As the sample size gets larger, what happens to the size of the correlation that is needed for significance?

Gets smaller

As the sample size gets larger, the size of the correlation needed for significance typically decreases. This is due to the effect of sample size on statistical power and the accuracy of the estimation.

To understand this concept, we should first clarify that the "size of the correlation needed for significance" refers to the strength of the relationship between two variables that is considered statistically significant. It is measured using a statistical test, such as the Pearson correlation coefficient.

When testing the significance of a correlation, researchers use a hypothesis test to determine if the observed correlation in the sample is statistically different from zero (no correlation). The result of this test provides a p-value, which indicates the probability of obtaining the observed correlation or a more extreme one, assuming that there is no true correlation in the population.

Now let's discuss how the sample size influences this process:

1. Statistical power: The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false. With larger sample sizes, the statistical power increases. This means that larger samples are more likely to detect true correlations, even if they are relatively small.

2. Precision of estimation: Larger sample sizes provide more reliable estimates of population parameters, such as the correlation coefficient. With more data, the estimate becomes more precise, reducing the uncertainty around the true correlation. Consequently, smaller correlations can be considered statistically significant because they are estimated with higher precision.

In summary, as the sample size increases, the statistical power improves and the precision of estimation increases. Consequently, the threshold for determining the size of the correlation needed for significance becomes lower, allowing smaller correlations to be considered statistically significant. However, it is important to note that the exact relationship between sample size and the size of the correlation needed for significance can also be influenced by other factors, such as the desired level of significance (alpha) and the specific statistical test used.