I am trying to understand what the Price equation says, based on the associated Wikepedia article at https://en.wikipedia.org/wiki/Price_equation. The equation they give is: $$ \Delta z = \frac{1}{w} cov(w_i,z_i) + \frac{1}{w} E(w_i \Delta z_i) $$ where z is the frequency of a trait and w is the fitness of the trait, which I assume means number of viable offspring.
Here is my stab at an explanation. The first term gives the change in frequency assuming that the $w_i$ terms stay constant and the second term gives the frequency change due to changes to $w_i$ caused by mutation. I am sure that is overly simplistic if not flat out wrong. Am I close? How does the equation relate to group selection?
Andy Gardner wrote a piece describing the equation for the journal Current Biology http://www.cell.com/current-biology/pdf/S0960-9822(08)00008-0.pdf. I reproduce some of his description of the equation below (note that I have written the Price equation as Gardner has it in his paper).
"The Price equation is a simple mathematical statement about change. In its usual formulation, it describes how the average value of any character — body weight, antler size, proclivity to altruism — changes in a biological population from one generation to the next.
Price denoted the individual’s character value as z, its number of offspring as w, and the discrepancy between the character values of itself and its offspring as ∆z, and showed that the change in the population average value of the character between parent and offspring generations is $$ \Delta \bar{z} = cov(\frac{w}{\bar{w}},z) + E(\frac{w}{\bar{w}}, \Delta z) $$
The Price equation separates the total change into two component parts. The first part is the change that can be ascribed to the action of selection, and this takes the form of a statistical covariance between individuals’ character values (z) and their relative reproductive success $(w/\bar{w})$.
For example, if individuals with larger values of the character of interest tend to have more offspring, then the covariance is positive and selection acts to increase the population average value of the character.
The remainder term takes the form of an expectation (arithmetic average) describing how offspring differ from their partners, and this is denoted the change due to transmission. If offspring are identical copies of their parents, then the transmission effect is zero and selection is the only factor involved in the evolution of the character.
However, offspring will often differ from their parents, perhaps because of mutation, or because their genes are recombined in a new way, or because of a change in their physical, biological or cultural environment, and in this case the transmission effect is non-zero."