In molecular biology, Tm is defined as the temperature at which 50% of dsDNA is converted to its single stranded form. Intuitively it would seem that the melting temperature should have been defined as the temperature at which the entirety(100%) of the dsDNA converts into ssDNA. What is the basis of using this ratio, i.e, 50% in the definition of Tm?
If you search for DNA melting temperature plots you will see they have a sigmoid shape. It's common for those kinds of plots to be described by the 50% value for example the therapeutic index and the 50% lethal dose of a substance (LD50). This is because at really low and really high temperatures all DNA molecules become dsDNA or ssDNA respectively so that information is not really useful. You could maybe talk about the first temperature at which DNA becomes 100% denatured but the curves are highly nonlinear at that region so it's difficult to quantify. The 50% is an easy way to compare different molecules since that region is usually linear and easy to compare in a meaningful way (basically all molecules have the same ends of the sigmoid shape, but it's the middle part that changes the most).
The rason for defining the Tm the way we do is because there is a Boltzmann distribution of energies of all DNA molecules in a sample (the DNA molecules have different energies due to thermal fluctuations). Thus, even at -10 degrees Celsius, some of the DNA will be ssDNA as a very small proportion of the population of DNA molecules will have a high enough thermal energy to be single stranded. Likewise, at higher temperatures, some of the DNA will be dsDNA as it has very little thermal energy. There is thus no single temperature at which all DNA molecules melt, so instead we use the definition of 50% melted as most DNA molecules have a similar thermal energy and so most melt at around the same temperature, giving a sharp phase transition, but not all DNA molecules.