# Units for log reaction times

**Posted:**November 28, 2010

**Filed under:**data analysis, mathematics Leave a comment

In psychological experiments, it is common to measure reaction time in the response of a subject (an experimental participant). In inferential statistics, reaction times are typically log-transformed because raw reaction times are skewed to the high. This is because there is a lower physical limit to how fast participants can respond, but not an upper one. Many statistical tests assume a normal distribution of the dependent variable, and thus, reaction times are log-transformed to reduce skew.

What happens to the units when we take the logarithm of a reaction time? A reaction time is measured in some unit of time, [T], e.g. seconds. But we cannot take the logarithm of a dimensioned quantity! The logarithm is defined as the inverse of exponentiation:

where .

Dimensional homogeneity must be preserved under equality; that is, the units of must be the same as the units of and the units of must be the same as the units of . Thus, must all be unitless: in particular, **the logarithmic function does not admit a dimensioned quantity as an argument, and a log-transformed quantity is unitless**. Note that dimensional analysis is essentially type checking, where the types are physical units.

So when we log transform a reaction time, e.g. , what we actually mean is which we can also write as . **Since the logarithmic function admits only unitless arguments, we must take the logarithm of a ratio of reaction times**. In physical systems, there is often a natural standard reference value to take the ratio to, as for pressure (standard atmospheric pressure), but there isn’t such a natural standard that I know of for reaction times. So one can take the ratio with respect to a unit quantity in the units the reaction time was measured in, as shown above.

Thus, in labeling a plot or table of log-transformed reaction times, it is incorrect to write *log RT (s)*. Instead, one should write * log (RT/s)* or * log (RT/[s])* or maybe *log RT (RT in s)*. We still want to know what units the raw reaction times were measured in, since they scale the log-transformed values!

For an expanded discussion of these topics, see Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions by Chérif F. Matta and Lou Massa and Anna V. Gubskaya and Eva Knoll, to appear in the *Journal of Chemical Education*.