I'm trying to get the function that lets me determine the value of y, where x is an arbitrary number. In this case, I'm looking at sales charts, and I'm making the assumption that the function is logarithmic. For example:
- I have a group of 10,000 sales people.
- They sell 50,000 widgets per day.
- The top sales person sells 75 widgets per day.
What does the mean sales person sell? What does sales person #x sell?
Just dividing 50,000/10,000 ( = 5) seems wrong, since I know that the top folks sell so much more.
This is not homework - or, I guess it's grown-up homework :)
I don't see a reason to fit logarithms here. Based on your description, I expect a power law distribution: few ultra-high achievers (Jon Skeets of sales), long tail of mediocrity.
Under the power law assumption, the agent number $x$ sells $C/x^p$ widgets. When $x=1$, this should be $75$. Thus, $C=75$. To find $p$, I used the information that all $10000$ together sell $50000$ widgets. Replacing the sum by integral for easy of calculation, this yields $$\int_1^{10000} \frac{75}{x^p}\,dx=50000$$ Hence $$\frac{75}{1-p}(10000^{1-p}-1) = 50000$$ I solved this numerically: $p\approx 0.34$. Thus, $y=75/x^{0.34}$ is a reasonable guess based on information in your question. Here is how it looks:
The mean sales volume is indeed $50000/10000=5$, by the definition of a mean; this is independent of the above assumption. The median sales volume is $75/5000^{0.34}\approx 4.14$, under the above assumption.