---
url: "https://questionstar.com/blog/correlation-vs-causation/"
title: "Correlation vs. causation: how to read r and regression"
description: "What the correlation coefficient measures, why correlation is not causation, and how to read a regression analysis. With a fully worked survey example."
lastmod: "2026-11-12"
type: "blog"
---

# Correlation vs. causation: how to read r and regression

*Two answers in my survey are related: what does that mean, and can I conclude cause and effect from it?*

What the correlation coefficient measures, why correlation is not causation, and how to read a regression analysis. With a fully worked survey example.

In short: a correlation tells you how closely and in which direction two characteristics vary together, but not why. To conclude cause and effect, three conditions have to be met: an association, the right order in time, and the elimination of other explanations (Malhotra and Birks 2007, p. 303). A single survey can usually check only the first (Schumann 2019, p. 127), and a regression, which predicts one answer from one or several others, doesn't change that.

This article answers five questions in order:

1. [What does the correlation coefficient measure, and why does the scatter plot come first?](#correlation)
2. [Pearson or Spearman, and when does a correlation count as strong?](#pearson-spearman)
3. [Why is a correlation no proof of cause and effect?](#causation)
4. [How do you read a regression, simple and with several independent variables?](#regression)
5. [How do I report a relationship without claiming too much?](#reporting)

In results presentations, "is related to" quickly turns into "leads to," and that turns into an action with a budget. This article works through one example from the first number to the regression and shows at every step which statement the data support.

## What does the correlation coefficient measure?

Tall people usually have bigger feet. Not every one of them, but by and large it holds: if you know a person's height, you can estimate their shoe size fairly well. With the house number, that doesn't work. It has nothing to do with height, and so nothing to do with feet either: very different people live behind the same house number.

Correlation is about exactly this, only with answers from a survey: can the answer to one question be estimated from the answer to another?

**The correlation coefficient r** sums this up in one number. It always lies between −1 and +1:

- **Close to +1:** Whoever scores high on one question almost always scores high on the other as well. Height and shoe size go in this direction. This is called a positive correlation ([Marx 2026, slide 210](/principles-of-survey-research/chapter-5/slide-210.html)).
- **Close to 0:** One answer hardly helps you estimate the other, at least not along the lines of "the more of one, the more of the other." That's the case with the house number.
- **Close to −1:** Whoever scores high on one question almost always scores low on the other. This is a negative correlation.

Stated precisely, Pearson's r measures how closely two [metric](/blog/levels-of-measurement/) variables are linearly related (Malhotra and Birks 2007, p. 575; [Marx 2026, slide 211](/principles-of-survey-research/chapter-5/slide-211.html)). Linear means: when one answer rises, the other rises or falls with it at an even rate on average. Individual answers can deviate considerably from this pattern, but overall, one answer varies predictably with the other.

There are two things r doesn't tell you (Schumann 2019, p. 229; Malhotra and Birks 2007, pp. 575-577):

- **Whether there is a different kind of relationship.** With r close to 0, only the even, linear relationship is missing. A different one, an arc for example, is still possible.
- **What depends on what.** r stays the same no matter which of the two answers you name first and in which units you measure.

### The example: satisfaction with support and likelihood to recommend

One invented example runs through the whole article. A software vendor writes to 200 randomly selected customers who submitted a support request in the past month. 48 respond. Three questions matter:

> "How satisfied are you with our support?"
> 1 (very dissatisfied) to 5 (very satisfied)
> "How likely are you to recommend us to a colleague?"
> 0 (not at all likely) to 10 (extremely likely)
> "Was your issue resolved at first contact?"
> yes · no

We set the third question aside for now. The means show the direction:

| Satisfaction with support | Respondents | Mean likelihood to recommend (0 to 10) |
|------|------|------|
| 1 very dissatisfied | 2 | 3.0 |
| 2 somewhat dissatisfied | 11 | 4.1 |
| 3 neither satisfied nor dissatisfied | 13 | 5.2 |
| 4 somewhat satisfied | 14 | 7.1 |
| 5 very satisfied | 8 | 7.6 |

*Invented data, n = 48, my own calculation.*

With every step up in satisfaction, the mean likelihood to recommend rises. The correlation coefficient condenses this to r = 0.61: those who are more satisfied with support report, on average, a clearly higher likelihood to recommend, but by no means every one of them.

## Why does the scatter plot come before the number?

A scatter plot draws one dot for each respondent: one answer on the horizontal axis, the other on the vertical axis. It shows the shape of the relationship, draws attention to patterns and problems, and lets you spot unusual combinations at once. That's why Malhotra and Birks put it first, before any regression calculation (2007, pp. 583-584). The article [Analyzing online survey results](/blog/analyzing-online-survey-results/) recommends the same.

In the example, the cloud of dots runs from the bottom left to the top right, without a curve and without single dots that dominate the picture. Because many answers on rating scales fall on the same spot (here 25 combinations for 48 respondents), the dot size shows how many people are behind each one. Something else stands out: the circles and the squares form two clouds at different heights. We'll come back to this under [causation](#causation).

### For the curious: where the name product-moment correlation comes from

In textbooks and statistics programs, r is also called the product-moment correlation. The name describes the calculation, and the scatter plot is a good place to show it. Two lines divide the picture: the average satisfaction (3.3) and the average likelihood to recommend (5.8).

For each person, the deviation from the average on one question is multiplied by the deviation on the other. That is the product. If both answers are above the average, or both below it, the product is positive. If they are on different sides, it is negative. In the example, 35 of the 48 respondents are in the two plus fields and only 13 in the minus fields. What matters, though, is not the number of respondents in the fields but the sum of the products: whoever lies far from the average counts more. Here the sum is positive, which is why r is positive too.

In statistics, the average of these products is called a moment. As a last step, it is divided by the spread of one question and by the spread of the other, that is, by how far apart the answers to a question lie. Otherwise the number would depend on the scales: if you counted the same answers from 0 to 10 instead of from 1 to 5, the products would be larger, although nothing about the relationship changes. This way, the result always lies between −1 and +1. If plus and minus cancel each other out, r ends up close to 0.

### Same r, four different pictures

How much a single number can conceal is shown by four invented data sets, each with ten teams of one company: for each team, the mean for team climate and the mean for intention to stay, both from 1 to 5. In all four, r = 0.70, and in all four the relationship is significant by the usual test (p = 0.02, more on this [below](#significance)).

| Data set | What the picture shows | What r = 0.70 conceals |
|------|------|------|
| A | a straight, even cloud of dots | Here the number matches the shape of the relationship. |
| B | a curve: intention to stay rises up to a team climate of about 3.7 and then falls slightly | A straight line would be the wrong model. |
| C | nine teams with no recognizable relationship, plus a single team far to the top right | Without this one team, r = −0.05. |
| D | two groups of five teams each, two locations for example | Within the groups, r = −0.20 and 0.00: r describes the difference between the locations. |

*Invented data, my own calculation. Idea based on Anscombe (1973, p. 19): four data sets with the same summary statistics and completely different pictures.*

You can't recognize any of these cases from the coefficient.

## Pearson or Spearman: which coefficient fits my data?

Pearson's r requires metric data (Malhotra and Birks 2007, p. 580). For ordinal data there are Spearman's rho and Kendall's tau, which use only the rank order of the answers and also lie between −1 and +1 (Malhotra and Birks 2007, p. 581).

| Your two variables | Suitable coefficient |
|------|------|
| both metric (for example age, a count, an amount, a scale score from several items) | Pearson's r |
| both ordinal (for example two single rating questions, ranks) | Spearman's rho, with Kendall's tau as an alternative when there are very many tied ranks |
| both nominal (categories without an order) or both yes/no | Cramér's V, the contingency coefficient or phi, see Cross-tabulation |

*My own summary based on Schumann 2019, p. 239, and Malhotra and Birks 2007, p. 581. Cramér's V is added.*

Which method fits other kinds of questions is shown in the overview [Which statistical test fits?](/blog/statistical-tests/). For surveys, the second row is the tricky one: whether a single rating question may be treated as metric is disputed (see [Levels of measurement](/blog/levels-of-measurement/#likert)). For correlation, this means in practice:

- **For single rating questions, calculate both Pearson and Spearman.** In the example, Pearson gives r = 0.61 and Spearman ρ = 0.62. If the two are this close, your statement doesn't hinge on the disputed question. If they differ clearly, look at the scatter plot to find out why.
- **Kendall's tau is the alternative with very many tied ranks.** If many respondents fall into a few categories, Malhotra and Birks prefer tau (2007, p. 581). For the same data, tau usually comes out smaller than r and rho, in the example tau-b = 0.48 (my own calculation). So compare tau only with tau.
- **Few scale points and imprecise measurement push the correlation down.** With fewer response categories, the coefficient tends to come out smaller (Malhotra and Birks 2007, pp. 352, 577), more on this in the article [Rating scale](/blog/rating-scale/#scale-points). The size of r also depends on the reliability of the measurements (Schumann 2019, p. 229), see [Reliability and validity](/blog/reliability-and-validity/).

## When is a correlation strong?

There is no universally accepted cutoff. Every discipline has its own traditional cutoffs, and they contradict one another. You don't have to read the table row by row: find the row of your discipline, and see in the right-hand column how differently the same number, r = 0.61, is labeled. For customer surveys and other market research it's the first row, for a thesis the row of your discipline.

| Discipline and source | Classification by the absolute value of r | What r = 0.61 is called there |
|------|------|------|
| Marketing (Marx 2026, slide 212) | 0 to 0.3 very weak · 0.3 to 0.5 weak · 0.5 to 0.7 moderate · 0.7 to 0.9 high · 0.9 to 1 very high | moderate |
| Behavioral sciences (Cohen 1988, pp. 79-80) | 0.10 small · 0.30 medium · 0.50 large effect | large |
| Differential psychology (Gignac and Szodorai 2016, p. 74) | 0.10 relatively small · 0.20 typical · 0.30 relatively large | relatively large |
| Psychology (Dancey and Reidy 2007, cited in Akoglu 2018, Table 1) | 0.1 to 0.3 weak · 0.4 to 0.6 moderate · 0.7 to 0.9 strong | moderate |
| Political science (Quinnipiac University, cited in Akoglu 2018, Table 1) | 0.1 negligible · 0.2 weak · 0.3 moderate · 0.4 to 0.6 strong · 0.7 to 0.9 very strong | strong |
| Medicine (Chan 2003, cited in Akoglu 2018, Table 1) | 0.1 to 0.2 poor · 0.3 to 0.5 fair · 0.6 to 0.7 moderate · 0.8 to 0.9 very strong | moderate |
| Medicine, anesthesiology (Schober et al. 2018, table) | 0.00 to 0.10 negligible · 0.10 to 0.39 weak · 0.40 to 0.69 moderate · 0.70 to 0.89 strong · 0.90 to 1.00 very strong | moderate |

*My own compilation from the sources named, with the labels as the sources use them. The three classifications in Akoglu are given in steps of 0.1, and r = 0.61 is assigned to the 0.6 step there.*

These classifications are traditions of the disciplines, not laws. The scale from my course slides is the one customary in marketing. Cohen explicitly calls his levels conventions, meant for the case in which no better ones suggest themselves (Cohen 1988, p. 79). Gignac and Szodorai recommend clearly lower values for differential psychology and derive them from published correlations: about a quarter of the correlations they analyzed were below 0.10, about half below 0.20 and about three quarters below 0.30 (2016, p. 74). Schober et al. call such cutoff points arbitrary and inconsistent (2018, section "Interpretation of the Correlation Coefficient").

The example shows the consequence: r = 0.61 is called "moderate" in marketing and in medicine, "strong" in political science and "large" by Cohen.

So report the number and not just a label: "r = 0.61" is unambiguous, "strong relationship" is not. If you use a label, name the convention it comes from.

## Is a significant correlation also a strong one?

No. Strength and significance answer two different questions. The absolute value of r tells you how close the relationship is in your sample. The p-value tells you how likely, under the assumptions of the test, a correlation at least this large in absolute value would be if there were no linear relationship in the population (t-test for r, Malhotra and Birks 2007, p. 578). More on this under [p-value](/blog/statistical-significance/#p-value) and [significance and practical importance](/blog/statistical-significance/#practical-significance).

Whether a correlation becomes significant depends heavily on the number of respondents. If there is no linear relationship in the population, chance can still produce a moderate r when there are only a few responses. With many responses, such chance values mostly lie close to zero. That's why, with many respondents, even a small r is enough to become significant:

| Respondents (n) | Smallest absolute value of r with p < 0.05 (two-sided, to three decimal places) |
|------|------|
| 30 | 0.362 |
| 100 | 0.197 |
| 400 | 0.099 |
| 1,000 | 0.062 |

*My own calculation, two-sided t-test for r based on Malhotra and Birks 2007, p. 578.*

So with 1,000 respondents, even r = 0.062 becomes significant. With an r this small, one answer hardly helps you estimate the other, at least not along the lines of "the more of one, the more of the other." Significant then means only this: an r this high would be rare if there were no linear relationship at all in the population. How strong the linear relationship is in your sample is still shown by r.

How precisely the strength is determined is shown by the [confidence interval](/blog/statistical-significance/#confidence-interval). In the example, the 95% confidence interval runs from 0.39 to 0.76 (p < 0.001): a clear relationship whose strength is only roughly determined with 48 responses. With r = 0.40 and 30 respondents, it runs from 0.05 to 0.66 (p = 0.03), that is, from almost zero to above the value of the example.

**How to calculate it yourself.** All you need is r and the number of respondents n. The usual method takes a detour: r is converted to a transformed value (Fisher's z), the interval limits are calculated on that scale, and the limits are then converted back (Bortz 2005, p. 194, cited in Schumann 2019, p. 237). The detour has a simple reason: r can't go beyond +1. With a high r there is therefore less room upward than downward, and the interval is not symmetric. In the example, the lower limit is farther from 0.61 than the upper one. Spreadsheet programs have the functions FISHER and FISHERINV for the two conversions:

> ℹ️
> 
> lower limit: `=FISHERINV(FISHER(r) - 1.96 / SQRT(n - 3))`
> 
> upper limit: `=FISHERINV(FISHER(r) + 1.96 / SQRT(n - 3))`
> 
> Example: r = 0.609, n = 48, that is from 0.39 to 0.76

*Invented data, my own calculation. The number 1.96 belongs to the 95% level, "n − 3" to this method.*

**Many pairs, many chance hits.** A questionnaire with 30 questions has 435 pairs. If there were no relationship between any of these questions, about 22 pairs would still be significant on average at α = 0.05 (my own calculation). So decide beforehand which relationships you want to test, and report everything else as exploration, see [Many tests, many false positives](/blog/statistical-significance/#multiple-testing). To generalize from the respondents to the population, this test, too, requires a random sample, see [online survey with an open link](/blog/statistical-significance/#random-sample).

## Why is correlation not causation?

Because the same relationship can come about in several ways. As my course slides explain, if the data don't come from a controlled experiment, you can only claim that a relationship between the variables exists, not its causal direction ([Marx 2026, slide 209](/principles-of-survey-research/chapter-5/slide-209.html)).

For r = 0.61 between satisfaction with support and likelihood to recommend, at least four explanations are possible:

| Explanation | In the example | How to check it |
|------|------|------|
| X affects Y | Good support makes customers willing to recommend. | An experiment, or the order in time and the elimination of the other explanations |
| Y affects X (reverse direction) | Those who value and recommend the product anyway also rate the support more leniently. | Order in time, repeated surveys of the same people |
| A third variable affects both | If the problem was resolved right away, satisfaction and recommendation rise together. | Collect the third variable and control for it |
| Selection of respondents | Of 200 people contacted, 48 respond, mainly the very satisfied and the very annoyed, and the middle is missing. | Check the selection procedure and the response rate, see Sampling methods |

*My own summary: possible explanations, not findings. Direction and third variables based on Schumann 2019, pp. 119-123.*

### The third variable in the example

Now the third question comes into play: was the issue resolved at first contact? The two groups differ clearly in both ratings:

|  | resolved (26 respondents) | not resolved (22 respondents) |
|------|------|------|
| Satisfaction with support, mean (1 to 5) | 3.9 | 2.6 |
| Likelihood to recommend, mean (0 to 10) | 7.6 | 3.8 |
| Correlation of satisfaction and likelihood to recommend within the group | 0.34 | 0.32 |

*Invented data, my own calculation.*

These are the two clouds of dots from the scatter plot. A large part of the correlation of 0.61 arises because the "resolved" group scores higher on both questions. Within the groups, the relationship is only about half as strong and, with 26 and 22 respondents, not significant on its own (p = 0.09 and p = 0.14). This separate analysis is the simplest way to control for a third variable (Schumann 2019, p. 123). It requires that you suspected the third variable and collected it, see [Third variables](/blog/cross-tabulation/#third-variables) in the article on cross-tabulation.

The **partial correlation** sums up the control in one number. It gives the correlation of two variables after the linear relationship of both with a third has been removed, and for that it needs only the three simple correlations (Malhotra and Birks 2007, pp. 578-579):

> **Example.** The three correlations 0.609, 0.555 and 0.801 give a partial correlation of 0.33 for satisfaction and likelihood to recommend once "resolved" has been removed.
> 
> ```math
> \begin{aligned} \text{satisfaction and likelihood to recommend} &: r = 0.609 \\ \text{satisfaction and “resolved” (yes = 1)} &: r = 0.555 \\ \text{likelihood to recommend and “resolved”} &: r = 0.801 \\[6pt] \text{partial correlation} &= \frac{0.609 - 0.555 \cdot 0.801}{\sqrt{(1 - 0.555^2)(1 - 0.801^2)}} = 0.33 \end{aligned}
> ```
> 
> *Invented data, my own calculation. Formula from Malhotra and Birks 2007, p. 579.*

Controlling for first-contact resolution, the correlation drops from 0.61 to 0.33 (p = 0.02). That is about as much as within the two groups (0.34 and 0.32). There, the relationship was not significant because each group on its own has only 26 or 22 respondents, and with so few responses chance, too, can produce an r of this size. The partial correlation still compares only within the groups, but it uses all 48 responses at once. With 48 responses, a relationship of this strength is significant. It is thus clearly smaller than 0.61, but it remains detectable (on interpreting such findings, Schumann 2019, p. 236).

What role the third variable plays is not decided by the calculation. If it affects both characteristics, Schumann speaks of a spurious correlation. If it lies as an intermediate link on the path from one characteristic to the other, he speaks of an intervention. In the data, the two cases look the same (Schumann 2019, pp. 120-123).

In the example, time helps to some extent: the issue was resolved or not before the respondents gave their ratings. That makes it unlikely that satisfaction influenced first-contact resolution. What remains open is whether first-contact resolution raises both ratings directly or whether it acts on the likelihood to recommend through satisfaction: first the resolution, then the satisfaction, then the recommendation. A one-time survey can't separate these readings. The answer "resolved" also comes from the same questionnaire and can be colored by the overall impression. The data support a conjecture that can be tested: does the likelihood to recommend improve when more issues are resolved at first contact? A trial answers that, not this survey.

A third variable can also hide a relationship or reverse it, see [Simpson's paradox](/blog/cross-tabulation/#simpson). Schumann's rule to remember, translated from the German: "Never trust a bivariate correlation!" (2019, p. 127).

### What conditions must be met to infer causation?

The literature names three conditions (Malhotra and Birks 2007, pp. 303-305; Hyman, cited in Schumann 2019, p. 127):

1. **Association.** Cause and effect vary together.
2. **Order in time.** The cause comes before the effect or occurs at the same time.
3. **Elimination of other explanations.** The relationship persists when you control for third variables that precede both in time.

The three conditions are necessary but not sufficient: even together they don't prove causation, which can only be supported more or less well (Malhotra and Birks 2007, p. 303). With a single survey, usually only the first can be checked. Anyone who interprets such a relationship causally is assuming that the direction is known (Schumann 2019, p. 127).

The statement gets stronger with an experiment: random assignment to the groups is meant to control for third variables (Schumann 2019, p. 120, fn. 35). In the example, a randomly selected part of the requests would go through a new resolution process. The order in time can be clarified by repeated surveys of the same people (Schumann 2019, p. 127). Why even questions about actions that were carried out don't prove their effect is explained in the article [Employee survey](/blog/employee-survey/#actions).

## What does a simple linear regression tell you?

Regression analysis examines associative relationships between a metric dependent variable and one or more independent variables (Malhotra and Birks 2007, p. 581; [Marx 2026, slide 213](/principles-of-survey-research/chapter-5/slide-213.html)). Simple linear regression fits a straight line through the cloud of dots so that the sum of the squared vertical distances of the dots from the line is as small as possible (Schumann 2019, p. 223, fn. 7). Unlike with correlation, you have to decide which variable is to be predicted. In the example, it's the likelihood to recommend:

```math
\begin{aligned} \text{in general: predicted value} &= a + b \cdot x \qquad (a \text{ intercept, } b \text{ slope}) \\ \text{example: predicted likelihood to recommend} &= 1.58 + 1.28 \cdot \text{satisfaction} \\ \text{coefficient of determination } R^2 &= 0.37 \qquad n = 48 \qquad p < 0.001 \\ \text{95\% confidence interval of the slope} &: 0.8 \text{ to } 1.8 \end{aligned}
```

*Invented data, my own calculation. Equation based on Schumann 2019, pp. 221-223.*

Regression treats both questions as metric data, so the distances between the scale points count as equal. The arguments for and against this are in the article [Levels of measurement](/blog/levels-of-measurement/#likert). The means per scale point in the table above rise by 1.1, 1.1, 1.9 and 0.5 points: the rise is not perfectly even, and it says nothing about the distances of the scale itself. Here is how to read the output:

**Slope (b = 1.28).** Respondents who rate the support one scale point better report, on average, a likelihood to recommend that is 1.28 points higher. This is a difference between different people at one point in time. What happens when one person's satisfaction changes is not something the equation observes: "dependent" and "independent" variable are terms from the mathematical relationship and don't imply causal dependence (Malhotra and Birks 2007, p. 581). The slope carries the units of your questions, here points from 0 to 10 per scale point from 1 to 5. With other scales the number changes, whereas R² does not (Schumann 2019, p. 228).

**Intercept (a = 1.58).** It is the predicted value at a satisfaction of 0, which doesn't exist on the scale from 1 to 5. It can only be interpreted if x = 0 is a meaningful value (Schumann 2019, p. 223, fn. 8). In general, the line holds only where there are data (Schumann 2019, p. 225).

**Coefficient of determination (R² = 0.37).** It is the share of the differences in likelihood to recommend that the line reproduces, and with a single independent variable it is the square of r (Schumann 2019, pp. 227-229). The usual wording is "explains 37% of the variance." Schumann clarifies: "explained" only means that the value was predicted on the basis of the regression equation. Whether there is a cause-and-effect relationship remains open (2019, p. 226, fn. 14).

**Residuals.** The residual is the distance between the actual answer and the predicted value (Schumann 2019, p. 226). For "somewhat satisfied" (4), the equation predicts 6.7. Someone who answered 10 there lies 3.3 points above it, and for 14 of the 48 respondents the answer deviates by more than two points: the line describes the average, not the individual person. Plotted against the independent variable, the residuals should scatter around zero without a pattern. An arc as in data set B shows that the straight line doesn't fit (Malhotra and Birks 2007, p. 597).

## Multiple regression: what does "holding the other variables constant" mean?

Multiple regression takes several independent variables into the equation at the same time. Each coefficient then states by how much the predicted values differ between respondents who are one unit apart on this variable and equal on the other variables in the equation (Malhotra and Birks 2007, pp. 592-593; Schumann 2019, p. 236).

In the example, first-contact resolution is added as a second variable (yes = 1, no = 0):

```math
\begin{aligned} \text{predicted likelihood to recommend} &= 2.45 + 0.50 \cdot \text{satisfaction} + 3.18 \cdot \text{first-contact resolution} \\ R^2 &= 0.68 \qquad n = 48 \end{aligned}
```

*Invented data, my own calculation.*

The slope of satisfaction falls from 1.28 to 0.50. Among respondents whose issues had the same resolution status at first contact, those whose satisfaction rating is one scale point higher thus report a likelihood-to-recommend score that is only half a point higher on average. The test of this coefficient checks the same thing as the partial correlation above and gives the same p-value (p = 0.02). At the same satisfaction rating, respondents whose issue was resolved report a likelihood-to-recommend score that is 3.18 points higher on average. "Holding the other variables constant" refers only to the variables in the equation. What you haven't collected remains uncontrolled.

**Key driver analysis.** In customer and employee surveys, multiple regression is often used as a key driver analysis: an overall measure such as likelihood to recommend is predicted from several individual ratings, and the rating with the coefficient that is largest in absolute value counts as the "strongest driver." What is compared here are standardized coefficients: they are converted so that the scales of the questions no longer play a role, similar to r. Two limits apply. The individual ratings are related to one another, and the more strongly they are, the less precise the coefficients become. If respondents almost always rate two topics similarly, the calculation can hardly tell which of the two the relationship should be attributed to: size and sign can change from sample to sample, and there is then no unambiguous measure of importance (Malhotra and Birks 2007, pp. 601-602). And the analysis is still cross-sectional: a "driver" is associated with the outcome when the other measured variables are held constant. This does not show that changing the driver changes the outcome.

## Correlation and regression with QUESTIONSTAR

In QUESTIONSTAR, you calculate correlations and regressions right in the reports: correlations by Pearson (r), Spearman (rho) and Kendall (tau) as well as regression analyses, each with the significance level reported ([Features](/features/)). All of this is included in the free plan as well.

- **Correlation:** for a single pair of questions or as a matrix of several questions. The report also shows the distributions of the answers as a chart.
- **Regression:** You can specify several independent variables and in this way include a suspected third variable in the equation. The report shows R² and the adjusted R². The adjusted R² helps when you compare equations with different numbers of variables. The reason: with the same respondents, R² itself never goes down when a variable is added, even if it contributes hardly anything.
- **Opt-out answers:** You decide in a setting whether "can't judge" and similar answers enter the calculation. Exclude them for correlation and regression: their numeric code is not a point on the scale, see [Levels of measurement](/blog/levels-of-measurement/#worked-example-what-an-incorrectly-coded-opt-out-answer-does).
- **Filters:** Filters apply in these reports as well. This is how you calculate a relationship separately for each group, as above for "resolved" and "not resolved."

The path from your responses to the result, in five steps:

1. **Choose the questions.** Pick the two questions whose relationship you want to check, if possible before the analysis.
2. **Exclude opt-out answers.** Set "can't judge" and similar answers so that they don't enter the calculation.
3. **Calculate the correlation.** In the report, calculate the correlation for this pair, for single rating questions by Pearson and by Spearman. Look at the distributions of the two answers as well.
4. **Check for a third variable.** If you suspect a third variable, calculate the relationship again with a filter for each group, or include the third variable as a further independent variable in a regression.
5. **Report.** Take the coefficient, its type and the significance level from the report. Count in the exported raw data how many respondents have a usable answer to both questions. Calculate the confidence interval [as shown above](#significance). A model sentence is under [How to report a relationship](#reporting).

An AI assistant that is connected to QUESTIONSTAR via the [MCP interface](/help/integrations/mcp-ai/) can start both calculations on your survey data. The calculation runs on QUESTIONSTAR's server.

- **Correlation:** pairwise for the questions you name, by Pearson, Spearman or Kendall. For each pair you get the coefficient, the two-sided p-value and the number of response pairs, because missing answers are excluded pair by pair. With Pearson, you also get the Fisher z value, from which the confidence interval can be determined.
- **Regression:** linear, with one dependent and one or more independent variables. As in the report, a third variable can be included this way as a further independent variable. You get R² and the adjusted R² and, for each coefficient, the estimate, standard error, t-value and p-value. Respondents who are missing an answer on one of the variables are left out. If the independent variables are very closely related to one another (collinearity), a warning points this out.

Free-text answers don't enter these calculations, and choice questions are calculated with their numeric code: check that a higher code really means "more" and that opt-out answers such as "can't judge" don't enter the calculation as a number. If you let the assistant search for the strongest relationships, have it keep count of how many pairs it calculated, and treat the findings as exploration. The same goes for a large matrix in the report: there, too, some of the many pairs stand out only by chance.

Questions you've marked as personal data are not passed to the assistant when the corresponding setting is switched on. Answers without that mark, especially free-text answers, can still contain personal data. So check beforehand which questions the assistant gets to see.

QUESTIONSTAR doesn't calculate a factor analysis. For that, for other methods and for your own charts, export the raw data as an SPSS, CSV or XLSX file.

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## How to report a relationship

A complete sentence about a correlation names the two characteristics, the coefficient and its type, the number of cases, the uncertainty and the kind of sample:

> "Satisfaction with support and likelihood to recommend are clearly related (Pearson's r = 0.61, 95% confidence interval 0.39 to 0.76, n = 48: 48 of the 200 randomly selected customers with a support request responded). After accounting for whether the issue was resolved at first contact, the relationship is weaker (partial correlation 0.33). This survey does not establish whether higher satisfaction with support increases the likelihood to recommend."

| Instead of | Better | Why |
|------|------|------|
| "Good support leads to more recommendations." | "Those who rate the support better report a higher likelihood to recommend on average." | What has been shown is a relationship, not an effect. |
| "A one-point increase in satisfaction produces a 1.28-point increase in likelihood to recommend." | "Respondents whose satisfaction rating is one scale point higher report a likelihood-to-recommend score that is 1.28 points higher on average." | This compares different people, not a change within one person. |
| "Leadership is the strongest driver of intention to stay." | "With ratings of the other topics held constant, the leadership rating has the strongest association with intention to stay." | The ranking holds only for the topics surveyed and is unstable when they are correlated. |
| "There is no relationship between age and satisfaction (r = 0.03)." | "No linear relationship between age and satisfaction is apparent (r = 0.03, n = 48)." | r captures only straight-line relationships, and the estimate is imprecise at n = 48. |

## Checklist: analyzing correlation and regression properly

- Scatter plot checked for a curve, conspicuous single dots and separate groups
- Coefficient chosen to match the level of measurement, Pearson and Spearman compared for single rating questions
- r reported with the number of cases and, if possible, with a confidence interval, unplanned findings marked as exploration
- Obvious third variables thought through in advance, collected and controlled for, reverse direction considered
- Regression: slope described as a difference between respondents, residuals inspected, independent variables checked for correlations with one another
- No statement about an effect without an established order in time and the elimination of other explanations

## Frequently asked questions about correlation, regression and causation

### What is the difference between correlation and causation?

Correlation means that two characteristics vary together. Causation means that one is the cause of the other. A causal conclusion requires an association, the right order in time and the elimination of other explanations, all three together (Malhotra and Birks 2007, p. 303). A single survey can usually check only the association (Schumann 2019, p. 127).

### How do I interpret the correlation coefficient?

The correlation coefficient r lies between −1 and +1. The sign shows the direction, the absolute value the strength of the linear relationship (Schumann 2019, p. 229). When a relationship is called "strong" is settled by each discipline according to its own tradition: by Cohen's conventions, 0.10 is a small, 0.30 a medium and 0.50 a large effect (Cohen 1988, pp. 79-80), while in marketing a relationship counts as high only from 0.7 ([Marx 2026, slide 212](/principles-of-survey-research/chapter-5/slide-212.html)). So report the number and not just a label.

### When do I use Pearson, and when Spearman?

Pearson requires metric data and measures straight-line relationships. Spearman works with ranks and fits ordinal data, such as single rating questions (Malhotra and Birks 2007, pp. 580-581). For rating questions, calculate both and compare.

### What is a spurious correlation?

A spurious correlation exists when two characteristics are related without one influencing the other, because both depend on a third variable (Schumann 2019, p. 120). If you control for this third variable, the relationship gets smaller or disappears.

### What does R² tell you in a regression?

The coefficient of determination R² is the share of the variance of the dependent variable that can be predicted from the independent variables. It lies between 0 and 1 (Malhotra and Birks 2007, p. 582). "Explained" here means predicted, not caused (Schumann 2019, p. 226, fn. 14).

### Can I prove causes with a regression on survey data?

As a rule, no. Regression describes the nature and strength of a relationship and doesn't assume causation (Malhotra and Birks 2007, p. 581). With several variables, it controls only for the variables you collected. For statements about effects, the design has to establish the order in time and rule out other explanations, ideally through an experiment.

## Sources

Akoglu, H. (2018). User's guide to correlation coefficients. Turkish Journal of Emergency Medicine, 18(3), 91-93. Table 1 (classifications by Dancey and Reidy, Quinnipiac University and Chan).

Anscombe, F. J. (1973). Graphs in Statistical Analysis. The American Statistician, 27(1), 17-21. p. 19 (four data sets with the same summary statistics).

Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates. pp. 79-80 (r as an effect size, conventions small, medium, large).

Gignac, G. E. and Szodorai, E. T. (2016). Effect size guidelines for individual differences researchers. Personality and Individual Differences, 102, 74-78. p. 74 (abstract: 0.10, 0.20 and 0.30 as relatively small, typical and relatively large).

Malhotra, N. K. and Birks, D. F. (2007). Marketing Research. An Applied Approach (3rd European edition). Harlow: Pearson Education. pp. 303-305 (concept of causality and conditions), p. 352 (number of response categories and correlation), pp. 575-578 (product moment correlation, significance test), pp. 578-581 (partial correlation, nonmetric correlation, regression analysis), pp. 582-585 (simple regression, scatter diagram), pp. 592-593 (multiple regression), p. 597 (residuals), pp. 601-602 (multicollinearity, relative importance).

Marx, P. (2026). Principles of Survey Research. Online course, chapter 5, slides 209 to 213. QUESTIONSTAR. [Slide 209](/principles-of-survey-research/chapter-5/slide-209.html), [slide 210](/principles-of-survey-research/chapter-5/slide-210.html), [slide 211](/principles-of-survey-research/chapter-5/slide-211.html), [slide 212](/principles-of-survey-research/chapter-5/slide-212.html), [slide 213](/principles-of-survey-research/chapter-5/slide-213.html).

Schober, P., Boer, C. and Schwarte, L. A. (2018). Correlation Coefficients: Appropriate Use and Interpretation. Anesthesia & Analgesia, 126(5), 1763-1768. Section "Interpretation of the Correlation Coefficient" with table (example of a common classification).

Schumann, S. (2019). Repräsentative Umfrage. Praxisorientierte Einführung in empirische Methoden und statistische Analyseverfahren [Representative surveys: a practical introduction to empirical methods and statistical analysis] (7th ed.). Berlin/Boston: De Gruyter Oldenbourg. In German. pp. 119-127 (experiment and survey, third variables, spurious correlation, intervention, conditions for causal statements), pp. 220-231 (simple regression, coefficient of determination, correlation coefficient), pp. 231-236 (partial correlation), pp. 236-239 (multiple regression, confidence intervals, further correlation coefficients).

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May the relationships you find stand up to a second look.

**Related articles:**

- [P-value and statistical significance, explained simply](/blog/statistical-significance/)
- [Cross-tabulation: how to read and test a contingency table](/blog/cross-tabulation/)
- [Levels of measurement: nominal, ordinal, interval and ratio](/blog/levels-of-measurement/)

*Dr. Paul Marx*

Author: Dr. Paul Marx
Dr. Paul Marx is the founder and Managing Director of QUESTIONSTAR and was a professor of marketing at the University of Siegen.

## Related articles

- [statistical-significance](/blog/statistical-significance)
- [cross-tabulation](/blog/cross-tabulation)
- [levels-of-measurement](/blog/levels-of-measurement)

*Author: Dr. Paul Marx — see [About](/about).*