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Suppose there are 10k people in the population. You want to find out how many of them brush their teeth. You take 1k of them entirely randomly. They all say "yes". But the rest 9k say "no".

So everyone in the sample is brushing their teeth, and everyone not in the sample is not brushing their teeth.

How is this sample representative of the entire population?

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    $\begingroup$ That sample isn't, but the probability of randomly drawing all 1,000 brushers without getting a single one of the 9,000 non-brusher is so small you could repeat this experiment every second and likely not see that happen before all protons in the known universe have decayed. It is very, very, very unlikely to get this as a 'random' sample. $\endgroup$
    – PBulls
    Commented Feb 27 at 18:53
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    $\begingroup$ And... how likely is this to occur? Not every sample will represent the population well, but with a sample of 1,000 people for a situation like this, it's almost certain that it won't be far off, e.g., if 70% of the people in the population brush their teeth, it's extremely probable that the sample fraction will lie between 650 and 750. $\endgroup$
    – jbowman
    Commented Feb 27 at 18:57
  • $\begingroup$ @jbowman does there exist a distribution for this? $\endgroup$
    – curiousCprogrammer1231
    Commented Feb 27 at 19:10
  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Commented Feb 27 at 19:53
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    $\begingroup$ Yes, it's called the Hypergeometric distribution. $\endgroup$
    – jbowman
    Commented Feb 27 at 20:22

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In the context of a random sample, the term "random" refers to the method used to select individuals or elements from a population for inclusion in the sample. Specifically, it means that every member of the population has an equal chance of being selected, and the selection process is not influenced by any systematic biases or preferences.

In your example, say the yes to no ratio in the population is $1:9$. If you select a random sample from this population it should contain approximately a $1:9$ ratio of yeses and nos.

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  • $\begingroup$ "If you select a random sample from this population it should contain approximately a 1:9 ratio of yeses and nos." Why? $\endgroup$
    – curiousCprogrammer1231
    Commented Feb 27 at 19:20
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    $\begingroup$ By definition, a random sample represents the population it was taken. So the characteristics (distribution, composition, etc) of the population should manifest through the lenses of the sample. $\endgroup$
    – Happy Cretine
    Commented Feb 27 at 19:29
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    $\begingroup$ @curiousCprogrammer1231 please read for example the Wikipedia entry on simple random sample. It also links to an answer to a question in one of your comments directly on your question: random sampling from a finite population (say, of 10000) without replacement (that is, anyone can only be selected at most once) when there are two exclusive outcomes (brush/doesn't) can be modeled by a hypergeometric distribution. $\endgroup$
    – EdM
    Commented Feb 27 at 19:52
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    $\begingroup$ There are several laws of large numbers that justify assertions about the representativeness of a random sample. Sure, a random sample--like any sample--could have properties departing appreciably from those of its parent distribution. But--uniquely for random sampling--we can compute the chances of those departures. The sample you posit has a chance of approximately $10^{-3246}$ of being selected randomly, a chance that is so incredibly small it can be entirely neglected. $\endgroup$
    – whuber ♦
    Commented Feb 27 at 20:15
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    $\begingroup$ @PBulls My apology: I misinterpreted a natural logarithm as a common logarithm. Dividing 3246 by the log of 10 produces your (correct) result. And yes, I used lchoose. Alternatives are lfactorial and lgamma. $\endgroup$
    – whuber ♦
    Commented Feb 28 at 17:45

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