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Suppose a dataset with observation values:

2
3
4
4
2

Variance of this observation, assuming this is the entire population, would be: enter image description here

Where the average is: 3. So the variance equals: 0.8.

Now, i read around that if I multiply the observation values by 5, the variance should increase by 25. But that doesn't seem to be the case, am I interpreting something erroneously?

10
15
20
20
10

The average is: 15, so the variance equals: 20.

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The variance increases by a factor of 25 (multiplication), it does not increase by 25 (addition). All of your calculations are correct. In sample 1, variance is 0.8 and in sample 2 variance is 20, which is 25 times larger than 0.8, i.e. 20=25*0.8.

In general, multiplying all observations of a random variable $X$ by a constant $c$ scales the variance up by $c^2$. Let $V(X)$ denote the variance operator. $$V(cX)=c^2V(X).$$ To see this more easily, note that the mean of your new sample is $c\mu$. Replace $X_i$ in your formula with $cX_i$ and $\mu$ with $c\mu$: $$\sum_{i=1}^N \frac{(cX_i-c\mu)^2}{N}=\sum_{i=1}^N \frac{c^2(X_i-\mu)^2}{N} =c^2 \sum_{i=1}^N \frac{(X_i-\mu)^2}{N} = c^2V(X).$$

In your example, $c=5$.

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