# SPSS Tutorials

BASICS REGRESSION T-TEST ANOVA CORRELATION

# SPSS Correlation Analyis Tutorial

## Correlation Test - What Is It?

A (Pearson) correlation is a number between -1 and +1 that indicates to what extent 2 metric variables are linearly related. It's best understood by looking at some scatterplots.

In short,

• a correlation of -1 indicates a perfect linear descending relation: higher scores on one variable imply lower scores on the other variable.
• a correlation of 0 means there's no linear relation between 2 variables whatsoever. However, there may be a (strong) non-linear relation nevertheless.
• a correlation of 1 indicates a perfect ascending linear relation: higher scores on one variable are associated with higher scores on the other variable.

## Null Hypothesis

A correlation test (usually) tests the null hypothesis that the population correlation is zero. Data often contain just a sample from a (much) larger population: I surveyed 100 customers (sample) but I'm really interested in all my 100,000 customers (population). Sample outcomes typically differ somewhat from population outcomes. So finding a non zero correlation in my sample does not prove that 2 variables are correlated in my entire population; if the population correlation is really zero, I may easily find a small correlation in my sample. However, finding a strong correlation in this case is very unlikely and suggests that my population correlation wasn't zero after all.

## Correlation Test - Assumptions

Computing and interpreting correlation coefficients themselves does not require any assumptions. However, the statistical significance-test for correlations assumes

• independent observations;
• normality: our 2 variables must follow a bivariate normal distribution in our population. This assumption is not needed for sample sizes of N = 25 or more.For reasonable sample sizes, the central limit theorem ensures that the sampling distribution will be normal.

## SPSS - Quick Data Check

Let's run some correlation tests in SPSS now. We'll use adolescents.sav, a data file which holds psychological test data on 128 children between 12 and 14 years old. Part of its variable view is shown below.

Now, before running any correlations, let's first make sure our data are plausible in the first place. Since all 5 variables are metric, we'll quickly inspect their histograms by running the syntax below.

*Quick data check: histograms over all relevant variables.

frequencies iq to wellb
/format notable
/histogram.

## Histogram Output

Our histograms tell us a lot: our variables have between 5 and 10 missing values. Their means are close to 100 with standard deviations around 15 -which is good because that's how these tests have been calibrated. One thing bothers me, though, and it's shown below.

It seems like somebody scored zero on some tests -which is not plausible at all. If we ignore this, our correlations will be severely biased. Let's sort our cases, see what's going on and set some missing values before proceeding.

*Inspect case with low iq / anxi scores.

sort cases by iq.

*One case has zero on both tests. Set as missing value before proceeding.

missing values iq anxi (0).

If we now rerun our histograms, we'll see that all distributions look plausible. Only now should we proceed to running the actual correlations.

## Running a Correlation Test in SPSS

Let's first navigate to Analyze Correlate Bivariate as shown below.

Move all relevant variables into the variables box. You probably don't want to change anything else here.

Clicking results in the syntax below. Let's run it.

## SPSS CORRELATIONS Syntax

CORRELATIONS
/VARIABLES=iq depr anxi soci wellb
/PRINT=TWOTAIL NOSIG
/MISSING=PAIRWISE.

*Shorter version, creates exact same output.

correlations iq to wellb
/print nosig.

## Correlation Output

By default, SPSS always creates a full correlation matrix. Each correlation appears twice: above and below the main diagonal. The correlations on the main diagonal are the correlations between each variable and itself -which is why they are all 1 and not interesting at all. The 10 correlations below the diagonal are what we need. As a rule of thumb, a correlation is statistically significant if its “Sig. (2-tailed)” < 0.05. Now let's take a close look at our results: the strongest correlation is between depression and overall well being : r = -0.801. It's based on N = 117 children and its 2-tailed significance, p = 0.000. This means there's a 0.000 probability of finding this sample correlation -or a larger one- if the actual population correlation is zero.

Note that IQ does not correlate with anything. Its strongest correlation is 0.152 with anxiety but p = 0.11 so it's not statistically significantly different from zero. That is, there's an 0.11 chance of finding it if the population correlation is zero. This correlation is too small to reject the null hypothesis.

Like so, our 10 correlations indicate to which extent each pair of variables are linearly related. Finally, note that each correlation is computed on a slightly different N -ranging from 111 to 117. This is because SPSS uses pairwise deletion of missing values by default for correlations.

## Scatterplots

Strictly, we should inspect all scatterplots among our variables as well. After all, variables that don't correlate could still be related in some non-linear fashion. But for more than 5 or 6 variables, the number of possible scatterplots explodes so we often skip inspecting them. However, see SPSS - Create All Scatterplots Tool.
The syntax below creates just one scatterplot, just to get an idea of what our relation looks like. The result doesn't show anything unexpected, though.

*Simple scatterplot for wellbeing by depression.

graph
/scatter wellb with depr
/subtitle "Correlation = - 0.8 | N = 128".

## Reporting a Correlation Test

The figure below shows the most basic format recommended by the APA for reporting correlations. Importantly, make sure the table indicates which correlations are statistically significant at p < 0.05 and perhaps p < 0.01. Also see SPSS Correlations in APA Format.

If possible, report the confidence intervals for your correlations as well. Oddly, SPSS doesn't include those. However, see SPSS Confidence Intervals for Correlations Tool.

# Let me know what you think!

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# This tutorial has 37 comments

• ### By Ruben Geert van den Berg on December 5th, 2016

Hi James, thanks for your comment! Which way to go really depends on your research questions and the structure of your data. I can't tell without looking into those in detail.

P.s. there's some new tutorials coming up regarding alternative correlation measures such as CramÃ©r's V, Kendall's Tau, Spearman and others. Follow us on facebook for staying up to date with our latest articles.

• ### By JAMES MUNGAI MUCHIRI on December 5th, 2016

i have staff and student data regarding how they rate a particular course is it possible to do some correlation test here.

• ### By J. Walsh on November 19th, 2016

Enlightening! Thank you very much!

• ### By rebecca on October 31st, 2016

great i got to understand it better with your explanation