You have a survey with 24 rating questions about a product, and a crosstab for every one of them would bury your readers. Principal component analysis (PCA) is the standard way to see what those 24 questions have in common, and it is the technique that most often sits behind the phrase “we grouped the questions.” This article covers what it does, how to read the output, and how it differs from the methods around it. For the wider picture, see our guide to factor analysis, PCA and cluster analysis in market research.
Key takeaways
- Principal component analysis (PCA) compresses many correlated variables into a few uncorrelated components, each a weighted blend of the original questions, ordered by how much variation it captures.
- Standardize variables first, because PCA is driven by variance and a question with a wider scale would otherwise dominate the first component.
- An eigenvalue is the variance a component captures, so with standardized variables an eigenvalue of 1 equals the information in one original question.
- Decide how many components to keep using the Kaiser rule of eigenvalues above 1, the scree plot elbow and whether you can name each component.
- PCA summarizes total variance with fewer scores, while factor analysis models the hidden traits behind the answers using only shared variance.
What does principal component analysis (PCA) do in plain terms?
PCA looks at how your variables move together and builds a new, shorter set of variables from them. If respondents who rate “good value for the price” highly also rate “fair prices” and “worth what I pay” highly, those three columns carry largely the same information. PCA replaces them with one score that captures most of what they share.
Each new variable is a principal component, and it is a weighted sum of the original variables. The first component is chosen to capture as much of the total variation in the data as any single combination can. The second captures as much as possible of what remains, with one condition: it must be uncorrelated with the first. The third does the same with what is left, and so on.
Components come out in order of importance, and because they are uncorrelated they do not repeat each other. You start with as many components as variables. The gain comes from keeping only the first few, which means accepting some information loss.
Why do you standardize variables before running PCA?
PCA is driven by variance, so a variable with a large spread dominates the first component simply because its numbers are bigger. A question on a 0 to 10 scale has more variance than one on a 1 to 5 scale by construction, which has nothing to do with your respondents.
The fix is to standardize each variable first: subtract its mean and divide by its standard deviation, so every variable has a mean of 0 and a variance of 1. This is the same as running PCA on the correlation matrix instead of the covariance matrix. Check your software’s default, because some tools use the covariance matrix unless told otherwise.
What are eigenvalues and variance explained?
Each component has an eigenvalue, which is the amount of variance that component captures. When variables are standardized, each one contributes a variance of 1, so the total variance equals the number of variables. An eigenvalue of 1 therefore means a component holds as much variance as a single original question.
The share of variance explained is the eigenvalue divided by the total. The table below uses eight standardized survey items, so the total is 8. The numbers are illustrative, not from a real study.
| Component | Eigenvalue | Variance explained | Cumulative |
|---|---|---|---|
| 1 | 3.2 | 40.0% | 40.0% |
| 2 | 2.0 | 25.0% | 65.0% |
| 3 | 0.8 | 10.0% | 75.0% |
| 4 | 0.6 | 7.5% | 82.5% |
| 5 | 0.4 | 5.0% | 87.5% |
| 6 | 0.4 | 5.0% | 92.5% |
| 7 | 0.4 | 5.0% | 97.5% |
| 8 | 0.2 | 2.5% | 100.0% |
The arithmetic: the eigenvalues sum to 8.0, component 1 is 3.2 / 8 = 40%, component 2 is 2.0 / 8 = 25%, and so on. The first two components together hold 65% of the variance, so two scores stand in for eight questions with 35% of the variation set aside.
How many components should you keep?
There is no single correct answer. Three tests are used together, and they do not always agree.
- Kaiser rule. Keep components with an eigenvalue above 1, since each one then explains more than a single original variable. In the table, that keeps components 1 and 2. The rule is simple but crude, so treat it as a starting point.
- Scree plot. Plot the eigenvalues in order and look for the elbow, the point where the line stops dropping steeply and flattens. In the table the drop from 2.0 to 0.8 is sharp and the values then level off, which points to two components.
- Interpretability. Keep the components you can name and explain. A component that no one can describe, or that is driven by a single question, is rarely worth reporting even if it passes the first two tests.
Some analysts also set a cumulative variance target, often 60% to 80%, but that is a convention and not a law.
How do you read component loadings?
A loading shows how strongly a variable relates to a component. In most software, with standardized variables, it is the correlation between the variable and the component, so it runs from -1 to 1. The sign gives direction and the size gives strength.
Here are illustrative loadings for six questions on the first two components of a customer survey.
| Question | Component 1 | Component 2 |
|---|---|---|
| Good value for the price | 0.82 | 0.10 |
| Prices are fair | 0.78 | 0.15 |
| Worth what I pay | 0.85 | 0.08 |
| Staff are helpful | 0.12 | 0.80 |
| Support responds quickly | 0.07 | 0.76 |
| Staff know the product | 0.18 | 0.72 |
Component 1 is clearly about price and value, and component 2 is about service. Those are the labels you would give them in the report. You choose the names; the software only produces the numbers.
Analysts often treat loadings of about 0.4 or higher as meaningful and 0.6 or higher as strong, but these cutoffs are rules of thumb. If the pattern is hard to read, a rotation such as varimax can make each question load mainly on one component. Rotation redistributes variance among the kept components; the total stays the same.
How is PCA different from factor analysis?
The two are often confused, and many tools put them in the same menu, but they answer different questions.
| PCA | Factor analysis | |
|---|---|---|
| Goal | Summarize many variables with fewer scores | Model hidden traits that cause the answers |
| Variance used | Total variance, including noise unique to each question | Shared variance only, with unique variance and error modeled separately |
| What you get | Components that are exact blends of your variables | Factors that are estimated, with questions as imperfect indicators |
| Typical use | Data reduction, scoring, removing overlap before another analysis | Building and testing a measurement scale |
If you believe an unobserved idea such as “perceived value” drives answers to several questions, factor analysis fits that belief. If you simply want fewer, cleaner variables, PCA is more direct. With many strongly correlated questions the two often give similar patterns. Our article on practical uses of factor analysis goes deeper on the factor side.
Where does PCA help with survey data?
PCA earns its place in a few recurring jobs.
- Shortening a battery. Reduce 30 brand-attribute ratings to four or five themes.
- Removing overlap before regression. Highly correlated predictors make coefficients unstable; component scores are uncorrelated.
- Preparing segmentation inputs. Clustering on component scores stops a topic with ten questions from dominating. See hierarchical clustering for the next step.
- Screening a questionnaire. A question that loads on nothing is worth reviewing before the next wave.
Halo Reports covers the reporting step with crosstabs and banner tables with weighting and significance testing, charts and interactive report pages, so component scores can be cut by segment like any other measure. For wide, complex studies, one governed platform for insights and research teams keeps them with your global brand trackers.
What are the common mistakes with PCA?
- Skipping standardization. The widest-scale question quietly takes over component 1.
- Treating components as proven constructs. PCA finds patterns in correlations; it does not show your questions measure a real trait.
- Running it on uncorrelated questions. If variables barely correlate, no small set of components summarizes them. Check the correlation matrix first.
- Following the Kaiser rule alone. Cross-check with the scree plot and interpretability.
- Using too small a sample. A few dozen respondents give components that shift with a handful of new interviews.
Before you run PCA on your next survey, open the correlation matrix and ask whether groups of questions really do move together. If they do, standardize, extract, check the scree plot and name only the components you can defend in a sentence. If those components describe your brand, track brand health, consideration and sentiment continuously so you know what changed and why.