# What happens with my variance if I multiply observations by 5?

Suppose a dataset with observation values:

2
3
4
4
2


Variance of this observation, assuming this is the entire population, would be: 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.

## 1 Answer

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, $$c=5$$.