Population - Sample

Now we can go back to Statistics. To begin with, recall the following:

Population: all of the entities (people, events, things etc.) that are the focus of a study
Sample: any subset of the population
Parameter: any numerical quantity associated with a population
Statistic: any numerical quantity associated with a sample

After our discussion of probability we can now be a little bit more precise

Example Say we roll a fair die until the first time we get a six. We always are in one of two situations:

Theoretical

We have not actually rolled any die, maybe we don't even have a die, we are just studying this exercise theoretically. That is we are studying the population of all possible outcomes of the experiment "Number of rolls of a fair die until a six". We now have a theoretical description of this experiment, namely the random variable X with X~G(1/6). That is, a random variable describes a population.

For the first couple of values of k we find
k: 1 2 3 4 5 ...
P(X=k): 0.167 0.139 0.116 0.096 0.080 ...

We have previously seen that if X~G(p), then m=1/p and s=(√q)/p, so here we find

• m=1/(1/6)=6.0
• s=(√(5/6))/(1/6) = 5.48

In fact there are formulas for all kinds of summary statistics. For example here we would find:

• third quartile Q3=8
• 95th Percentile P95=16

and so on. But because they are computed for the whole population they are parameters.

Practical

On the other hand we can study this exercise by actually rolling a fair die many times and observing what happens. Let's say we do this 500 times and we find

6 1 7 4 1 9 9 13 30 1 3 1 15 14 5 7 5 11 6 8 6 5 12 2 10
2 2 25 2 3 8 4 1 5 1 2 2 15 8 6 5 10 3 3 5 1 12 6 2 5
8 19 1 4 1 18 4 1 1 2 10 4 8 5 8 11 2 2 4 3 2 4 1 23 4
18 1 5 19 1 5 1 4 6 4 1 17 3 16 1 3 11 10 1 2 5 1 1 5 1
4 16 5 2 1 1 2 7 6 4 2 3 6 12 15 1 17 8 6 4 9 2 1 2 21
7 9 18 28 5 26 8 1 12 10 11 1 11 21 3 3 8 8 5 8 7 1 6 6 1
1 9 20 12 1 11 2 1 7 1 20 1 13 1 2 31 2 2 1 1 8 10 1 4 10
2 1 3 1 12 9 1 4 3 1 5 5 5 2 3 4 3 2 3 6 6 4 13 3 1
5 6 14 4 5 6 3 8 1 2 4 20 4 11 4 12 5 1 19 6 3 9 17 10 3
3 18 2 2 8 20 8 3 9 5 12 5 5 6 9 5 1 2 2 4 1 12 5 9 1
6 1 19 22 3 5 3 7 8 10 12 8 1 5 2 1 6 13 6 12 1 2 1 4 8
7 5 19 5 2 7 1 1 7 1 13 2 6 1 7 1 6 11 2 1 2 2 8 2 2
9 1 2 15 2 18 12 4 5 12 1 31 4 3 3 11 3 3 6 2 7 1 6 25 2
5 4 15 3 9 11 5 11 3 9 10 6 6 2 1 11 2 1 9 14 15 2 19 2 2
8 5 8 4 9 7 2 1 2 2 4 1 12 11 6 4 8 14 6 10 3 5 5 2 9
1 1 4 7 4 5 11 13 8 5 2 3 1 5 2 13 2 25 3 8 4 4 15 4 1
2 10 10 8 8 9 1 1 7 8 3 5 4 4 2 5 1 6 2 2 2 1 15 3 1
7 8 1 18 3 7 5 9 3 10 5 5 4 5 9 19 2 7 2 4 6 5 7 1 12
2 2 2 1 1 1 1 4 12 3 1 11 1 1 10 1 1 7 17 4 6 3 5 9 7
1 1 4 1 6 6 7 3 4 8 6 14 14 5 2 8 2 6 4 9 5 6 9 1 1

Now we can use this dataset to find probabilities:
k: 1 2 3 4 5 ...
P(X=k): 0.182 0.132 0.078 0.086 0.098 ...

and of course we can find the summary statistics:

• mean =6.192
• standard deviation s=5.50
• third quartile Q3=8.0
• 95th Percentile P95=18.0

Because these are computed from a sample they are statistics

Population - Sample

The real powerful idea here is to combine these two approaches: Say we have a die that we suspect is not a fair die and we wish to test this. So we roll the die and then compare the practical results with the theoretical ones. For our die we find
k: 1 2 3 4 5
P(X=k) (theory) 0.167 0.139 0.116 0.096 0.080
P(X=k) (sample) 0.182 0.132 0.078 0.086 0.098

or we can do this by looking at some summaries:
  s Q3 P95
Population 6.0 5.48 8 16
Sample 6.192 5.50 8 18

It seems our die is pretty much a fair die.

The most important feature of the scientific method is that any scientific theory has to be falsifyable, that is it has to be possible to carry out experiments and compare the results of these experiments to predictions made by the theory. If they agree, the theory looks good, if not we need to change the theory or even find a new one. But how do we decide whether or not they "agree"? That is one place where Statistics comes into play.

• Theory: our die is fair

• predictions made using this theory: P(X=1)=0.167, m=6.0, ...

• carry out an experiment (6, 1, 7, 4, 1, ...)

• compare predictions with results of the experiment
P(X=1)=0.168 (theory), P(X=1)=0.182 (experiment)
m=6.0, =6.192

• do they agree or is the theory bad?

Note: most "theories" we look at are not big scientific theories but simple things like "Our new drug works better than the currently available one"