Statistics

There are some ``classic'' examples of the scientific method at work. Mendeleev noticed some patterns in particular elements and proposed the periodic table, which contained gaps that were later filled by the discovery of new elements. Einstein started with a theoretical hypothesis on the nature of gravity which was only years later confirmed by experimental observations. Often though it is hard to discern from experimental data whether or not some sort of scientific principle is at work or not, and if so, what it might be. Suppose you asked a group of 20 students to tell you the season they were born in and also whether or not they own a car. You might get some data as below.

 

Table 1.1: 20 student sample size

Birth month	Owns a car	Does not own a car
Fall	3	1
Winter	1	2
Spring	6	0
Summer	6	1

 

From this data we might conclude that

• a student is more likely to be born in the spring or summer,
• a good majority of students own cars,
• a student born in the spring or summer is more likely to own a car than one born in the fall or winter.

However, if we went out and questioned 20,000 students, we might obtain the following data:

 

Table 1.2: 20,000 student sample size

Birth month	Owns a car	Does not own a car
Fall	2334	2425
Winter	2456	2534
Spring	2581	2533
Summer	2625	2512

 

The apparent trends we saw with 20 students have now disappeared, and we now find a more homogeneous mixture of birth months and owning or not owning cars. This probably agrees with our expectations on this question (although one should be objective!), but in any case this data is contrived, being in fact generated by a random sequence of numbers via a computer program. The lesson to be learned is that one should be very careful in interpreting statistics for trends, and that a small sample base can lead one to misdirected conclusions. One less trivial example of this that we shall encounter later concerns the dangers of living near power lines. Recent statistical studies have come to opposite conclusions, but in this case there are some plausible scientific reasons that might explain a correlation between a possible health hazard and living near power lines. Another example is the link between lung cancer and smoking - initially this too started as a statistical study, which was dismissed by many as being statistically insignificant, but today the dangers are of course accepted by the vast majority.