What is Principal Component Analysis (PCA)?

Principal Component Analysis (PCA) is a dimensionality reduction technique that transforms a set of possibly correlated variables into a smaller set of uncorrelated variables called principal components. Each component is a linear combination of the original variables and is ordered by the amount of variance it explains in the data.

Should I standardize variables before running PCA?

You should standardize variables before PCA when they are measured on different scales (e.g., kilograms and centimeters) or when you want each variable to contribute equally regardless of its variance. Standardization is usually done by subtracting the mean and dividing by the standard deviation, which is equivalent to performing PCA on the correlation matrix instead of the covariance matrix.

How many principal components should I keep?

There is no single rule, but common criteria include: keeping enough components to explain a target percentage of variance (e.g., 90%), using the Kaiser rule (eigenvalues greater than 1 when using standardized data), and inspecting the scree plot to find the elbow where additional components add little extra variance.

What is the difference between loadings and scores in PCA?

Loadings are the coefficients of the eigenvectors that define each principal component as a linear combination of the original variables. Scores (or principal component scores) are the coordinates of each observation in the new component space, obtained by multiplying the centered (and possibly standardized) data matrix by the loading matrix.

PCA Calculator – Principal Component Analysis

Upload or paste your data and instantly compute principal components, eigenvalues, explained variance, scores and loadings.

1. Input data

Paste a numeric data matrix (rows = observations, columns = variables). Use commas or tabs as separators. Optionally include a header row and/or a first column with observation labels.

First row contains variable names

First column contains observation labels

2. Options

Scaling Center only (covariance PCA) Standardize (z-scores, correlation PCA)

Target cumulative variance (%)

Used to suggest how many components to keep.

4. Eigenvalues & explained variance

Component	Eigenvalue	% Variance	Cumulative %

Use this table as scree-plot data: look for the “elbow” where additional components add little extra variance.

5. Loadings (eigenvectors)

Each loading is the coefficient of a variable in a principal component. Large absolute values indicate strong contribution.

6. Scores (coordinates in PC space)

These are the transformed coordinates of each observation on the principal components.

How this PCA calculator works

This tool performs a standard Principal Component Analysis (PCA) on your numeric data matrix. You can choose between:

Covariance PCA (center only) – appropriate when variables are on comparable scales.
Correlation PCA (standardize) – recommended when variables have different units or variances.

Step-by-step algorithm

Build the data matrix \(X\) of size \(n \times p\) (n observations, p variables).
Center each column: subtract its mean \(\mu_j\).
Optionally standardize each column by its standard deviation \(s_j\): \(z_{ij} = (x_{ij} - \mu_j)/s_j\).
Compute the covariance or correlation matrix:
\[ S = \frac{1}{n-1} X_c^\top X_c \]
Eigen-decomposition of \(S\):
\[ S v_k = \lambda_k v_k \]
where \(\lambda_k\) are eigenvalues and \(v_k\) the eigenvectors (loadings).
Sort eigenvalues in descending order and reorder eigenvectors accordingly.
Compute scores (principal components):
\[ Z = X_c V \]
where \(V\) is the matrix of eigenvectors.

Explained variance

Each eigenvalue \(\lambda_k\) measures the variance captured by component \(k\). The proportion of variance explained is:

\[ \text{Var}_k = \frac{\lambda_k}{\sum_{j=1}^{p} \lambda_j}, \quad \text{CumVar}_k = \sum_{i=1}^{k} \text{Var}_i \]

The calculator uses your target cumulative variance (e.g. 90%) to suggest how many components to retain.

Interpreting loadings and scores

Loadings show how strongly each original variable contributes to a component.
Scores are the coordinates of each observation in the reduced-dimensional space.
Observations close together in the first 2–3 components are similar in terms of the original variables.

When to use PCA

To reduce dimensionality before clustering or regression.
To visualize high-dimensional data in 2D or 3D.
To remove multicollinearity between predictors.

Frequently asked questions

Do I need to normalize my data?

If all variables are measured in the same units and have similar variance, centering may be enough. If not, select Standardize (z-scores) so that each variable has mean 0 and variance 1.

What if my dataset has missing values?

This calculator currently ignores rows that contain non-numeric values in any variable. For serious analysis, consider imputing missing values (e.g. mean imputation, k-NN, or model-based methods) before running PCA.

Is this PCA the same as in R, Python, or SPSS?

Yes, the core math is the same: eigen-decomposition of the covariance or correlation matrix. Minor numerical differences can arise from rounding and implementation details, but results should closely match functions like prcomp in R or sklearn.decomposition.PCA in Python.