The edit distance is a basic string similarity measure used in many applications such as text mining, signal processing, bioinformatics, and so on. However, the computational cost can be a problem when we repeat many distance calculations as seen in real-life searching situations. A promising solution to cope with the problem is to approximate the edit distance by another distance with a lower computational cost. There are, indeed, many distances have been proposed for approximating the edit distance. However, their approximation accuracies are evaluated only theoretically: many of them are evaluated only with big-oh (asymptotic) notations, and without experimental analysis. Therefore, it is beneficial to know their actual performance in real applications. In this study we compared existing six approximation distances in two approaches: (i) we refined their theoretical approximation accuracy by calculating up to the constant coefficients, and (ii) we conducted some experiments, in one artificial and two real-life data sets, to reveal under which situations they perform best. As a result we obtained the following results: [Batu 2006] is the best theoretically and [Andoni 2010] experimentally. Theoretical considerations show that [Batu 2006] is the best if the string length n is large enough (n >= 300). [Andoni 2010] is experimentally the best for most data sets and theoretically the second best. [Bar-Yossef 2004], [Charikar 2006] and [Sokolov 2007], despite their middle-level theoretical performance, are experimentally as good as [Andoni 2010] for pairs of strings with large alphabet size.