Associate uncertainities to variance's estimate of a distribution with heavy tails

I have a set of $$M$$ variables $$\{ \lambda_N^{(i)} \}_{i=0}^{M(N)}$$. The distribution of $$\lambda$$ depends on the $$N$$ parameter; for $$N \rightarrow \infty$$ the distribution shrinks around the mean value, but for finite $$N$$ the distribution is characterized by the presence of outliers which determine a fat tails decays. The presence of this outliers gradually (and slowly) disappear increasing $$N$$. I want to verify if the shrinking of my distribution follow the theoretical prediction. So, starting from a set $$\{ \lambda_N^{(i)} \}_{i=0}^{M(N)}$$ I look for an estimate of the variance, which is the variable I want to study, and a confident interval for it.
So I have to repeat it for different values of the $$N$$ parameter (say $$n$$ times) and perform a fit with the $$n$$ couples $$(N, Var_{\lambda_N})$$.
My doubt is how to associate the uncertainties with my variables $$Var_{\lambda_N}$$
starting from the set $$\{ \lambda_N^{(i)} \}_{i=0}^{M(N)}$$ I perform the 2 following transformation:

• $$\lambda_N \rightarrow w_N \equiv \lambda_N- \overline{\lambda_N}$$, with $$\overline{\lambda_N} = \frac{1}{M(N)} \sum_{i=0}^{M(N)} \lambda_N^{(i)}$$
• $$w_N \rightarrow z_N \equiv w_N^2$$
With the distribution of $$z_N$$ I estimated $$Var_{\lambda_N} = \overline{z_N}$$ and the standard deviation of $$Var_{\lambda_N}$$ with the standard deviation of $$z_N$$.
The problem with this approach is that, due to the outlier's presence the standard deviation's estimate is too large.
My second attempt was to use bootstrap method:
Starting from the set of $$M(N)$$ values I perform $$n_r$$ resampling (with repetition), each one with $$n_s$$ element.
The problem with this approach is that the estimation of the confident interval (always for the presence of outliers) is strongly dependent on the choice of $$n_s$$
In fact, using a big $$n_s$$'s value, I increase the probability to find at least one outlier in a generic resampled set, which deeply increase the variance estimation of that set.
So, with big $$n_s$$ I have a small variability between different resampled set (small confident interval), while decreasing $$n_s$$ I reduce the probability to extract an outlier, increasing the variability between different resampled set (bigger confident interval).
I don't want that my confidence interval's estimate depend on the choice of $$n_s$$.
Is there any reasonable reasonable basis to determine the right choice of $$n_s$$?
Should I use a different method in situations like that (jack knife or others)?