Thermodynamic Temperature

Hello, new to these forums

I'm trying to complete a basic thermodynamics problem, calculating the thermodynamic temperature of a gas immersed in water at triple point.

The problem begins with a table of pressures at varying temperatures of a gas under pressure P, and pressure P(tp) - triple point pressure.



Now, I know the following equation;


So I have calculated the average value of P/P_TP as 1.53555

But I also have access to a solution to this problem, and in the solution an extra step is also taken:



I don't understand the source of the 2.5 * avg change between P and P(tp). Where has this come from? Why is it necessary?

Thanks