Choosing the Right Mean Procedure

Most errors in inference for means happen before any calculation starts. The formulas for one-sample, paired, and two-sample t procedures look similar, but they answer different questions, and picking the wrong one gives you a confident, precise, wrong answer. So the real skill is reading the design of the study, not crunching numbers.

Start by asking how many measurements each individual contributes. If every individual gives one measurement, compared against a fixed number, use a one-sample t procedure. If individuals are measured twice, or matched into pairs that share something important, reduce each pair to a single difference and run a one-sample t procedure on those differences: this is the paired t procedure. If two separate, unrelated groups are each measured once and compared to each other, use a two-sample t procedure, which never combines individuals across groups into a single value.

This sits right after you've learned the three t procedures individually; now the job is matching design to method without being told which one to use. It also connects back to why t replaces z at all: whenever the population standard deviation sigma is unknown and you estimate it with the sample standard deviation s, you must use t, no matter how large n is. Large samples make t and z nearly identical in value, but that convenience does not change which one is technically correct. The same logic that tells you t beats z also tells you paired beats two-sample when pairing exists: use the structure that matches how the data actually arose.