How to NOT use statistics This is sort of an open ended question but I wanna be clear. Given a sufficient population you might be able to learn something (this is the open part) but whatever you learn about your population, when is it ever applicable to a member of the population?
From what I understand of statistics it's never applicable to a single member of a population, however, all to often I find myself in a discussion where the other person goes "I read that 10% of the world population has this disease" and continue to conclude that every tenth person in the room has this disease.
I understand that ten people in this room is not a big enough sample for the statistic to be relevant but apparently a lot don't.
Then there's this thing about large enough samples. You only need to probe a large enough population to get reliable statistics. This though, isn't it proportional to the complexity of the statistic? If I'm measuring something that's very rare, doesn't that mean I need a much bigger sample to be able to determine the relevance for such a statistic?
The thing is, I truly question the validity of any newspaper or article when statistics is involved, they way it's used to build confidence. 
That's a bit of background.
Back to the question, in what ways can you NOT or may you NOT use statistics to form an argument. I negated the question because I'd like to find out more about common misconceptions regarding statistics.
 A: Unless the people in the room are a random sample of the world's population, any conclusions based on statistics about the world's population are going to be very suspect.  One out of every 5 people in the world is Chinese, but none of my five children are...
A: *

*To address overapplication of statistics to small samples, I recommend countering with well-known jokes ("I am so excited, my mother is pregnant again and my baby sibling will be Chinese." "Why?" "I have read that every fourth baby is Chinese.").

*Actually, I recommend jokes to address all kinds of misconception in statistics, see http://xkcd.com/552/ for correlation and causation.

*The problem with newspaper articles is rarely the fact that they treat a rare phenomenon.

*Simpsons's paradox comes to mind as example that statistics can rarely be used without analysis of the causes.
A: There is an interesting article by Mary Gray on misuse of statistics in court cases and things like that...  
Gray, Mary W.; Statistics and the Law. Math. Mag. 56 (1983), no. 2, 67–81
A: When it comes to logic and common sense, be careful, those two are rare.   With certain "discussions" you might recognize something......the point of the argument is the argument.
http://www.wired.com/wiredscience/2011/05/the-sad-reason-we-reason/
A: To make conclusions about a group based on the population the group must be representative of the population and independent.  Others have discussed this, so I will not dwell on this piece.
One other thing to consider is the non-intuitivness of probabilities.  Let's assume that we have a group of 10 people who are independent and representative of the population (random sample) and that we know that in the population 10% have a particular characteristic.  Therefore each of the 10 people has a 10% chance of having the characteristic.  The common assumption is that it is fairly certain that at least 1 will have the characteristic.  But that is a simple binomial problem, we can calculate the probability that none of the 10 have the characteristic, it is about 35% (converges to 1/e for bigger group/smaller probability) which is much higher than most people would guess.  There is also a 26% chance that 2 or more people have the characteristic.
A: Statistical analysis or statistical data?
I think this example in your question relates to statistical data: "I read that 10% of the world population has this disease". In other words, in this example some one is using numbers to help communicate quantity more effectively than just saying 'many people'. 
My guess is that the answer to your question is hidden in the motivation of the speaker on why she is using numbers. It could be to communicate some notion better or it could be to show authority or it could be to dazzle the listener. The good thing about stating numbers rather than saying 'very big' is that people can refute the number. See Popper's idea on refutation.
A: Hypothesis: $A$
(Textbook) Result: Do no reject $A$ ($\sigma = c$)
Your Statement: $A$ holds with probability $\sigma$!
Correct would be: In this case, you know nothing. If you want to "prove" $A$, your hypothesis has to be $\neg A$; reject it with $\sigma$ to get the desired statement.
A: 
From what I understand of statistics it's never applicable to a single member of a population

It's not true. It depends on the application.
Example: nuclear decay in physics. The rate of decay, defines the probability of a decay of every single nucleus. You take any nucleus and it'll have exactly the same probability of decay, which you established by experimentation on the sample.
