PCA Calculator – Free Online Principal Component Analysis

PCA, principal component analysis, turns many correlated variables into a few uncorrelated ones called principal components, ordered so the first one carries the most variance.

You standardise the variables, build their correlation matrix, and take its eigenvalues and eigenvectors. Each eigenvalue is the variance carried by one component. Divide an eigenvalue by the number of variables and you get the share of total information that component holds. Components with an eigenvalue above 1 are usually the ones worth keeping. PCA does not test a hypothesis and returns no p-value: it re-describes your data in fewer dimensions.

Rule of thumb: keep every component with an eigenvalue above 1.0, aim for a cumulative variance of at least 70%, and only name a component from variables that load above 0.40 on it. In the worked example on this page four exam subjects reduce to two components that together explain 94.72% of the variance.

Principal component analysis is a dimensionality-reduction technique that transforms correlated variables into a smaller set of uncorrelated variables, called principal components, that capture the maximum possible variance. It was set out by Pearson in 1901 and given its modern algebraic form by Hotelling in 1933.

Picture your data as a cloud of points in space, one axis per variable. If two variables are correlated, the cloud is a stretched cigar shape rather than a ball. PCA finds the long axis of that cigar and calls it component one. It then finds the longest direction at right angles to it and calls that component two, and so on. Nothing is thrown away at this stage: p variables give p components. The reduction happens when you decide the last few components hold so little variance that you can drop them.

What the calculator gives you:

• Eigenvalues, the variance carried by each component. On a correlation matrix they sum to the number of variables.
• Explained variance, each eigenvalue as a percentage, plus the running cumulative percentage.
• Loadings, the correlation between each original variable and each component. These are what you interpret.
• Communalities, how much of each variable is captured by the components you kept.
• Component scores, a new value for every case on every component, ready to paste into a regression or a cluster analysis.
• Eigenvectors, the weights the calculator used to build each score from the standardised data.
• Varimax rotation, an optional rotation that pushes each loading toward 0 or 1 so the components are easier to name.
• KMO and Bartlett's test, the two standard checks that your data are suitable for PCA at all.

A worked one-liner: twelve students sit four exams. Maths and Physics correlate at r = 0.90, English and History correlate at r = 0.89, and the two pairs barely relate to each other. PCA returns eigenvalues of 2.368 and 1.421, which is 59.20% and 35.51% of the variance. Two components replace four subjects and keep 94.72% of the information.

Who uses it and for what: psychologists reduce long questionnaires to a handful of underlying scales. Ecologists compress dozens of environmental measurements into two axes they can plot. Finance uses PCA on a yield curve, where the first three components are read as level, slope and curvature. Genomics uses it on expression matrices to see population structure. Image compression and face recognition, the eigenfaces method, are PCA applied to pixels. In machine learning it is the standard way to fight multicollinearity and shrink a feature set before fitting a model.

PCA against its nearest neighbour. The question people ask most is factor analysis vs principal components. They look alike and answer different questions.

In practice, when communalities are high and you have more than about 20 variables, PCA and factor analysis give nearly the same loadings. The difference matters most with few variables and modest correlations, which is exactly the situation in the worked example above.

Each variable is one comma-separated list. Every list must hold the same number of observations, because one position in every list is one case. PCA needs at least three variables and works best when you have at least five cases per variable.

4.1 Read the eigenvalues before anything else

An eigenvalue is the amount of variance one component carries. When you analyse the correlation matrix, every original variable has been standardised to a variance of exactly 1, so the eigenvalues sum to the number of variables. That single fact makes the whole table readable. With four variables the eigenvalues add to 4. A component with an eigenvalue of 2.4 is doing the work of 2.4 original variables. A component with an eigenvalue of 0.11 is doing almost nothing.

Divide any eigenvalue by the number of variables and you get the percentage of variance that component explains. This is the number you will quote in a paper. The cumulative column matters more than any single row, because it answers the only question a reader really has: how much of my data survives if I keep just these components.

Eigenvalues are always reported in descending order, and they always will be, because that is how PCA is defined. A rising eigenvalue further down the table is a sign the software failed, not a discovery.

4.2 How many components should you actually keep?

There is no single correct answer, which is why this calculator gives you four rules and shows you all of them at once.

• The Kaiser rule keeps components with an eigenvalue above 1.0. The logic is simple: on a correlation matrix a component below 1 explains less than one original variable, so keeping it is a step backwards. It is the default in SPSS and the most commonly reported rule. Its weakness is that it over-extracts when you have many variables, often keeping a third of them.
• The scree test asks you to look at plot 1 and find the elbow, the point where the curve flattens into scree. Keep the components above the elbow. It is subjective, but with a clear structure it is usually obvious and it usually beats Kaiser.
• The cumulative variance rule keeps components until you reach a target, commonly 70% or 80%. This is the rule to use when PCA is a pre-processing step and you care about how much information you are willing to lose.
• The broken-stick model compares each eigenvalue with what you would expect if the total variance were split at random. It is the strictest of the four and the least likely to over-extract, which is why ecologists favour it.

When the rules disagree, and they often do, prefer the interpretable solution. A three-component answer where every component has a name beats a four-component answer with an uninterpretable fourth. State in your write-up which rule you used and why, because a reviewer who cannot tell will assume you picked the flattering one.

4.3 What a loading actually is

A loading is the correlation between an original variable and a component. It runs from minus 1 to plus 1 and is read exactly like any other correlation. A loading of 0.85 means that variable moves almost in lockstep with the component. A loading of minus 0.85 is just as strong, in the opposite direction.

The conventional cut-offs are 0.30 for a loading worth mentioning, 0.40 for one worth interpreting, and 0.70 for one that defines the component. Some fields use 0.32, following Tabachnick and Fidell, because 0.32 squared is roughly 10% shared variance. The exact threshold matters less than applying the same one to the whole table and saying which you used.

Do not confuse loadings with eigenvector weights. The eigenvector is the recipe used to build the score; the loading is the correlation you interpret. On a correlation matrix loading equals eigenvector times the square root of the eigenvalue, which is why they look similar but never match.

4.4 Naming your components

This is the part software cannot do for you. Take one component, list every variable loading above 0.40 on it, look at what those variables have in common, and give it a short name. Then check the variables loading negatively, because they belong to the same component pointing the other way.

In the worked example, Maths and Physics load one way on component one while English and History load the other way. That component is not "maths ability", it is a quantitative versus verbal contrast, and naming it as a contrast is the honest reading. Component two has all four variables loading positively at a similar size, which is the classic signature of a general "overall performance" component.

Two warnings. A component you cannot name is usually a component you should not keep. And a name is a hypothesis, not a finding, so write "component one was interpreted as" rather than "component one measured".

4.5 The sign of a component is arbitrary

If you multiply an eigenvector by minus 1 it is still an eigenvector of the same matrix with the same eigenvalue. Nothing in the mathematics fixes the direction. That is why the same data can give loadings of plus 0.77 in R and minus 0.77 in Python, and neither is wrong.

This calculator uses the convention that the largest loading on each component is made positive, which is what scikit-learn does. R's prcomp applies no convention at all and returns whatever the underlying decomposition produced. If your signs flip between software, the interpretation flips with them and the conclusion is unchanged. Never treat a sign difference as a discrepancy, and never report one without saying which direction you called positive.

4.6 Communalities tell you which variables belong

A communality is the sum of squared loadings for one variable across the components you kept. It is the share of that variable explained by the retained solution, and it runs from 0 to 1.

High communalities, above about 0.60, mean the retained components represent that variable well. A communality below 0.40 is a warning: that variable is largely independent of everything else, and keeping it in the analysis does nothing except dilute the components. The usual response is to drop it and rerun, then say so in the methods.

Uniqueness is simply 1 minus communality. Some software reports one, some the other, and they carry identical information.

4.7 Correlation matrix or covariance matrix?

This choice changes the answer more than any other setting on the page, and it is the mistake beginners make most often.

The covariance matrix uses the raw variances. If one variable is measured in millimetres and another in kilometres, the millimetre variable has a numerically enormous variance and will dominate component one entirely. The result is arithmetically correct and scientifically useless.

The correlation matrix standardises every variable to a variance of 1 first, so each one enters the analysis on equal terms. This is the right default and it is what almost every published PCA uses. Choose covariance only when all variables share a unit and the differences in their spread are part of what you want the analysis to see, for example repeated measurements of the same quantity or a set of log-transformed variables.

4.8 Rotation, and why it does not cheat

An unrotated solution spreads variance so component one grabs as much as possible. That is optimal mathematically and often terrible for interpretation, because many variables load moderately on component one and nothing defines anything.

Varimax rotation turns the retained components as a rigid frame, pushing each loading toward either 0 or 1 while keeping the components uncorrelated. The total variance explained by the retained set does not change by a single decimal place, only its distribution across components changes. Nothing is manufactured.

Rotate when you have kept two or more components and want to name them. Do not rotate when PCA is a pre-processing step feeding a regression or a clustering, because then the ordering by variance is the point. And never rotate a one-component solution: there is nothing to rotate against.

One honest warning about software. Varimax is an iterative search, not a closed-form calculation, so different programs can land on different rotations of the same solution. This calculator sweeps every pair of retained components repeatedly until the varimax criterion stops improving. R's varimax() starts from the identity matrix and on small problems, four variables and two components in particular, it can stop at a local optimum with a visibly less simple structure. Both answers preserve the total variance and every communality exactly, so neither is wrong in the sense of being miscalculated, but they are not the same rotation. If you are comparing this page with R and the rotated loadings differ, compare the simplicity of the two tables rather than assuming one is broken.

4.9 KMO and Bartlett, the two entry checks

The Kaiser-Meyer-Olkin measure asks whether your correlations are widespread or scattered. It compares ordinary correlations with partial correlations. If variables share common factors, partial correlations are small and KMO approaches 1. Kaiser's own labels: below 0.50 unacceptable, 0.50 to 0.59 miserable, 0.60 to 0.69 mediocre, 0.70 to 0.79 middling, 0.80 to 0.89 meritorious, above 0.90 marvellous. KMO is unreliable with very few variables and is not worth reporting below about six.

Bartlett's test of sphericity tests whether the correlation matrix is an identity matrix, meaning no variable correlates with any other. A significant result, p below 0.05, says the correlations are strong enough to be worth reducing. It is a low bar and it is almost always significant with a decent sample, so a non-significant Bartlett is a genuine red flag and a significant one is not evidence of much.

4.10 Sample size: how much is enough?

The old guidance was a fixed ratio, five or ten cases per variable, or an absolute floor of 100 or 300 cases. Those rules are simple and they are not very good.

The modern position, following MacCallum and colleagues, is that the required sample depends on the communalities. When communalities are high, above 0.60, even 60 cases can give a stable solution. When they hover around 0.50 you want 100 to 200. When they are low and you have few variables per component, you may need 500 and still get an unstable answer. Report your n and your communalities together and let the reader judge.

A separate hard limit applies whatever the guidance says. If you have fewer cases than variables, the correlation matrix is singular, some eigenvalues are exactly 0, and the last components are meaningless noise. The calculator warns you when this happens.

4.11 What PCA cannot tell you

PCA has no p-value for the components themselves, no confidence intervals as standard, and no significance test, because it is a description of the data and not an inference about a population. It cannot tell you that a component is real.

It finds linear structure only. Data lying on a curved manifold, a horseshoe or a ring, will be badly represented by straight axes, and the components can produce the arch effect that ecologists know well. It is not robust to outliers: one extreme case can define a whole component on its own, which is why robust principal component analysis exists.

It also gives you no causal claim whatsoever. A component is a weighted sum of what you measured. If you did not measure it, PCA cannot find it.

4.12 Practical judgement, in order

Check suitability first: KMO, Bartlett, sample size, and a scan of the correlation matrix for a variable that correlates with nothing. Then decide the number of components using the scree plot and one stated rule, not four rules chosen after the fact. Then rotate if you are naming components. Then read the loadings and write the names down before you look at anything else, so you are not fitting the story to a later result.

Finally, sanity-check the communalities. If every variable is well represented and the components have names you would defend out loud, the analysis is done. If one variable will not fit anywhere, drop it, rerun, and report both solutions.

▶ Run the analysis above to auto-fill all five examples with your own results.

5.1 What a complete PCA write-up must contain

Reviewers reject principal component analysis sections for a small, predictable list of omissions. Include all nine of these and the section will pass.

1. The number of variables entered and the number of cases, with any exclusions.
2. Whether you analysed the correlation or the covariance matrix, and why.
3. The suitability evidence: KMO, Bartlett's test with its chi-square, degrees of freedom and p-value.
4. The rule you used to decide how many components to keep, stated before the result.
5. Eigenvalues and the percentage of variance for every retained component, plus the cumulative percentage.
6. The loading table, with the cut-off you applied and whether small loadings were suppressed.
7. The rotation used, or a clear statement that none was used.
8. Your interpretation and name for each component.
9. The software and version. This calculator's citation is in Section 19.

5.2 Five ready-to-use templates

5.3 Table and figure conventions

The loading table is the centrepiece. Put variables in rows and components in columns, sort rows by their highest loading so the block structure is visible, and suppress loadings below your stated threshold. Add a communality column on the right and an eigenvalue and percentage-of-variance row at the bottom. Give the table a note stating the extraction method, the rotation and the suppression threshold.

The scree plot is the figure. Plot component number on the x axis and eigenvalue on the y axis, draw a horizontal reference line at 1.0, and mark the retained components. A biplot is a good second figure when you have a two-component solution and want to show cases and variables together, but it becomes unreadable above about 200 cases.

5.4 The Discussion sentence

Link each component back to your research question, not to the algebra. Write "the quantitative-verbal contrast explained most of the variation between students, which supports the two-domain view of attainment" rather than "component one had an eigenvalue of 2.368". The eigenvalue belongs in Results. The meaning belongs in Discussion.

5.5 Reporting checklist

Seven formulas take you from raw data to a named component. They run in this order, and every number this calculator prints comes from one of them.

Ten steps, start to finish. The worked example used throughout this page is four exam subjects measured on twelve students.

Worked example: paste 52, 48, 55, 61, 47, 70, 66, 58, 44, 63, 57, 50 into variable 1, name it Maths, then add Physics, English and History from the sample dataset. Run it. You should see eigenvalues of 2.3681 and 1.4206, which is 59.20% and 35.51% of the variance and 94.72% cumulative.

Excel can do a complete principal component analysis, but it will not hand you one. There is no =PCA() function and the Analysis ToolPak has no PCA entry, so you have to build the eigenvalues yourself. The good news is that the two hard steps, the correlation matrix and the matrix multiplication, both have direct functions, and the eigenvalues fall out of a short iteration you can drag down a column.

8.1 The functions you need

8.2 Step by step

8.3 The awkward cases

There is no eigenvalue function. This is the only genuinely hard part. Power iteration with MMULT works and is what step 3 shows, but it is slow to set up for more than about six variables and it needs deflation for each extra component. For a serious analysis with many variables, install a free add-in that has PCA built in, or move the job to R or Python. Do not try to solve the characteristic polynomial by hand above three variables.

MMULT is an array formula. In Microsoft 365 it spills automatically. In Excel 2019 and earlier you must select the whole output range first, type the formula, then press Ctrl+Shift+Enter. If you press plain Enter you get one number instead of a matrix and nothing downstream works.

The ToolPak correlation output is triangular. Data Analysis → Correlation fills only the lower triangle and leaves the upper half blank. MMULT needs a full square matrix, so mirror the values across the diagonal before you use it, or build the matrix with CORREL as in step 2.

KMO needs a matrix inverse. Use =MINVERSE(G2:J5), then convert to partial correlations with −ajk ÷ √(ajj akk). It is doable but fiddly, and Excel will refuse if the matrix is singular. If your determinant from =MDETERM() is close to zero, the inverse is unreliable and KMO cannot be trusted.

Rotation is not practical in Excel. Varimax is an iterative procedure over pairs of components. Nobody should build it in a spreadsheet. Use the rotation dropdown on this page, or R's varimax().

8.4 Why your Excel answer might differ from R or Python

8.5 Charting the result in Excel

Select the eigenvalue column with its component labels, then Insert → Line with Markers. That is your scree plot. Two formatting changes make it publication ready: set the y-axis minimum to 0 so the drop is not exaggerated, and add a horizontal line at 1.0 by plotting a second series of 1s and formatting it as a dashed grey line. Label it "Kaiser criterion" so a reader knows what it is.

For a loading plot, select the PC1 and PC2 loading columns and insert a Scatter chart, then add data labels from the variable name column using the "Value From Cells" option. Set both axes to run from −1 to 1 and add axis lines through zero, because a loading plot is only readable when the origin is visible.

8.6 Excel errors you will probably hit

R gives you principal component analysis in one line with prcomp(dat, scale. = TRUE), and that one line is almost never what you should report. It returns standard deviations rather than eigenvalues, no loadings in the correlation sense, no communalities and no suitability checks. The script below does the full job in base R with no packages at all, so it runs on any installation.

All three ship with R. Nothing to install. eigen(), cor() and pchisq() live in stats, and png() lives in grDevices.

9.1 The complete script

# =====================================================================
# Principal Component Analysis (PCA) in R, base R only
# Worked example: 12 students, 4 exam subjects
# Saves pca_plot.png at 300 dpi. Run with: Rscript pca_script.R
# =====================================================================

set.seed(42)                       # reproducible, so the figure is identical every run

# ---- 1. YOUR DATA. Change only this block. --------------------------
dat <- data.frame(
  Maths   = c(52, 48, 55, 61, 47, 70, 66, 58, 44, 63, 57, 50),
  Physics = c(59, 42, 54, 60, 49, 71, 70, 68, 48, 71, 64, 51),
  English = c(70, 81, 55, 79, 71, 61, 59, 77, 67, 71, 60, 61),
  History = c(72, 79, 55, 72, 68, 57, 64, 72, 61, 69, 56, 65)
)
# From a file instead:
# dat <- read.csv("exam_scores.csv")
# dat <- dat[complete.cases(dat), ]        # drop any row with a missing value

n <- nrow(dat); p <- ncol(dat)

# ---- 2. Correlation matrix and eigen-decomposition ------------------
R  <- cor(dat)
eg <- eigen(R)                      # eigen() returns values already sorted, largest first
evals <- eg$values
evecs <- eg$vectors

# Sign convention: make the largest weight in each column positive.
# R applies no convention of its own, so without this your signs may
# differ from Python, SPSS or this page. The interpretation is identical.
for (k in seq_len(p)) {
  j <- which.max(abs(evecs[, k]))
  if (evecs[j, k] < 0) evecs[, k] <- -evecs[, k]
}
rownames(evecs) <- names(dat)
colnames(evecs) <- paste0("PC", seq_len(p))

prop <- 100 * evals / sum(evals)
cum  <- cumsum(prop)

loadings <- evecs %*% diag(sqrt(evals))          # loading = eigenvector * sqrt(eigenvalue)
dimnames(loadings) <- dimnames(evecs)

Z      <- scale(dat)                              # scale() uses sd(), divisor n - 1
scores <- Z %*% evecs

keep <- max(sum(evals > 1), 1)                    # Kaiser rule
comm <- rowSums(loadings[, seq_len(keep), drop = FALSE]^2)

# ---- 3. Suitability checks ------------------------------------------
detR   <- det(R)
bart   <- -((n - 1) - (2 * p + 5) / 6) * log(detR)
df     <- p * (p - 1) / 2
bart_p <- pchisq(bart, df, lower.tail = FALSE)

Rinv    <- solve(R)
D       <- diag(1 / sqrt(diag(Rinv)))
partial <- -D %*% Rinv %*% D
diag(partial) <- 0
Roff <- R; diag(Roff) <- 0
kmo  <- sum(Roff^2) / (sum(Roff^2) + sum(partial^2))

# ---- 4. Printed summary ---------------------------------------------
cat("PRINCIPAL COMPONENT ANALYSIS\n")
cat(sprintf("Cases n = %d   Variables p = %d   Matrix: correlation\n\n", n, p))

cat("Eigenvalues and variance explained\n")
print(round(data.frame(Eigenvalue = evals, Percent = prop, Cumulative = cum,
                       row.names = colnames(evecs)), 4))

cat("\nLoadings and communality\n")
print(round(cbind(loadings[, seq_len(keep), drop = FALSE], h2 = comm), 4))

cat("\nSuitability\n")
cat(sprintf("  KMO               = %.4f\n", kmo))
cat(sprintf("  Bartlett chi2(%d)  = %.4f, p = %.3g\n", df, bart, bart_p))
cat(sprintf("  Determinant of R  = %.6f\n", detR))

cat(sprintf("\nVERDICT: Bartlett is significant and KMO is %.2f, so the variables are correlated enough for PCA. Keeping %d component(s) retains %.2f%% of the variance.\n",
            kmo, keep, cum[keep]))

# ---- 5. One 300 dpi figure: scree plot + biplot ----------------------
png("pca_plot.png", width = 11, height = 4.6, units = "in", res = 300)
par(mfrow = c(1, 2), mar = c(4.4, 4.4, 3, 1.4))

plot(seq_len(p), evals, type = "b", pch = 19, cex = 1.4, lwd = 2.2, col = "#16a34a",
     xlab = "Component", ylab = "Eigenvalue", main = "Scree plot",
     ylim = c(0, max(evals) * 1.2), xaxt = "n")
axis(1, at = seq_len(p))
abline(h = 1, lty = 2, col = "#9ca3af", lwd = 1.4)
text(p, 1.08, "Kaiser criterion", adj = 1, cex = .8, col = "#6b7280")
grid(col = "#e5e7eb")

sc <- max(abs(scores[, 1:2])) / max(abs(loadings[, 1:2])) * 0.85
plot(scores[, 1], scores[, 2], pch = 19, col = "#2563eb", cex = 1.2,
     xlab = sprintf("PC1 (%.2f%%)", prop[1]), ylab = sprintf("PC2 (%.2f%%)", prop[2]),
     main = "Biplot: cases and variables")
abline(h = 0, v = 0, col = "#d1d5db")
grid(col = "#e5e7eb")
arrows(0, 0, loadings[, 1] * sc, loadings[, 2] * sc, col = "#c2410c", lwd = 2, length = .1)
text(loadings[, 1] * sc * 1.14, loadings[, 2] * sc * 1.14, rownames(loadings),
     col = "#7c2d12", font = 2, cex = .9)

mtext(sprintf("Principal Component Analysis, n = %d, p = %d", n, p),
      outer = TRUE, line = -1.4, font = 2)
dev.off()
cat("\nFigure saved to pca_plot.png at 300 dpi\n")

9.2 What each part does

9.3 What the figure shows

The left panel is the scree plot. Eigenvalues fall from 2.37 to 1.42, then collapse to about 0.11. The dashed line at 1.0 is the Kaiser criterion, and the elbow sits clearly between components 2 and 3, so both rules agree on a two-component solution.

The right panel is the biplot. Blue points are the twelve students positioned by their PC1 and PC2 scores. Orange arrows are the four subjects, drawn from the origin with their length set by their loadings. Maths and Physics point one way, English and History the other, and the near-90-degree angle between the two pairs is the picture of two independent components. A student sitting far along an arrow scored highly on that subject.

9.4 Common R problems and fixes

9.5 Useful one-liners

# The fast version, correlation matrix PCA
p1 <- prcomp(dat, scale. = TRUE); summary(p1)

# Eigenvalues from prcomp: square the standard deviations
p1$sdev^2

# Loadings, the way SPSS reports them
p1$rotation %*% diag(p1$sdev)

# Varimax rotation on the first two components
varimax(p1$rotation[, 1:2] %*% diag(p1$sdev[1:2]))

# Component scores for every case
head(p1$x)

# Scree plot in one call
screeplot(p1, type = "lines", main = "Scree plot")

# Built-in biplot
biplot(p1, cex = 0.8)

# Write the loadings to CSV
write.csv(round(loadings, 4), "pca_loadings.csv")

Most people reach for sklearn.decomposition.PCA, and for a machine-learning pipeline that is the right choice. For a paper it is the wrong shape: scikit-learn gives you explained_variance_ratio_ and components_, but no loadings, no communalities, no KMO and no Bartlett test. The script below uses NumPy and SciPy directly so every number a reviewer asks for is printed.

Install with pip install numpy scipy matplotlib pandas. NumPy does the eigen-decomposition, SciPy supplies the chi-square distribution for Bartlett's test, matplotlib draws the figure, and pandas is needed only if you read a CSV.

10.1 The complete script

# =====================================================================
# Principal Component Analysis (PCA) in Python
# Worked example: 12 students, 4 exam subjects
# Saves pca_plot.png at 300 dpi. Run with: python pca_script.py
# =====================================================================
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from scipy.stats import chi2

np.random.seed(42)                       # reproducible jitter in the biplot

# ---- 1. YOUR DATA. Change only this block. --------------------------
labels = ["Maths", "Physics", "English", "History"]
X = np.array([
    [52, 59, 70, 72], [48, 42, 81, 79], [55, 54, 55, 55], [61, 60, 79, 72],
    [47, 49, 71, 68], [70, 71, 61, 57], [66, 70, 59, 64], [58, 68, 77, 72],
    [44, 48, 67, 61], [63, 71, 71, 69], [57, 64, 60, 56], [50, 51, 61, 65],
], dtype=float)
# From a file instead:
# import pandas as pd
# df = pd.read_csv("exam_scores.csv").dropna()   # drop rows with any missing value
# labels, X = list(df.columns), df.to_numpy(dtype=float)

n, p = X.shape

# ---- 2. Standardise, then build the correlation matrix --------------
Z = (X - X.mean(axis=0)) / X.std(axis=0, ddof=1)   # ddof=1 matches R's sd()
R = np.corrcoef(X, rowvar=False)

# ---- 3. Eigen-decomposition ----------------------------------------
evals, evecs = np.linalg.eigh(R)                   # eigh: R is symmetric
order = np.argsort(evals)[::-1]                    # largest eigenvalue first
evals, evecs = evals[order], evecs[:, order]

for k in range(p):                                 # sign convention: biggest weight positive
    j = np.argmax(np.abs(evecs[:, k]))
    if evecs[j, k] < 0:
        evecs[:, k] *= -1

prop = 100 * evals / evals.sum()
cum = np.cumsum(prop)
loadings = evecs * np.sqrt(evals)                  # loading = eigenvector * sqrt(eigenvalue)
scores = Z @ evecs                                 # one score per case per component

keep = int((evals > 1).sum())                      # Kaiser rule
keep = max(keep, 1)
comm = (loadings[:, :keep] ** 2).sum(axis=1)       # communality over kept components

# ---- 4. Suitability checks -----------------------------------------
detR = np.linalg.det(R)
bart = -((n - 1) - (2 * p + 5) / 6) * np.log(detR)
df = p * (p - 1) / 2
bart_p = chi2.sf(bart, df)

Rinv = np.linalg.inv(R)
D = np.diag(1 / np.sqrt(np.diag(Rinv)))
partial = -D @ Rinv @ D
np.fill_diagonal(partial, 0)
Roff = R.copy()
np.fill_diagonal(Roff, 0)
kmo = (Roff ** 2).sum() / ((Roff ** 2).sum() + (partial ** 2).sum())

# ---- 5. Printed summary --------------------------------------------
print("PRINCIPAL COMPONENT ANALYSIS")
print("Cases n = %d   Variables p = %d   Matrix: correlation" % (n, p))
print("\nEigenvalues and variance explained")
print("%-6s %12s %12s %12s" % ("PC", "Eigenvalue", "% variance", "Cumulative %"))
for k in range(p):
    print("%-6s %12.4f %12.2f %12.2f" % ("PC%d" % (k + 1), evals[k], prop[k], cum[k]))

print("\nLoadings (first %d components) and communality" % keep)
head = "%-10s" % "Variable" + "".join("%10s" % ("PC%d" % (k + 1)) for k in range(keep)) + "%12s" % "h2"
print(head)
for j in range(p):
    row = "%-10s" % labels[j] + "".join("%10.4f" % loadings[j, k] for k in range(keep))
    print(row + "%12.4f" % comm[j])

print("\nSuitability")
print("  KMO                     = %.4f" % kmo)
print("  Bartlett chi2(%d)        = %.4f, p = %.3g" % (df, bart, bart_p))
print("  Determinant of R        = %.6f" % detR)

verdict = ("Bartlett is significant and KMO is %.2f, so the variables are correlated enough for PCA. "
           "Keeping %d component%s retains %.2f%% of the variance." %
           (kmo, keep, "" if keep == 1 else "s", cum[keep - 1]))
print("\nVERDICT: " + verdict)

# ---- 6. One 300 dpi figure: scree plot + biplot ---------------------
fig, ax = plt.subplots(1, 2, figsize=(11, 4.6))

ax[0].plot(range(1, p + 1), evals, "o-", color="#16a34a", lw=2.2, ms=8)
ax[0].axhline(1, ls="--", color="#9ca3af", lw=1.4)
ax[0].text(p, 1.06, "Kaiser criterion", ha="right", fontsize=9, color="#6b7280")
ax[0].set_xticks(range(1, p + 1))
ax[0].set_xlabel("Component"); ax[0].set_ylabel("Eigenvalue")
ax[0].set_title("Scree plot"); ax[0].set_ylim(0, max(evals) * 1.2)
ax[0].grid(alpha=.3)

ax[1].axhline(0, color="#d1d5db", lw=1); ax[1].axvline(0, color="#d1d5db", lw=1)
ax[1].scatter(scores[:, 0], scores[:, 1], s=46, color="#2563eb", alpha=.75, zorder=3)
scale = np.abs(scores[:, :2]).max() / np.abs(loadings[:, :2]).max() * 0.85
for j in range(p):
    ax[1].arrow(0, 0, loadings[j, 0] * scale, loadings[j, 1] * scale,
                color="#c2410c", width=.012, head_width=.11, length_includes_head=True, zorder=4)
    ax[1].text(loadings[j, 0] * scale * 1.14, loadings[j, 1] * scale * 1.14, labels[j],
               color="#7c2d12", fontsize=9.5, fontweight="bold", ha="center")
ax[1].set_xlabel("PC1 (%.2f%%)" % prop[0]); ax[1].set_ylabel("PC2 (%.2f%%)" % prop[1])
ax[1].set_title("Biplot: cases and variables"); ax[1].grid(alpha=.3)

fig.suptitle("Principal Component Analysis, n = %d, p = %d" % (n, p), fontweight="bold")
fig.tight_layout()
fig.savefig("pca_plot.png", dpi=300, bbox_inches="tight")
print("\nFigure saved to pca_plot.png at 300 dpi")

10.2 What each part does

10.3 What the figure shows

The same two panels as the R figure, and by design the same numbers. The scree plot on the left drops steeply from 2.37 to 1.42 and then flattens near 0.11, which is a textbook elbow. The biplot on the right places the twelve students by their PC1 and PC2 scores in blue and draws the four subject loadings as orange arrows.

Read the biplot by angle. Two arrows pointing the same way are positively correlated variables. Two arrows at right angles are uncorrelated. Two arrows pointing opposite ways are negatively correlated. Here Maths and Physics sit almost on top of each other, English and History sit almost on top of each other, and the two pairs point in opposite directions along PC1.

10.4 Common Python problems and fixes

10.5 Useful one-liners

# The fast version with scikit-learn, standardised
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
Z = StandardScaler().fit_transform(X); fit = PCA().fit(Z)

# Eigenvalues and percentage of variance
fit.explained_variance_; fit.explained_variance_ratio_ * 100

# Loadings the way SPSS reports them
fit.components_.T * np.sqrt(fit.explained_variance_)

# Keep only enough components for 80 percent of the variance
PCA(n_components=0.80).fit_transform(Z)

# Component scores for every case
fit.transform(Z)[:5]

# Varimax rotation using factor_analyzer
from factor_analyzer.rotator import Rotator
Rotator(method="varimax").fit_transform(loadings[:, :2])

# KMO and Bartlett in one import
from factor_analyzer.factor_analyzer import calculate_kmo, calculate_bartlett_sphericity

# Save the loadings to CSV
import pandas as pd; pd.DataFrame(loadings, index=labels).round(4).to_csv("pca_loadings.csv")

Run it with python pca_script.py from a terminal. In Jupyter, delete the matplotlib.use("Agg") line and add plt.show() at the end so the figure appears inline as well as on disk.

Eight lookup tables covering every threshold this calculator applies. Each one says what to conclude, not only what the number is.

11.1 The four component retention rules compared

What to conclude: when the rules disagree, report the one you chose in advance and mention the alternative in a sentence. Parallel analysis is the most accurate in simulation studies (Zwick and Velicer, 1986) but Kaiser remains the most widely reported.

11.2 Interpreting the size of a loading

What to conclude: these bands follow Comrey and Lee (1992) and the 0.32 floor follows Tabachnick and Fidell. A variable loading above 0.32 on two or more components is cross-loading, and the usual response is to drop it or to rotate.

11.3 Kaiser-Meyer-Olkin sampling adequacy

What to conclude: KMO is unstable with fewer than about six variables, because there are too few partial correlations to average over. With four variables a KMO near 0.50 is normal and is not by itself a reason to abandon the analysis. Check the per-variable MSA values too: a single variable below 0.50 is a candidate for removal.

11.4 How many cases do you need?

What to conclude: report n, p and your communalities together. A small n with high communalities and few components is far safer than a large n with poor communalities. The old ratio rules are a starting point, not a standard.

11.5 Broken-stick expected eigenvalues

What to conclude: keep component k only while its observed eigenvalue exceeds the value in this table, and stop at the first component that fails. In the worked example, p = 4 and the observed eigenvalues 2.3681 and 1.4206 both beat 2.083 and 1.083, so the broken-stick rule also keeps two components. Values are the expected lengths of pieces of a randomly broken unit stick, rescaled to p.

11.6 Cumulative variance targets by field

What to conclude: a cumulative variance that looks low in one field is normal in another. State your target before you extract, and never move it after seeing the answer.

11.7 Chi-square critical values for Bartlett's test

What to conclude: if your Bartlett chi-square exceeds the .05 column, the correlation matrix differs significantly from an identity matrix and PCA is worth running. In the worked example, chi-square(6) = 29.008 beats even the .001 value of 22.458. Note that with a large n this test is significant almost automatically, so passing it is a floor, not a recommendation.

11.8 PCA against the methods people confuse it with

What to conclude: PCA is the default first move for numeric, roughly linear data. Move to one of the alternatives only when you can name the specific assumption PCA is breaking.

Eight worked analyses, every number computed with the same engine that runs above. They span the range on purpose: a clean textbook case, a single dominant component, a borderline eigenvalue, no structure at all, a sample that is too small, a component the Kaiser rule would wrongly discard, a matrix broken by a duplicated variable, and a field study where rotation rescues the interpretation.

Design this PCA calculator assumes: a single sample of independent cases, each measured on the same set of continuous or near-continuous numeric variables, with no missing values and no grouping structure. PCA has no groups, no treatment and no outcome. Every variable enters on the same footing.

13.1 Twelve steps to data a principal component analysis can use

13.2 The data layout the calculator expects

One row per case, one column per variable, a header row of real variable names, and no ID column mixed into the numbers. On the Upload tab you click the four numeric columns you want and each becomes its own variable in the calculator. The case ID column is simply not clicked.

13.3 Minimum sample and pre-registration

The hard algebraic minimum is that the number of cases must exceed the number of variables, otherwise the correlation matrix is singular. The practical minimum for a stable solution is 5 cases per variable with an absolute floor around 50, and 10 per variable with a floor around 100 for anything you intend to publish.

Pre-register four things: the exact variable list, correlation or covariance, the retention rule, and the loading threshold for interpretation. Also state in advance what you will do if a communality falls below 0.40, because deciding that after seeing the result is where most PCA papers lose credibility.

13.4 Common mistakes that make data unusable for PCA

• Mixing units without standardising. Fixed by the correlation matrix default, but fatal if you switch to covariance without thinking.
• Entering a sum alongside its parts. Creates an exact dependency and a zero eigenvalue.
• Different numbers of observations per variable. Each row must be one case across every variable, so ragged columns cannot be analysed.
• Coding missing values as 0 or 999. These become real numbers and destroy the correlations. Leave them blank and delete the row.
• Including an ID or a date column as a variable. Case numbers correlate with collection order and produce a meaningless component.
• Analysing groups together that should be separate. Two populations mixed into one PCA produce a first component that just separates the groups.

This free PCA calculator is designed for researchers, students and analysts who have several correlated numeric variables and want fewer of them without losing the information they carry.

14.1 Use it when

• You have three or more numeric variables measured on the same cases.
• The variables are correlated, and you suspect they measure overlapping things.
• You want a small number of scores to feed into a regression, a clustering, or a plot.
• You want to see the structure in a correlation matrix rather than read 45 correlations one by one.
• You need to fight multicollinearity before fitting a model.
• You want a two-dimensional picture of a many-dimensional dataset.

14.2 Do not use it when

• Your variables are categorical or binary. Use correspondence analysis or multiple correspondence analysis.
• You have fewer cases than variables. The matrix is singular and the answer is not defined.
• You want to test a specific factor structure. Use confirmatory factor analysis.
• You believe in a latent construct that causes the variables. Use exploratory factor analysis.
• You have known groups and want to separate them. Use linear discriminant analysis.
• The relationships are curved rather than linear. Use kernel PCA, or transform the variables first.
• Your variables are uncorrelated. Bartlett's test will tell you, and there is nothing to reduce.

14.3 Four real situations

14.4 Decision tree: is PCA the right tool?

16.1 Assumptions

PCA does not assume multivariate normality. Normality matters only for Bartlett's test and for any inference about the components, not for the decomposition itself, which is pure linear algebra.

16.2 Limitations

1. No significance test for the components. PCA is descriptive. For a formal test of dimensionality use parallel analysis, or confirmatory factor analysis if you have a hypothesis.
2. Component names are interpretation, not measurement. Two competent researchers can name the same component differently. If the name matters, validate it against an external criterion.
3. Sensitive to the variable set. Adding or removing one variable changes every loading. Solutions are not comparable across studies that measured different things.
4. Linear only. Use kernel PCA, t-SNE or UMAP for curved structure, accepting that those axes are not interpretable.
5. Not robust to outliers. Use robust principal component analysis when a few extreme cases are distorting the solution.
6. Sign indeterminacy. Components are defined only up to a sign, so a convention must be stated whenever loadings are compared across studies.
7. Cannot handle missing data natively. Use probabilistic PCA or multiple imputation when gaps are unavoidable.
8. The retained components may not be the useful ones. PCA orders components by variance, not by relevance to your question. In principal component regression the component that predicts the outcome best is sometimes the fourth.
9. Not a causal model. A component is a weighted sum of measured variables. It says nothing about what causes what.
10. Scale-dependent when using covariance. Changing a variable from metres to centimetres changes the entire covariance solution, though it leaves the correlation solution untouched.

What principal component analysis is for

A PCA calculator answers one question: how few numbers can carry your data without losing what matters. It does that by rebuilding your variables as uncorrelated components ordered by the variance each one holds, then letting you keep the first few. In the worked example on this page, four exam subjects become two components holding 94.72% of the information, and both components have a name a teacher would recognise.

What it means in practice

The eigenvalue table is the finding, the loading table is the interpretation, and the scree plot is the evidence for how many components you kept. Everything else on this page supports one of those three. Once you have named your components you have a shorter dataset that a regression, a cluster analysis or a two-dimensional plot can actually work with.

Cautions

Three mistakes account for most bad PCA. Running it on the covariance matrix when the variables have different units, which lets one variable take over. Trusting the Kaiser rule without looking at the scree plot or the loadings. And naming a component from a loading of 0.35 because it was the only thing on the component. None of these produce an error message, which is why all three are so common.

Next steps

Run your own variables above, check KMO and Bartlett before anything else, choose your retention rule before you look at the eigenvalues, and turn on varimax rotation when a two-component solution is hard to name. Then export the component scores and use them. Principal component analysis is only finished when the components are doing work somewhere else in your analysis.

If this PCA calculator produced numbers that appear in your work, cite it alongside the statistical sources in Section 22.

Run the analysis above and the Methods paragraph fills itself with your own values.

Tools on StatsUnlock that pair naturally with a principal component analysis.

Twenty peer-reviewed journal articles behind every rule and threshold this principal component analysis calculator applies. Each has a resolvable DOI.

1. Abdi, H., & Williams, L. J. (2010). Principal component analysis. WIREs Computational Statistics, 2(4), 433–459. https://doi.org/10.1002/wics.101
2. Bartlett, M. S. (1950). Tests of significance in factor analysis. British Journal of Statistical Psychology, 3(2), 77–85. https://doi.org/10.1111/j.2044-8317.1950.tb00285.x
3. Candès, E. J., Li, X., Ma, Y., & Wright, J. (2011). Robust principal component analysis? Journal of the ACM, 58(3), 1–37. https://doi.org/10.1145/1970392.1970395
4. Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1(2), 245–276. https://doi.org/10.1207/s15327906mbr0102_10
5. Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4(3), 272–299. https://doi.org/10.1037/1082-989X.4.3.272
6. Gabriel, K. R. (1971). The biplot graphic display of matrices with application to principal component analysis. Biometrika, 58(3), 453–467. https://doi.org/10.1093/biomet/58.3.453
7. Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179–185. https://doi.org/10.1007/BF02289447
8. Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24(6), 417–441. https://doi.org/10.1037/h0071325
9. Jackson, D. A. (1993). Stopping rules in principal components analysis: A comparison of heuristical and statistical approaches. Ecology, 74(8), 2204–2214. https://doi.org/10.2307/1939574
10. Jolliffe, I. T., & Cadima, J. (2016). Principal component analysis: A review and recent developments. Philosophical Transactions of the Royal Society A, 374(2065), 20150202. https://doi.org/10.1098/rsta.2015.0202
11. Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23(3), 187–200. https://doi.org/10.1007/BF02289233
12. Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20(1), 141–151. https://doi.org/10.1177/001316446002000116
13. Kaiser, H. F. (1974). An index of factorial simplicity. Psychometrika, 39(1), 31–36. https://doi.org/10.1007/BF02291575
14. MacCallum, R. C., Widaman, K. F., Zhang, S., & Hong, S. (1999). Sample size in factor analysis. Psychological Methods, 4(1), 84–99. https://doi.org/10.1037/1082-989X.4.1.84
15. Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 559–572. https://doi.org/10.1080/14786440109462720
16. Ringnér, M. (2008). What is principal component analysis? Nature Biotechnology, 26(3), 303–304. https://doi.org/10.1038/nbt0308-303
17. Tipping, M. E., & Bishop, C. M. (1999). Probabilistic principal component analysis. Journal of the Royal Statistical Society: Series B, 61(3), 611–622. https://doi.org/10.1111/1467-9868.00196
18. Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41(3), 321–327. https://doi.org/10.1007/BF02293557
19. Zou, H., Hastie, T., & Tibshirani, R. (2006). Sparse principal component analysis. Journal of Computational and Graphical Statistics, 15(2), 265–286. https://doi.org/10.1198/106186006X113430
20. Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99(3), 432–442. https://doi.org/10.1037/0033-2909.99.3.432