# Calculating "annualized" standard deviation from monthly returns and the different month lengths

I have the monthly returns and want to estimate an "annualized" standard deviation.

An industry-standard way seems to be the following:

$$\sigma_a = \sqrt{12} \sigma_m,$$

where $$\sigma_m$$ is "monthly" standard deviation calculated from the monthly returns $$r_i$$:

$$\sigma_m = \sqrt{\frac{1}{n - 1} \sum_{i=1}^n (r_i - \bar{r})^2}.$$

My question: the formulae above do not take into account the different lengths of the calendar months. It's clear that 1% returns in January and in February are a bit different effective returns.

Is there a commonly used way that takes into account the number of days per month?