    Chapter 6: Analysing the Data Part III: Common Statistical Tests # Critical values

From Pearson's table (reproduced here), the column headed "Level of significance for two-tailed test" gives the critical values for Pearson's r that are needed to be surpassed to achieve significance. Using the .05 level and travelling down the column we can see that as the sample size gets larger, the size of the correlation that is needed to achieve significance gets smaller. For a sample of size 5 (df = 3), the critical value is .878 (or -.878). So the previous example of -.65 is not significant at the .05 level for a sample of size 5. We would therefore report the result as r(3) = -.65, p > .05.

If we move to a sample of size 10 (df = 8) we find that a correlation of .632 or greater (or -.632 or less) is significant at the .05 level. The critical value for the .02 level is .716, so our example value is significant at the .05 level but not significant at the .02 level. We would report the result as r(8) = -.65, p < .05.

For samples of size 20 (df = 18), the critical value is .444 for the .05 level of significance, .516 for the .02 level, and .561 for the .01 level. The value we were examining of -.65 can therefore be considered significant at the .01 level for samples of size 20. We would report this as r(18) = -.65, p < .01. For samples of size 100, a correlation of .65 is highly significant.

For the example on logical reasoning and creativity a correlation of .736 was obtained. Looking at the line for df = 18 we would report that r(18) = .736, p < .01.

These days we usually do not have to refer to statistical tables to determine the significance of a test-statistic, the computer will produce the significance level with the value of the test statistic itself. The SPSS printout in Figure 6.5 reports that the "Sig. (2-tailed)" value is .000 and puts two asterisks next to the r value of .736. This is just SPSS's way of reporting the p-value. Here, ".000" does not actually mean zero. This is just a rounding problem. If it was actually zero, that would mean that the probability of an r-value of this size or greater occurring is impossible. However, we were working with real numbers and obviously the result is possible because we just did it! Of course the actual p-value is probably something like .000376, but when this is rounded to nearest 3 decimal places, it comes out as .000.

From the SPSS output which is more accurate than the tables, we would report r(18) = .736, p < .001

Output 6.1 Correlate -> Bivariate . . . Output 6.5 SPSS output of the correlation between logical reasoning and creativity.

SPSS uses one asterisk to indicate that p < .05 and two asterisks to imply that p < .01. SPSS never uses more than two asterisks, however, you will occasionally read published articles that use three and even four asterisks to indicate higher levels of significance. (Three asterisks usually indicates p < .001 and four indicates p < .0001).

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