Is this the correct equation for the mean free path? $\lambda=\frac{KT}{\sigma p}$ is the equation given in my textbook but everywhere else I look it is different. Is it wrong? The derivation is as follows:

$$\lambda = \frac{v_{rel}}{z}$$ where $\lambda$ is the mean free path and $z$ is the collision frequency. 

Since $z=\frac{\sigma v_{rel}}{KT}$ where $\sigma$ is the area where it is possible for two particles to collide.

Then substituting for $z$ gives the equation $\lambda=\frac{KT}{\sigma p}$. Have I gone wrong somewhere. I often see a factor of root two in the denominator but I don't see where that might come from.
Your derivation is correct. The $\sqrt{2}$ comes from $v_{rel}=\sqrt{2}v_{mean}$, where $v_{rel}$ is the mean relative speed (relative to the other molecules in the gas) and $v_{mean}$ is just the mean speed of the molecules in the gas.

In conclusion: note what the subscript says about the variable!