Is it valid to divide data by a constant to make the estimated beta larger / more interpretable? In analyzing my data, I find a similar problem to the one described in interpreting-this-regression-coefficient. I'm using regression analysis to build a human energy expenditure estimation model using a motion sensor. Depending on the intensity of the motion, the output of the sensor may be up to 1,000 to 2,000 counts per minute, and my data show an average of 900-1,100 counts per minute. I used these data to develop a model, with energy expenditure as the dependent variable and motion sensor output as the independent variable. The result was that the model was statistically significant ($p<0.01$), but the unstandardized coefficient for my independent variable was shown as $0.000$. I have tried dividing the output by 100 and the coefficient become $0.021$. I wonder if what I did is statistically / mathematically valid? If so, is there a reference (academic paper, etc.) that establishes this? 
 A: Whatever software you used was evidently reporting coefficients to 3 d.p. So 0.000 just meant <0.0005. 
It makes perfect sense to use units of measurement that yield coefficients that aren't inconveniently large or small. No statistical principle is violated thereby. You don't need a reference or authority to back this up: it is fine to choose (e.g.) mm or m or km or miles or feet or inches for lengths depending on a problem and which units are familiar in your field. 
In your case, how about dividing by 60 to get counts per second? Do people in your field ever use that as a unit? 
A: One way to look at it is this.  If you change the units of all independent variables (while keeping the same units for the dependent variable) then you should expect the regression coefficients to change.  The smaller the units, implying larger values, the smaller the coefficients.  No academic paper would make such a basic point, but if you need a justification you could refer to the matrix-form coefficient formula:
$\beta = (X^{T}X)^{-1}X^{T}{\bf{y}}$
Here the matrix of values of independent variables $X$ appears as the inverse of a squared term times an unsquared term, the net effect being that smaller units, implying larger values within $X$, reduce the value of coefficients.
In practice things are a little more complicated where the regression includes a constant term, requiring a column of 1's within $X$. Changing the units of the independent variables will not change the constant (which would change only if the units of the dependent variable were changed).
By contrast, the p-value of a coefficient is a unit-free measure of significance (of its difference from zero) and will not change when the units of the variable are changed.
A: I like to think of this as a problem in choosing representative or otherwise sensible values of x1 and x2 for a predictor X, and getting predicted Y when X=x2 minus predicted Y when X=x1.  It is easy to get a confidence intervals for such a difference (in R this is a feature of the rms package).  This handles nonlinearities in the X effect.  I don't like to recode data just to make it easier to get predicted values.  I like to fully specify exactly what I'm estimating.  Often I choose x1=25th percentile of X, x2=75th percentile.
