Relating the Different Types of Intervals and Error Bars

Relating the different types of intervals

The standard error of the mean (SEM) is the basis for intervals that are relating to the means. Similarly, the standard error of the difference (SED) is the basis when the focus is on the differences between means. Therefore, if we know how the SED relates to the SEM, we know how the different intervals relate to each other, too. For example, we can mentally sketch approximate least significant differences (LSDs) onto graphs that show the means and SEMs (or the means and the CIs). In the plots we’ve looked at previously, 95% CIs were set as the wider intervals (drawn with thin lines) and the LSDs as the narrower intervals. Let’s imagine that we’ve only been presented with the 95% CIs. Because we know how the SEM and SED are related, we can estimate the approximate LSDs ourselves. The three panels of the figure below show three scenarios:

1. We know the means are clearly not significantly different ($p > 0.05$).
2. The means are significantly different ($p ≈ 0.01$).
3. The means are different at about $p ≈ 0.05$.

General principles

The general principles presented in the figure below work because at a level of significance where $p < 0.05$ (and with large sample sizes, the SED is approximately $1.4$ SEMs, which makes the LSD intervals approximately $2.8$ SEMs wide. Because the LSD interval is centered on each mean, the upper and lower halves of the LSD bars are approximately $1.4$ SEMs long. In other words, for an eyeball test, the bounds of an LSD interval lie just under midway between the ends of a conventional error bar (which shows $±1$ SEM) and the ends of a 95% CI (which shows $±2$ SEM). Knowing how one interval relates to another enables us to perform these eyeball tests using the following general principle.