Correlation

In univariate statistics, analysis is confined to the characteristics of a single variable (such as national income, factory wages, or wheat prices). However, in economic life, variables rarely exist in isolation; they continuously interact. When two variables vary together in such a manner that a change in one is accompanied by an equivalent or systematic change in the other, they are said to be correlated.

Prominent statisticians have defined correlation as follows:

• Croxton & Cowden: "When the relationship is of a quantitative nature, the appropriate statistical tool for discovering and measuring the relationship and expressing it in a brief formula is known as correlation."
• A. M. Tuttle: "Correlation is an analysis of the covariation between two or more variables."
• Boddington: "Whenever some definite connection exists between two or more groups, classes, or series of data, there is said to be correlation."

Data involving two simultaneous characteristics measured on the same statistical unit (e.g., height and weight of an individual, price and quantity demanded of a commodity, fertilizer dosage and crop yield per acre) constitute a bivariate distribution, represented as pairs of observations $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$. Correlation measures the strength and direction of linear association within such paired observations.

Correlation is classified according to three distinct analytical dimensions:

The Scatter Diagram (or Scattergram) is the simplest, most intuitive graphical technique for visually inspecting the nature, direction, and strength of correlation between two variables. It requires no complex mathematical computations and provides an immediate visual summary of bivariate relationships.

Construction Procedure:

1. Take the independent or base variable on the horizontal axis ($X$-axis) and the dependent or related variable on the vertical axis ($Y$-axis).
2. Select convenient and proportionate scales for both axes so that the entire range of data is clearly distributed across the graph.
3. Plot each pair of values $(x_i, y_i)$ as a discrete coordinate dot $(\bullet)$ on the Cartesian plane.
4. The resultant cluster of plotted dots constitutes the scatter diagram. The general shape, slope, and dispersion of the dot cloud reveal the underlying correlation pattern.

• Merits:
  • Extremely simple and non-mathematical; easily understood by non-statisticians.
  • Instantly reveals whether the association is linear or non-linear, and whether positive or negative.
  • Visual inspection immediately exposes extreme outliers or recording anomalies that could distort algebraic averages.
• Limitations:
  • Purely qualitative and descriptive; it does not provide an exact numerical coefficient indicating the precise magnitude of correlation.
  • Subjective visual interpretation: different observers may judge the tightness of the scatter band differently.
  • Incapable of measuring multivariate relationships involving more than two variables simultaneously.

Formulated by the eminent British biometrician and statistician Karl Pearson in 1896, the Product-Moment Correlation Coefficient (denoted by the symbol $r$ or $r_{xy}$) is the most widely utilized mathematical measure of linear association between two quantitative variables.

Pearson defined $r$ as the ratio of the covariance between variables $X$ and $Y$ to the product of their individual standard deviations:

Where:

• $\text{Cov}(X, Y) = \frac{1}{N}\sum (X - \bar{X})(Y - \bar{Y}) = \frac{\sum xy}{N}$ (with $x = X - \bar{X}$ and $y = Y - \bar{Y}$)
• $\sigma_X = \sqrt{\frac{1}{N}\sum (X - \bar{X})^2} = \sqrt{\frac{\sum x^2}{N}}$
• $\sigma_Y = \sqrt{\frac{1}{N}\sum (Y - \bar{Y})^2} = \sqrt{\frac{\sum y^2}{N}}$

Substituting these into the ratio cancels out the sample size $N$, yielding the classic product-moment formula:

In WBCHSE examinations, candidates can compute $r$ using three standard methods depending on whether the arithmetic means are whole integers or fractions:

Karl Pearson's coefficient of correlation satisfies five essential mathematical theorems that frequently form the subject of theoretical questions in Higher Secondary examinations:

The numerical value of the correlation coefficient is strictly bounded within the closed interval $[-1, +1]$:

Mathematical Rationale: This property derives directly from the Cauchy-Schwarz Inequality: for any two real vectors $\vec{u}$ and $\vec{v}$, $(\vec{u} \cdot \vec{v})^2 \le |\vec{u}|^2 |\vec{v}|^2$. Applying this to centered deviations, $[\sum (x-\bar{x})(y-\bar{y})]^2 \le [\sum (x-\bar{x})^2][\sum (y-\bar{y})^2]$, which guarantees that $r^2 \le 1$, meaning $-1 \le r \le +1$. If a calculation yields $r = 1.25$ or $r = -1.40$, an arithmetic error is certain.

The value of $r$ is completely independent of the choice of origin (addition or subtraction of constants) and choice of scale (multiplication or division by positive constants).

If original variables $X$ and $Y$ are transformed into new variables $U$ and $V$ by:

where $a, b$ are arbitrary origin shifts and $h, k$ are non-zero scale constants, then:

• If $h$ and $k$ have the same sign (both positive or both negative), then $r_{UV} = r_{XY}$.
• If $h$ and $k$ have opposite signs (one positive and one negative), then $r_{UV} = -r_{XY}$.

Because $r$ is the ratio of covariance (measured in units of $X \times$ units of $Y$) to the product of standard deviations (measured in units of $X \times$ units of $Y$), all physical measurement units cancel out entirely. Thus, $r$ is a pure dimensionless number, permitting direct comparison between relationships measured in completely different dimensions (e.g., height in inches vs. weight in kg, or price in rupees vs. quantity in quintals).

Correlation is a symmetric relationship: $r_{XY} = r_{YX}$. The correlation between price and demand is algebraically and numerically identical to the correlation between demand and price.

A value of $r = 0$ indicates the total absence of a linear relationship between $X$ and $Y$. However, it does not prove that the two variables are statistically independent. A perfect non-linear or curvilinear relationship (such as $Y = X^2$ over symmetric domains $[-3, 3]$) produces $r = 0$ even though $Y$ is completely deterministic from $X$.

To evaluate whether a sample correlation coefficient $r$ reflects a genuine relationship in the underlying population or arose merely from random sampling fluctuations, statisticians utilize the Probable Error ($PE$):

Decision Criteria:

• If $r < PE_r$, the correlation is statistically insignificant and should be discounted.
• If $r > 6 \times PE_r$, the correlation is highly significant and points to a definitive relationship in the population.
• The population correlation coefficient is expected to fall within limits: $r \pm PE_r$.

In many economic and social inquiries, variables cannot be measured directly on a quantitative cardinal scale (e.g., intelligence, beauty, managerial competence, honesty, consumer preference, brand loyalty). However, statistical individuals can easily be arranged in serial order or ranked (1st, 2nd, 3rd, $\dots$).

To measure the association between two sets of qualitative rankings, the British psychologist and statistician Charles Edward Spearman developed the Rank Correlation Coefficient (denoted by $\rho$ or $R$) in 1904. It is a non-parametric metric that evaluates monotonic relationships without assuming a bivariate normal distribution.

When each individual receives a unique, distinct rank in both series ($R_1$ and $R_2$), the formula is:

Where:

• $D = R_1 - R_2$ is the difference between the ranks assigned to the same individual in variables $X$ and $Y$.
• $\sum D^2$ is the sum of the squares of rank differences.
• $N$ is the total number of paired observations / individuals ranked.

When two or more items have identical values in a series, they are assigned the average (arithmetic mean) of the ranks they would have occupied if they were slightly different. For example, if two students tie for the 3rd rank, they occupy positions 3 and 4; each is awarded rank $\frac{3 + 4}{2} = 3.5$. The next student receives rank 5.

Because tying reduces rank variance, a correction factor of $\frac{m(m^2 - 1)}{12}$ must be added to $\sum D^2$ for each tied group, where $m$ is the number of times a value repeats:

If value $X = 40$ repeats 3 times ($m_1 = 3$), its correction factor is $\frac{3(9 - 1)}{12} = \frac{24}{12} = 2.0$. If another value repeats twice ($m_2 = 2$), its correction factor is $\frac{2(4 - 1)}{12} = 0.5$.

Correlation analysis is embedded throughout empirical microeconomics, macroeconomic policymaking, and financial management:

• Law of Demand & Elasticity: Economists test whether the empirical correlation between price and demand is consistently negative ($r < 0$). In commodity markets, high negative correlation confirms normal goods, while a positive correlation signals exceptional Giffen goods or Veblen goods.
• Keynesian Consumption Function: John Maynard Keynes posited that consumption expenditure ($C$) is directly related to disposable income ($Y$). Empirical studies across Indian macroeconomic data reveal a high positive correlation ($r > +0.90$), validating the fundamental psychological law of consumption.
• Phillips Curve & Monetary Trade-offs: The historical Phillips Curve examined the inverse correlation between the rate of unemployment and the rate of wage inflation ($r < 0$), providing central banks with critical policy insights regarding macroeconomic stabilization.
• Modern Portfolio Diversification: In financial economics, Harry Markowitz demonstrated that constructing an investment portfolio of securities with negative or zero correlation minimizes overall investment variance (risk) without sacrificing expected returns.

In analyzing correlation, economists and students must remain vigilant against common analytical traps:

1. Spurious / Nonsense Correlation: Two independent series may exhibit high correlation simply because both happen to share a common secular upward trend over time (e.g., both cumulative mobile phone sales in India and life expectancy have risen steadily over the past 30 years, but one does not cause the other).
2. Third-Variable Distortion (Confounding Factors): Ice cream sales and drowning accidents display a strong positive correlation in summer months. Neither causes the other; both are driven by a confounding third variable: high ambient summer temperatures.
3. Masked Non-Linearity: A low or zero Pearson coefficient ($r = 0$) does not mean variables are unrelated. The relationship between tax rates and total tax revenue (Laffer Curve) is inverted U-shaped: revenue rises with tax rates up to an optimal peak, then declines. Linear $r$ across the entire range would be close to zero, completely masking the profound non-linear relationship.
4. Truncation of Range: Restricting the sample to a very narrow range (e.g., studying the correlation between intelligence and economic performance only among Nobel laureates) drastically reduces the observed correlation coefficient relative to the broader population.

When confronted with an empirical problem, select the appropriate method using this diagnostic rule:

• Quantitative cardinal data with linear relationship? $\rightarrow$ Use Karl Pearson's Direct or Short-cut Method ($r$).
• Qualitative attributes (honesty, beauty, debate rankings) or ordinal data? $\rightarrow$ Use Spearman's Rank Correlation ($R$).
• Initial exploratory visual inspection or non-linear trend detection? $\rightarrow$ Use Scatter Diagram.