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Why small effects need big samples

That’s a question that periodically comes up in class. Suppose someone is planning a study. As demanded by her teacher, she computes the needed sample size upfront. So the question arises: Given some to-be-achieved level of power (80%), some effect size, and some other details: How large does my sample need to be?

Some students are puzzled by the fact that small effects need larges samples. Why is that?

To get a grip, suppose you are farmer in the Normandie, en France, bien sur. Your are in the truffle business (o la la). You own several large land fields where your truffle pigs search these lucrative amuse-guelles.

You own a handsome herd of such truffle pigs. Beautiful and intelligent animals.

Now, one morning you are looking to a certain piece of land, asking yourself: “How many truffle pigs should I admission to this piece of land?”.

What’s the answer?

The answer of course depends on (among other things) on the size of the truffle you expect to find. If this truffle is gargantuan, large as giant pumpkin, one truffle pig might suffice for this job. However, is the truffle is small as a pie, then you will need a large herd, and even that might now yield a high probability of finding the treasure.

The number of truffle pigs in this tale corresponds to the sample size, \(n\). The size of the truffle corresponds to the effect size.

As can easily be seen: Large truffle demands small pig herds, and vice versa. In other words: Large effects – small \(n\) suffices; in contrast small effects demand large \(n\).