According to the central limit theorem, as sample size grows large, the distribution of the sum (or average) tends to:

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According to the central limit theorem, as sample size grows large, the distribution of the sum (or average) tends to:

The central idea here is that combining many independent, identically distributed random contributions tends to produce a normal shape, no matter what the original distribution looks like, as long as the mean and variance are finite. When you add up a large number of such variables, their fluctuations tend to cancel out in a way that smooths the overall distribution into a bell curve. Specifically, the sum of n observations has mean nμ and variance nσ², while the average has mean μ and variance σ²/n. As n grows, the distribution of the average (and of the standardized sum) approaches a normal distribution. That’s why you see normal behavior emerge in many real-world samples, even if individual observations come from a skewed or irregular distribution. So the best answer is that the distribution tends to a normal distribution, regardless of the original shape. It isn’t generally true that the sum or average preserves the original distribution, nor that it becomes uniform or remains skewed.

The central idea here is that combining many independent, identically distributed random contributions tends to produce a normal shape, no matter what the original distribution looks like, as long as the mean and variance are finite. When you add up a large number of such variables, their fluctuations tend to cancel out in a way that smooths the overall distribution into a bell curve.

Specifically, the sum of n observations has mean nμ and variance nσ², while the average has mean μ and variance σ²/n. As n grows, the distribution of the average (and of the standardized sum) approaches a normal distribution. That’s why you see normal behavior emerge in many real-world samples, even if individual observations come from a skewed or irregular distribution.

So the best answer is that the distribution tends to a normal distribution, regardless of the original shape. It isn’t generally true that the sum or average preserves the original distribution, nor that it becomes uniform or remains skewed.

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