Rate and order of a reaction
The rate of reaction is defined in IUPAC's Gold Book as:

$$r = - \frac{1}{a} \frac{\mathrm d[\ce{A}]}{\mathrm dt} = - \frac{1}{b} \frac{\mathrm d[\ce{B}]}{\mathrm dt} = \frac{1}{p} \frac{\mathrm d[\ce{P}]}{\mathrm dt} = \frac{1}{q} \frac{\mathrm d[\ce{Q}]}{\mathrm dt}$$

where $[\ce{X}]$ denotes the concentration of the substance $\ce{X}$, and the lower case letters represent the coefficients of those substances in the balanced equation. The rates for $\ce{P}$ and $\ce{Q}$ are positive, as these are products in the reaction.

You are right in that you can determine the reaction rate by studying the change in concentration of a particular reactant (or product) over time. But, to determine the rate r for the reaction, not just for a particular species, you must take into consideration the coefficients in the balanced equation. The stoichiometric numbers are included in the definition of the rate, $r$, so that it is independent of which reaction species you study. For example if $a = 1$ and $b = 3$ then $\ce{B}$ is consumed three times more rapidly than $\ce{A}$, but $r = -\mathrm d[\ce{A}]/\mathrm dt = -(1/3)\mathrm d[\ce{B}]/\mathrm dt$ is uniquely defined.

Source: https://en.wikipedia.org/wiki/Reaction_rate