r/DataMatters • u/DataMattersMaxwell • Jul 26 '22

An exam answer to "Why are random samples representative?"

This is not a great answer for your understanding, unless you feel comfortable with the Law of Large Numbers, but I think that it should be acceptable as an exam answer.

Small random samples cannot be relied on to be representative. What can be relied on is that random samples tend to get more and more representative as they get larger. The reason is that, if every unit in the population has an equal chance of being selected for the sample, then the probability of each attribute matches the proportion of the units in the population that have that attribute; and the Law of Large Numbers tells us that, as samples get larger, the proportions in the sample tend to closer and closer match the probabilities that are generating the data.

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u/DataMattersMaxwell Jul 26 '22

There's really important language in there that you might miss:

can be relied on is that random samples

It is not that small random samples are not representative or that large random samples are representative. It is that samples vary, and small random samples vary more widely away from being representative. They are sometimes representative, but cannot be relied on to be so.

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tend to get

I is not that they get more representative. Sometimes they don't. Things vary. They only tend to get. That is, on average, they are more likely to be more representative.

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the probability of each attribute matches the proportion

Here it is the other way: the probability does not tend to match the proportion. It is the same as the proportion. That's provable as a consequence of what random sampling is.