Variance estimation for small sample size 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?

If we assume that the underlying distribution is normal, than MAD and variance are related as
$$
\hat{\sigma}=k\text{MAD}, k\approx 1.4826
$$
Mean absolute deviation (MAD) is defined as the median of the absolute deviation from the sample median. If the underlying distribution is normal, then the median is an estimator for mean and MAD, scaled as above, can serve as an estimate for the scale (i.e., \sigma$). However, unlike sample mean and sample variance, median and MAD are robust estimators of location and scale (respectively), that is they are less distorted by the outliers that might contaminate the data.
See, e.g., Robust Statistics: Theory and Methods
