# Difference between population and sample

I have a general understanding of the difference between a population (set of entities under study) and a sample (a subsection selected from the population). However, I've been doing some work in PPC (Pay-Per-Click) and AdWords recently, and can't seem to grasp the population/sample difference in regards to that.

For example, let's say there are two Google AdWords ads. Users will click the ad and it takes them to a form which they can fill out if they choose to. Therefore, I have data on the number of clicks and the number of forms filled out. The question I'm trying to answer is which Ad was more effective at getting more clicks and forms filled out.

        Clicks      Forms


Initially, I thought that my sample was the two ads (Ad1 and Ad2), but that wouldn't be right as I'm really examining the number of clicks and forms filled out. So it would seem that the population that I'm examining is the clicks and forms associated with the two ads (Ad1 and Ad2) and my sample would be the number of clicks and number of forms filled out. Is that right/wrong? Thus, would clicks and forms be considered two seperate samples taken from the same population? Or is my population the same as my sample in this case?

The phrase "Population" is an abstract concept you use to define what type of question you want to answer. You could consider the "population of ads" or the "population of clicks" - they are just two different forms of inference (on about ads, one about clicks). I would suggest that in either case the notion of "population" is not helpful to you from the way you have asked the question. Suppose I write two hypothesis:

$$\begin{array}{c c} H_{1}:\text{Ad1 is more effective at getting clicks and forms filled out} \\ H_{2}:\text{Ad2 is more effective at getting clicks and forms filled out} \end{array}$$

Now I also write out the data, denoted $D$ in your table, and your prior information by $I$. Then we calculate:

$$P(H_{1}|D,I)=P(H_{1}|I)\frac{P(D|H_{1},I)}{P(D|I)}$$

$P(H_{1}|I)$ is the prior probability - how plausible is $H_{1}$ without considering the data? Note that because you only have two ads, then their probabilities must sum to 1, under this prior information. The main quantity to calculate is the likelihood $P(D|H_{1},I)$ - how plausible is the data you actually observed given that $H_{1}$ is true? $P(D|I)$ is sometimes called the evidence, and is usually not explicitly calculated, but divided out by considering odds ratios $\frac{P(H_{1}|D,I)}{P(H_{2}|D,I)}$. It can be thought of as asking the question how plausible is the data you actually observed, regardless of the hypothesis in question? ("does any hypothesis being considered explain these data well?")

You don't need a notion of the population to answer the question you have asked This can be seen quite clearly from the absence of such a notion in your question which Ad was more effective at getting more clicks and forms filled out?.

I've personally decided to call this "the golden rule" for inference using probability theory as extended logic. You write a proposition which if true or false would answer your question (such as $H_{1}$ and $H_{2}$). Then you simply calculate the probability of that proposition, conditional on whatever evidence you have (i.e. what you know), marginalising (or averaging) out what is unknown.