I have $N$ samples for the time a (same) person runs a distance $d$, this was measured by hand (i.e. the time has an error). And the person does not perform equally on each measurement.
What would be the conceptual difference of $\bar{v}$ and $\hat{v}$:
$$\bar{v} = \frac{1}{N}\sum_{i=1}^N \frac{d}{t_i}$$
versus doing:
$$ \bar{t} =\frac{1}{N}\sum_{i=1}^N {t_i} \rightarrow \hat{v} = \frac{d}{\bar{t}}$$
I was asked for the average speed of the person to run a distance $d$, and for that they asked me for $N$ samples. However it is not clear to me what should be the metric I am supposed to use. I guess the first one is the expected value of the velocity of the person, and the later the average velocity of the $N$ experiments. I am really confused on which one I am supposed to use.
Let's plug in some numbers. Let $d=10$ meters. If you have two samples, one of $1$ sec and one of $2$ sec, the runner did $10$ m/s once and $5$ m/s once. If you average the speeds you get $7.5$ m/s. If you average the times and get $1.5$ sec, the speed will be $6\frac 23$ m/s. Which do you want? If the times are close it won't matter.