This chapter explores the concept of correlation and its significance in understanding relationships between variables in economics.
Correlation – Formula & Equation Sheet
Essential formulas and equations from Statistics for Economics, tailored for Class 11 in Economics.
This one-pager compiles key formulas and equations from the Correlation chapter of Statistics for Economics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
r = Cov(X,Y) / (σX * σY)
r is the correlation coefficient, Cov(X,Y) is the covariance between variables X and Y, σX is the standard deviation of X, and σY is the standard deviation of Y. This formula quantifies the degree of linear relationship between two variables.
Cov(X,Y) = Σ[(Xi - X̄)(Yi - Ȳ)] / N
Cov(X,Y) represents the covariance between X and Y. Xi and Yi are the individual sample points, X̄ and Ȳ are the means of X and Y respectively, and N is the number of observations. Covariance indicates the direction of the relationship.
X̄ = ΣXi / N
X̄ is the mean of variable X, where ΣXi is the sum of all observations of X and N is the number of observations. The mean provides a central value for the dataset.
Ȳ = ΣYi / N
Ȳ is the mean of variable Y, defined similarly to X̄. It serves as a central reference point for variable Y.
σX = √[Σ(Xi - X̄)² / N]
σX is the standard deviation of X, which measures the dispersion of the data around the mean. A higher standard deviation indicates greater variability in X.
σY = √[Σ(Yi - Ȳ)² / N]
σY is the standard deviation of Y, indicating how much the values of Y spread out from the mean Ȳ.
r = (ΣXY - (ΣX)(ΣY)/N) / √[(ΣX² - (ΣX)²/N)(ΣY² - (ΣY)²/N)]
This alternate formula for r computes the correlation coefficient using the sums of products of X and Y. It highlights the relationship's strength and direction.
Spearman’s rank correlation: r_s = 1 - (6Σd²)/(n(n²-1))
r_s is Spearman’s rank correlation coefficient, where d is the difference between ranks for each observation. It assesses the strength and direction of the association between ranked variables.
d = rank(X) - rank(Y)
d represents the difference between the ranks of corresponding data points in X and Y. It is crucial for calculating Spearman’s rank correlation.
N = number of pairs of observations
N is the count of paired values in the correlation analysis, necessary for integrating into formulas for correlation.
Equations
Cov(X,Y) = [ΣXY - (ΣX)(ΣY)/N]
This represents the covariance between two variables, showing the extent to which they change together.
σ²X = [ΣX² - (ΣX)²/N]
σ²X is the variance of X, indicating how variable the observations are around the mean. Variance is the squared standard deviation.
σ²Y = [ΣY² - (ΣY)²/N]
σ²Y denotes the variance of Y, a key measure for understanding the data's dispersion.
X̄ = μ + (σX/σY)(Ȳ - μ)
This linear prediction equation shows the relationship of Y in terms of X, illustrating how changes in Y can influence X.
r^2 represents the coefficient of determination
This value explains the proportion of the variance in the dependent variable that is predictable from the independent variable. A higher r² indicates a better fit of the model.
N = Σ(1 for all pairs)
This summation gives the total number of pairs as needed for computing correlation, reflecting all included observations.
When r = 0, variables X and Y are uncorrelated.
This indicates that there is no linear relationship between the two variables, though other types of relationships could exist.
When r = ±1, there is a perfect linear correlation.
This indicates a deterministic relationship between the two variables; when one changes, the other changes proportionally.
If r is close to +1 or -1, the correlation is considered strong.
A high absolute value of r indicates a close relationship, either positive or negative, respectively.
d = 0 indicates a perfect match in ranks.
When there is no difference in ranks, it highlights a perfect correlation where observations are directly aligned.
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