Principal component analysis (PCA) calculator

About this tool

Principal component analysis rotates a table of correlated variables into a new set of uncorrelated axes — the principal components — ordered so the first one captures as much of the spread in the data as possible, the second as much of what is left, and so on. It is the standard first move for dimensionality reduction, for spotting which measurements move together, and for compressing a wide table into two or three plottable coordinates.

Paste your matrix and this calculator does the whole decomposition in the browser: it centers each column, standardizes it by default so the analysis runs on the correlation matrix, diagonalizes that matrix with the Jacobi eigenvalue algorithm, and reports

• the eigenvalue of every component (the variance along that axis) with its proportion, percentage and cumulative share of the total variance;
• a text scree plot, so the drop-off you pick the component count at is visible without a chart;
• how many components you need for 90%, 95% and 99% of the variance, plus the Kaiser count (components with an eigenvalue above 1);
• the loadings — each original variable's weight in each component, which is what you read to name a component;
• the scores — every observation projected onto the components, ready to scatter-plot.

Nothing is uploaded; the maths runs entirely in your browser as WebAssembly, so the same numbers come out on the page, in the CLI and in chat.

Worked example — body measurements

Paste six people measured on three variables (the first row is read as a header of column names):

height,weight,age
170,65,30
180,80,42
165,59,25
175,72,35
190,95,50
160,54,22

The report opens with

PCA on 6 observations × 3 variables (correlation matrix — columns standardized to unit variance)

Explained variance:
  component   eigenvalue   proportion    percent   cumulative
  PC1           2.992685     0.997562   99.7562%     0.997562
  PC2           0.004537     0.001512    0.1512%     0.999074
  PC3           0.002778     0.000926    0.0926%            1

PC1 holds 99.76% of the variance, and its loadings — height 0.577, weight 0.578, age 0.577 — are near-identical and all positive. That is the signature of a single "size" factor: in this sample, taller people are also heavier and older, so one number per person replaces three with almost no loss. The scores column then ranks the six observations along that axis, from -2.01 to +2.69.

Worked example — keeping the top 2 components

Set Components to report to 2 on exam marks in four subjects:

maths,physics,english,history
78,74,55,58
92,95,61,60
55,52,80,84
61,58,88,90
84,88,64,62
49,45,91,95

PC1 takes 94.24% and PC2 5.64%, and PC1's loadings split by sign — maths -0.505, physics -0.496 against english 0.492, history 0.507. A component whose loadings oppose each other like this is a contrast: it measures the science-versus-humanities tilt of each student rather than overall ability. The percentages are always computed against the full set of components, so trimming the report to 2 does not inflate them.

Covariance vs correlation

Leave Standardize columns to unit variance ticked when the columns are in different units — otherwise the variable with the biggest raw numbers simply wins. Untick it to run PCA on the covariance matrix, which is what you want when every column is already in the same unit and the differences in magnitude are real signal. Compare 0.1,240 / 0.3,610 / 0.2,395 / 0.5,980 / 0.4,760 with and without the box: unstandardized, PC1's loadings are 0.00054 and 1 — the second column's variance of ~85,420 buries the first one entirely.

Reading the output formats

text is the formatted report and prints the first 20 score rows. csv emits just the scores as row,PC1,PC2,… so you can paste them straight into a plotting tool. json returns the full result — every score row, plus the column means and sample standard deviations used to center and scale, which are exactly what you need to project new observations onto the same components later.

Limits and edge cases

• Shape. At least 2 rows and 2 columns; up to 20,000 observations and 100 variables. Every row must have the same number of columns — a ragged row is an error, not a silent truncation.
• Separators. Commas, tabs, semicolons and runs of spaces all work, so a paste from a spreadsheet, a CSV or a fixed-width dump is fine. Blank lines are skipped.
• Header detection. The first row is treated as column names only if its tokens are not all numeric. Fill in Column names to override it, or to name columns in a headerless file.
• Constant columns. A column with zero variance cannot be standardized (it would divide by zero), so standardized runs reject it by name — drop the column or switch to covariance PCA, which handles it fine.
• More variables than observations. Allowed, but at most n − 1 components carry real variance; the rest come back at essentially zero and are noise, not structure.
• Component signs are arbitrary. Flipping every loading and every score in a component gives an equally valid answer, so a convention is fixed: each component is flipped, if needed, so its largest-magnitude loading is positive. Other packages may show the opposite sign — the analysis is identical.
• Rounding. All reported numbers are rounded to 6 decimal places; eigenvalues that should be exactly 0 can appear as tiny values in unstandardized runs.

FAQ