Two reasons we can be assured that a normal distribution is a reasonable assumption for many variables are:

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Tchebychev's Inequality Theorem states that for any distribution, regardless of its shape, at least 75% of the data falls within two standard deviations of the mean. This gives us reason to believe that many variables may be reasonably normal. The Central Limit Theorem states that for large enough sample sizes, the sampling distribution of the mean will be approximately normal, even if the population distribution is not normal. This also gives us reason to believe that a normal distribution may be a reasonable assumption for many variables.

Two Reasons We Can Be Assured That a Normal Distribution