = 4 |
rs = 1 − (6 × 4) ÷ (5 × 24) = 1 − 24 ÷ 120 = 0.8: the judges largely agree.
When values are given instead of ranks, rank each series first (highest value = rank 1), then apply the formula. When two items have equal values (tied ranks), give each the average of the ranks they would have occupied, and add a correction factor (m³ − m) ÷ 12 to ΣD² for each tie, where m is the number of items tied.
5. Index numbers: meaning and problems
An index number is a statistical device that measures the change in a group of related variables over time, relative to a base period (index = 100).
Types: price index numbers (changes in prices), quantity index numbers (changes in quantities produced or sold) and value index numbers.
Problems in constructing index numbers: choosing a base year (it should be a normal year, free from wars, droughts and booms, and not too far back); choosing the items (they should represent the group); choosing the prices (retail or wholesale, and from where); assigning appropriate weights; and choosing the formula.
6. Constructing price index numbers
Data for the examples
Commodity | Base year price p0 | Current year price p1 | Base year quantity q0 | Current year quantity q1 |
|---|---|---|---|---|
A | 10 | 12 | 5 | 6 |
B | 20 | 25 | 4 | 4 |
C | 40 | 44 | 2 | 3 |
Simple aggregative method
P01 = (Σp1 ÷ Σp0) × 100 = (81 ÷ 70) × 100 = 115.71
It ignores the relative importance of items, so a costly item dominates.
Weighted aggregative methods use quantities as weights.
Laspeyres' index (base-year quantities as weights): P01 = (Σp1q0 ÷ Σp0q0) × 100 = (248 ÷ 210) × 100 = 118.10
Paasche's index (current-year quantities as weights): P01 = (Σp1q1 ÷ Σp0q1) × 100 = (304 ÷ 260) × 100 = 116.92
Workings: Σp1q0 = 12 × 5 + 25 × 4 + 44 × 2 = 60 + 100 + 88 = 248; Σp0q0 = 50 + 80 + 80 = 210; Σp1q1 = 72 + 100 + 132 = 304; Σp0q1 = 60 + 80 + 120 = 260.
Weighted average of price relatives (used for the consumer price index)
Price relative (R) = (p1 ÷ p0) × 100; Index = ΣRW ÷ ΣW
Example. A family's index with weights by share of expenditure:
Item | Price relative R | Weight W | RW |
|---|---|---|---|
Food | 120 | 50 | 6,000 |
Clothing | 110 | 20 | 2,200 |
Fuel | 130 | 15 | 1,950 |
Others | 105 | 15 | 1,575 |
Total | 100 | 11,725 |
Index = 11,725 ÷ 100 = 117.25: the family's cost of living has risen 17.25% since the base year.
7. Important index numbers in India
Index | Measures | Notes |
|---|---|---|
Consumer Price Index (CPI) | Change in retail prices of a basket of goods and services bought by households (cost of living) | Published for rural, urban and combined; CPI (combined) is used by the RBI for inflation targeting. Separate CPIs exist for industrial workers and agricultural labourers |
Wholesale Price Index (WPI) | Change in prices at the wholesale level | Doesn't cover services |
Index of Industrial Production (IIP) | Change in the volume of industrial output (mining, manufacturing, electricity) | A quantity index; base year 2011-12 |
Sensex | Movement of share prices on the Bombay Stock Exchange | Based on 30 large companies; base year 1978-79 = 100 |
Nifty 50 | Share prices on the National Stock Exchange | Based on 50 large companies |
8. Uses of index numbers
Measuring inflation. Rate of inflation = ((Current index − Previous index) ÷ Previous index) × 100. Example: CPI rises from 150 to 159: inflation = (9 ÷ 150) × 100 = 6%.
Real wages. Real wage = (Money wage ÷ CPI) × 100. Example: a worker earns ₹15,000 when CPI is 120. Real wage = 15,000 × 100 ÷ 120 = ₹12,500 at base-year prices. If wages rise by less than prices, real wages fall.
Other uses: fixing dearness allowance; formulating economic policy; deflating national income to real terms; and judging trends in output and share markets.
A reliable problem-solving routine
For Karl Pearson's r, find the means, deviations, xy, x² and y², and apply the formula; check that r lies between −1 and +1.
For Spearman's rs, rank both series, find D and D², and apply the formula (with tie correction if needed).
For an index, decide the method: simple aggregative, Laspeyres (q0), Paasche (q1) or weighted price relatives.
Use inflation and real wage formulas with the right index values.
Connections that matter
Measures of central tendency provide the means used in Karl Pearson's coefficient and the weighted mean used in index numbers.
Consumer equilibrium and demand describes the negative relationship between price and quantity demanded.
Class 12 economics: national income uses the GDP deflator, a price index.
Common mistakes
Treating correlation as proof of causation.
Getting r outside the range −1 to +1. Recheck calculations.
Using current-year quantities in Laspeyres' index. Laspeyres uses base-year quantities; Paasche uses current-year quantities.
Choosing an abnormal year as the base.
Dividing real wage by money wage. Real wage = money wage ÷ index × 100.
Revision checklist
I can explain types of correlation and read a scatter diagram.
I can calculate Karl Pearson's r and state its properties.
I can calculate Spearman's rank correlation, including with values instead of ranks.
I can construct simple, Laspeyres and Paasche indices and a weighted price-relative index.
I can explain CPI, WPI, IIP and Sensex.
I can calculate inflation and real wages.