Standard deviation is a measure that can be calculated on any set of data irregardless of its actual distribution. It is simply a measure of value dispersion in relation to the data set's mean.

Any normality assumption to which you are referring usually is only a concern when doing statistical inference. For instance, if you need to test whether a sample standard deviation is 'large' or if two sample standard deviations are the same, then the underlying distribution of the data is important.

So, if all you need for your analysis is to do is compute the standard deviation, then the standard deviation formula you have been using is sufficient.

Standard deviation is a measure that can be calculated on any set of data regardless of its actual distribution. It is simply a measure of value dispersion in relation to the data set's mean.

Any normality assumption to which you are referring usually is only a concern when doing statistical inference. For instance, if you need to test whether a sample standard deviation is 'large' or if two sample standard deviations are the same, then the underlying distribution of the data is important.

So, if all you need for your analysis is to do is compute the standard deviation, then the standard deviation formula you have been using is sufficient.