The following variance estimator of a set of data points $x = (x_1, ..., x_N)$ $$ var(x) = \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x}) $$ has itself a large variance when $N$ is small (in my case $N<10$). Looking around for better estimators, I found the median absolute deviation (MAD) that seems to be similar. However, comparing the var values to the MAD values, the latter are smaller. So, they don't really correspond to a variance estimate.

Are there other variance estimators that are robust against small sample sizes?

The following variance estimator of a set of data points $x = (x_1, ..., x_N)$ $$ \text{Var}\,(x) = \frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2 $$ has itself a large variance when $N$ is small (in my case $N<10$). Looking around for better estimators, I found the median absolute deviation (MAD) that seems to be similar. However, comparing the var values to the MAD values, the latter are smaller. So, they don't really correspond to a variance estimate.

Are there other variance estimators that are robust against small sample sizes?