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Why would increasing the sample size decrease the confidence interval width?

Wouldn't increasing sample size lead to an increase in the variability (and thus the standard deviation) of the data?

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  • $\begingroup$ Increasing sample size leads to more information about the unknown quantity, hence a narrower confidence interval for it. That is to say your interval estimate is getting more accurate. $\endgroup$ – StubbornAtom Nov 28 '19 at 11:24
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Consider the extreme case:

If the sample size is close to the entire population size, how much variability would there be for your point estimate?

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It is important to understand the difference between the variability from observation to observation, and the variability of the mean of observations. They are not the same.

To illustrate, suppose I have a fair six sided die. I roll the die ten times, and I might get the sample $$\{1, 3, 1, 6, 4, 3, 2, 5, 4, 5 \}.$$ If I roll the die $100$ times, I can continue to get any number between $1$ and $6$ inclusive. It doesn't matter how many times I roll the die; these numbers will occur with approximately equal frequency.

But now, suppose I decide to take the arithmetic mean of the ten rolls. I get $\bar x = 3.4$. If I roll the die another ten times, I might get $\bar x = 3.7$, and so forth. The variability of this mean value I am calculating depends on the number of observations in the sample--the sample size, and this is the variability that is used when computing the confidence interval. We call this the standard error of the sample mean (and in practice, abbreviated somewhat imprecisely as "standard error" or "SE").

Why is this the case? Well, back to the die example, ask yourself what it would take for $\bar x = 1$ for a sample of size $n = 10$. For the sample mean to equal $1$, every single observation would need to be $1$. If any die roll is not $1$, the mean would exceed $1$. So clearly, this has a probability of only $1/6^{10} \approx 1.65 \times 10^{-8}$ of occurring. But for a value like $\bar x = 3$, there are many more ways for the sample to give you such a mean. This fact is related to what is sometimes referred to as the "law of large numbers."

So, when we are calculating an interval estimate for a parameter, what we are doing is in essence constructing a statistic from a sampling distribution, rather than the distribution for the observations from which the sample is taken. The standard deviation of the sampling distribution of the sample mean is the standard error of the sample mean.

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