No matter what the original shape is, the sample mean becomes bell-shaped
The distributions on the left can be very different (uniform, exponential, or extreme). If you take several numbers (a sample) from one of them, calculate the average, and repeat that hundreds of times, what shape will you get? No matter what the original distribution looks like, the distribution of the sample mean eventually becomes bell-shaped (normally distributed).
Why does this happen? An average has the effect of having “multiple random values cancel out one another’s skew.” When values that are far to one side are mixed with values that lean the other way, the resulting average tends to cluster near the middle. The larger the sample size (n), the stronger this effect becomes.
Original distribution (population)
Distribution of sample means
Sample size (n, how many values to take for each average)5
💡 Thanks to this theorem, even if we do not know what distribution the actual data follows, we can use the normal distribution for statistical calculations involving the “sample mean” (such as confidence intervals and hypothesis tests). It is one of the fundamental theorems supporting statistics as a whole.