MODULE 05

Correlation

Pearson's r summarises two things about a linear association between two variables: its direction and its strength, on a scale from −1 to +1.

Research question
Variables
Variable roles
Measurement levels (trained here)
Method choice (trained here)
Statistical logic (trained here)
SPSS workflow (trained here)
Output interpretation (trained here)
Defensible conclusion

Understand

Correlation playground

Change the trend and the noise. Notice that r depends on how tightly the points hug a line, not on how steep the line is: a steep trend with lots of noise can have a smaller r than a gentle trend with little noise.

Pearson's r in this sample

r = .69

positive strong

−10+1

Strength of the trend1.00

Noise (scatter around the trend)1.00

Jump to a population correlation of

r gives you

the direction of a linear association (sign)
the strength of a linear association (|r| from 0 to 1)

r does not give you

causality (correlation ≠ causation)
the slope in original units
a complete description of nonlinear relationships

Try

Guess the correlation

Training your eye makes output easier to sanity-check.

Round 1. Look at the cloud and estimate r.

Your guess.00

Understand

When r = 0 but the relationship is obvious

Pearson's r only looks for straight-line patterns. Always look at the scatterplot.

Pearson's r

r = .05

Working hours rise from young adulthood, peak in middle age and fall towards retirement. The relationship is obvious, yet r is close to zero because the rising and falling halves cancel out.

r = 0 means no linear association. It does not necessarily mean no association.

Understand

Pearson or Spearman?

Spearman's rank correlation is a rank-based measure of monotonic association: it replaces values with their ranks and correlates the ranks.

1Are both variables at least ordinal (not purely nominal)?

Answer the questions to get a recommendation.

Pearson r

.82

Spearman ρ

.98

Y always increases with X, but not along a straight line. Spearman works on ranks, so it only asks whether higher X goes with higher Y: it captures this monotonic pattern better than Pearson.

Advanced note

In large social-science samples, mild departures from normality may be less consequential than in very small samples. Outliers and clearly curved patterns matter more than small deviations from a bell shape.

SPSS

Correlation in SPSS

Before you open SPSS

1What is the research question?Is education associated with trust in parliament?
2What are X and Y?Xyears of education Ytrust in parliament (0–10)
3Measurement levels?Both treated as scale.
4Compare or associate?Associate: two scale variables, no groups.
5Which method?Pearson (scatterplot roughly linear); Spearman as a robustness check.

Analyze›Correlate›Bivariate

Bivariate Correlations

age

gender

polint

happy

hours

1Variables:

eduyrs

trustprl

PearsonKendall's tau-bSpearmanTwo-tailedFlag significant correlations

PasteResetCancelOK

1
Variables:
Move both variables here. The order does not matter: correlation is symmetric.
Pearson
Tick Pearson, Spearman, or both, depending on the decision above.
Two-tailed
Test of significance: two-tailed unless you had a directional hypothesis in advance.
Click Paste to save the syntax, or OK to run.

Correlations
		Years of education	Trust in parliament
Years of education	Pearson Correlation	1	.31**
	Sig. (2-tailed)		.000
	N	1412	1412
Trust in parliament	Pearson Correlation	.31**	1
	Sig. (2-tailed)	.000
	N	1412	1412

1Correlation coefficient: sign = direction, size = strength.
2Sig. (2-tailed): the p-value for H₀: no correlation in the population. .000 means p < .001.
3N: cases with valid values on both variables (pairwise).

** Correlation is significant at the 0.01 level (2-tailed). The table is symmetric: read one half.

Read this output

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Interpret SPSS output

What is the direction of the relationship?

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Test yourself

Mastery check

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Interpret SPSS output

Daily hours of TV viewing and trust in other people (0–10). How do you read this output?

Correlations
		TV hours	Social trust
TV hours	Pearson Correlation	1	-.42
	Sig. (2-tailed)		<.001
	N	850	850

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