# Different formulas of empirical variance

I came across two formulas of empirical variance. My professor used this one : $$s^2=\sum_{i=0}^k n_i(x_i-m)^2$$ with $$m$$ being the mean of the series. But it didn't make much sense to me and when I looked on the internet I found this more common formula: $$s^2=\sum_{i=0}^n(x_i-m)^2$$ Now I'm confused as to what formula to use and what's the difference between the two.

• What are $k$ and $n_i$? Commented Dec 27, 2020 at 10:54
• @RichardHardy $n_i$ is the number of $x_i$ sample and there are no information on k but for sure it's not the size of the whole sample. Commented Dec 27, 2020 at 11:21

Of course, $$k$$ will be the number of groups (repeated measures) and $$n_i$$ the number of repeated measures from that observation.

For example, consider the following vector $$X = (1,2,2,3,3,3)$$. The sum of squares by the second formula will be $$s ^ 2 = \sum_ {i = 6} (x_i- \mu) = 1.778 + 0.111 + 0.111+ 0.444 + 0.444 + 0.444 = 3.333$$ Do you notice the repeated measures in the formula? The first formula follows from the grouping of these measures. $$s ^ 2 = \sum_ {k} n_i(x_i- \mu) = 1.778 + 2 \times 0.111 + 3 \times 0.444 = 3.333$$

The second formula is the general one, where $$x_i$$’s are the raw data, e.g. every for every $$i$$-th person you record their age and there’s $$n$$ people in your sample.

You can use the first formula for aggregated data, where $$x_i$$’s are the district age categories and $$k_i$$’s are their counts. For example, you observed $$n_i=137$$ people of $$x_i=26$$ years old in your sample. In this case, there’s $$n$$ distinct age categories and the total number of observed samples is $$\sum_{i=1}^k n_i$$.

Other case for first formula is when $$n$$ is the number of samples and $$k_i$$ are sample weights, where you want to have some samples have more impact on the result than others.

Notice that variance is an average squared deviation from the mean, so in case of both formulas you need to divide by the total sample size, i.e. by $$n$$ in second case, or by $$\sum_{i=1}^k n_i$$ in second case. For the sample variance, you would divide by $$n-1$$.

• It would be worth noting that neither formula is for the variance: they are sums of squares.
– whuber
Commented Dec 27, 2020 at 15:32
• @whuber good point, I missed it. Fixed.
– Tim
Commented Dec 27, 2020 at 16:23