1.5 Chance and variation in the response

Natural (chance) variation refers to variation that cannot otherwise be explained: even repeating a study exactly the same way every time will not always produce the same values of the response variable. This is called natural variation, chance variation, or just chance.

Natural variation makes the influence of the explanatory variable (which we are wanting to study) hard to detect, so minimising chance variation is important. Minimising the amount of the chance variation, requires using good design principles, and measuring as many other extraneous variables that may explain variation in the response variable as is reasonable.

Chance can impact the values of the response variable in different ways: each individual can produce different values of the response variable each time the individual repeats the study (within-individuals variation); each individual in the study can produce different values of the response variable compared to other individuals (between-individuals variation):

To estimate the amount of variation within individuals: Many observations are needed from each unit of analysis (individual).
To estimate amount of variation between individuals: Many units of analysis (individuals) are needed.

Since between-individual variation is usually more variable than the within-individual variation, using many individuals is usually more important than using a smaller number of individuals many times.

Lemma 1.2 (Chance) Consider the letter-typing study (Example here) again. What are the advantages and disadvantage of:

measuring one female 30 times?
measuring 30 different females once each?
measuring 10 different females three times each?