[![enter image description here](https://i.stack.imgur.com/QCFy5.png)](https://i.stack.imgur.com/QCFy5.png)

This sample follows a Normal Distribution with Mean $= 280 / 20 = 14$, and Variance $= (3977.57 / 20) - 14^2 = 2.88$. To find the unbiased variance, we can divide it by $19$ to get $3.03$.

However, in the following question: 

[![enter image description here](https://i.stack.imgur.com/bVRgb.png)](https://i.stack.imgur.com/bVRgb.png)

[![enter image description here](https://i.stack.imgur.com/3a9QB.png)](https://i.stack.imgur.com/3a9QB.png)

I used $\displaystyle\int\frac{3x^3 + 2x^2}{10} \, dx$ and $\displaystyle\int \frac{3x^4 + 2x^4}{10}\,dx - \text{mean}^2$ to get the mean and variance that the sample follows. The answer gives the variance as $\displaystyle\int \frac{3x^4 + 2x^4}{10}\, dx - \text{mean}^2$ divided by $n$.

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> I understand that this is the definition given by the Central Limit Theorem. What I don't understand is why we don't divide by $n$ in the first example if we do in this case. Is it because we only divide by $n$ when using the central limit theorem, and the first example had only 20 samples taken, meaning that the central limit theorem could not be used? In this way, does it depend on the size of $n$?
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Any help will be greatly appreciated, thanks in advance.