Understanding PMCC Hypothesis Testing

PMCC yields us the number for the correlation, but it doesn’t tell us if the result is statistically significant. Could it be a mere coincidence? That’s where hypothesis testing helps us.

First, we set up our hypotheses. The null hypothesis, H₀, states there is no linear correlation — r equals zero. The alternative hypothesis, H₁, suggests that a correlation does exist. One common confusion is whether to use a one-tailed or two-tailed test. If you only care about a positive or negative direction — for example, checking whether more revision increases marks — that’s one-tailed. But if any correlation is of interest, regardless of direction, that’s a two-tailed test.

After that, we derive the test statistic using the conversion formula for ‘r’ to ‘t,’ which is influenced by the sample size. You do not have to worry about the formula; the concept is quite straightforward: the higher the ‘r’ value, the greater the sample size, which makes it easier to identify a significant relation between the two factors in a correlation. Taking an example where ‘r’ is 0.6 for fifteen children in a study would identify if the relation is significant using the ‘t’ value obtained.

In teaching, I personally resort to exemplifying using shoe size versus height. Although `r` appears rather high, samples can’t be significant with small values for `n`. 

Once we have the t-value, we compare it with the critical value from the t-distribution, using n – 2 degrees of freedom. If the absolute t-value exceeds the critical value, we reject the null hypothesis — meaning the correlation is likely real. Otherwise, we fail to reject the null, which tells us the evidence isn’t strong enough to trust the correlation.

Creating Clear Conclusions

“The correlation is significant,” is a poor formulation likely to appear in student work, where it is corrected to “There is strong evidence of a positive correlation between hours studied and exam scores.” Of course, the difference is simply the addition of a word, but it is an addition that gives your answer a tone of assurance and humanity by connecting it to its specific, contextual setting.

Visualising Correlations

Scatter plots are the most useful tool for understanding PMCC in operation. Upwardly sloping points signify positive correlations. Downward-sloping points signify negative correlations. When the points appear to be scattered randomly, it means that there is little to no linear correlation. You can also plot a line of best fit in an attempt to recognise any trends more easily. I recommend that students plot one even before they calculate the value for “r”.

Here’s a little exercise: take five friends’ revision hours and their exam scores. Plot the points. Can you estimate r before doing any calculation? Practicing this builds a natural intuition for correlation, which is far more helpful than memorising formulas alone.