Error in thermal time constant from measurement

I would like to know how to calculate the error in thermal time constant ($tau$) based on temperature measurements. For example, lets say I have a plate with a thermocouple and I provide it with a step input. The step power input will cause a $1$st order temperature change at the location of the thermocouple.

The temperature at time $t$ is
$$
T(t) = T_s + (T_i – T_s) e^{-t/tau}.
$$
where $T_s$ the static-state (equilibrium) temperature, $T_i$ the initial temperature.

At $t=tau$, we have $$T(tau) = T_s + frac{T_i – T_s}{e}$$
I can find the temperature at time $tau$ and from the temperature vs time measurement data I can determine the time constant $tau$ ($T_i$ and $T_s$ are both not $0 C/K$).

If my thermocouple has a fixed error of lets say $pm 1 C$, then what would be the error in the derived time constant, $tau$?

Thanks for your help in advance!