GIFSQ Tools · Module 16 — Statistics, Experimental Design & Research · Tool 314 of 500
Reduce two correlated measurements to their principal components: the eigenvalues, the variance each component explains, and the loadings that show what each component means.
Try sample values — fills the form with the worked example below.
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PC1 variance explained (%)
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PC1 eigenvalue
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PC2 eigenvalue
Equation: Covariance matrix C = [[sₓ², sₓᵧ], [sₓᵧ, s_y²]] (sample covariance, n − 1). Eigenvalues λ = (tr C ÷ 2) ± √(((sₓ² − s_y²) ÷ 2)² + sₓᵧ²). Variance explained by PC1 = λ₁ ÷ (λ₁ + λ₂) × 100. Loadings are the unit eigenvector of PC1 (first loading made positive).
Method: Principal component analysis of the sample covariance matrix for two variables (standard textbook method; Jolliffe, Principal Component Analysis). The 2 × 2 eigen-decomposition is closed-form, so no iteration is involved.
Assumptions: the variables are measured on comparable scales — standardise first if one variable is in very different units; the components are linear combinations of the original variables.
Limitations: two variables only; with more variables use a full statistics package. PCA finds directions of variance, not of group separation — it is not a classification method.
Rounding: eigenvalues to 4 decimal places; variance explained to 2 decimal places; loadings to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Method: Principal component analysis of the sample covariance matrix for two variables (standard textbook method; Jolliffe, Principal Component Analysis). The 2 × 2 eigen-decomposition is closed-form, so no iteration is involved.
Assumptions: the variables are measured on comparable scales — standardise first if one variable is in very different units; the components are linear combinations of the original variables.
Limitations: two variables only; with more variables use a full statistics package. PCA finds directions of variance, not of group separation — it is not a classification method.
Rounding: eigenvalues to 4 decimal places; variance explained to 2 decimal places; loadings to 4 decimal places; intermediates unrounded.
This is a calculation aid, not a laboratory report — confirm against your accredited method before citing a value.
Worked example: X 2,4,4,4,5,5,7,9 with Y 1,2,3,3,4,5,6,8: PC1 eigenvalue 9.5801, PC2 0.1342, PC1 explains 98.62% of the variance, loadings (0.6854, 0.7282).
Enter values above to see results.
If PC1 explains most of the variance, the two measurements are nearly one measurement — consider dropping one. Tool 314 of 500 — GIFSQ
