I need to calculate the standard deviation on the entire population. The population size is 7, but for others it may be between 5-14. I see a few variance algorithms listed on Wikipedia. What stdev algorithms should I use for this population size?
With a small amount of data, you should have no problems with any of the algorithms unless the variance is small and the values are large. In that case, calculate the mean first and then subtract it from the values, since a second pass costs you little.
Much more importantly to avoid a common error, since you have the whole population you should always divide by the population size $n$ rather than $n-1$ (using $n-1$ might make sense if you only had a sample, could not find the population mean, and wanted an unbiased estimate of the population variance).
Then, as you say, take the square root of the variance to get the standard deviation.