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It may be well-known that when we take statistics, we essentially need a large number of samples. Because I am taught this fact before studying the math, I have been here without exploring the reason. What is the intuition behind larger number of samples are better for statistics?

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Intuitively we can imagine statistics as trying to see an image using only finitely many pixels/samples. If you have more pixels/samples and they're well-distributed, then the image is more clear.

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In statistics, there are two concepts, population and sample. We say, for instance, that a random phenomenon can be described using a distribution. For instance, the grades students achieve at school have Normal distribution. This is satisfied for other random natures. There is a point, whenever you study a phenomenon, you have a limited size of data. This data, sample, is taken from a larger group, population. if you pick each data independently and all the data that you've picked are in their routine condition, these are called iid condition which stands for independent and identical distribution, your data which is going to be called sample will be like your population. This means the statistics of the population, expected value and other descriptive things, can roughly be found using a bunch of formulas. This means by studying some samples of the population, you can find out how the entire population works. This is clear that if you have more data which is more representative, you can have better approximations. In my answer, I've avoided referring to formulas for finding population statistics out of sample statistics, but if you study them, you will see that but increasing the size of sample, usually noted by $n$, you can have better approximations to describe the entire population by just having a very small subsets of that.

I refer to my first words about distribution. Depending on the phenomenon you are attempting to deal with, it has a distribution which that specifies the appropriate formulas to go from samples to population.

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What is the intuition behind larger number of samples are better for statistics?

This is because the larger the sample, the most likely it is to be a faithful representation of the full population. Formally this is a consequence of the law of large numbers:

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

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  • $\begingroup$ What about the rest of the distribution, not just the mean? $\endgroup$
    – Dave
    Jul 7, 2020 at 2:25
  • $\begingroup$ @Dave I'm not sure what you mean, I'd suggest you ask a new question rather than having a discussion in the comments of an old answer. $\endgroup$
    – Erwan
    Jul 7, 2020 at 12:04
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To exemplify: let's say you have a coin, made by yourself, hence you know it is fair and returns heads and tails with same 1/2 probability (frequency).

You wish to check that by flipping the coin n number of times. If you flip it one time, n=1, you will get a single heads or tails observation (sample) and cannot differentiate it from a biased coin. For a very large number of flips (large n = large sample size), you are very likely to see similar number of heads and tails. This allows you to convincingly show people that the coin is fair.

The idea is that reading the fairness out of a sample becomes more and more reliable as the sample size increases (provided flips/observations are not biased).

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