In order to quantify those qualitative and graphical reasonnings, we need a way of measuring the width of those "bell-shaped curves": we could take the variance of each sample but that would give us four numbers, one for each as we prefer a single number; so we take the mean of those variances -- if the three samples actually come from the same population, the mean of the variances is indeed an estimator of the variance of the whole population. We also need a measure of the dispersion of the means: we could take the variance of the means, but in order to compare it with the mean of the variances, we multiply it by the size of the samples -- we get another estimator of the variance of the whole population.