The Size of Samples
The purpose of sampling is to allow us to infer some property of a population without having to examine every member of the population. This raises the question of how big the sample should be.
Suppose we were interested in identifying the average height of people in developed countries. Roughly a billion people live in developed countries, so measuring everyone’s height would be impractical. How many people would we need to include in our sample in order to produce a good estimate of the true average for this large population?
In general, people get taller with age until around 20 years of age. With old age, people begin to shrink. With improved nutrition, in most developed countries people have been getting taller over time. Measurements made between 2003 and 2006 indicate that the average height of females (twenty years of age and older) is 1.622 meters (around 5 ft, 4 inches). The average height of males is is 1.763 meters (around 5 ft 9.5 inches).
There is no guarantee that every random sample will be representative of the population. Even if we sample a thousand people, we might, simply by chance, have sampled people who are shorter than average, and so our estimate will be biased. This is highly improbably, but nevertheless possible.
Roughly 95% of all people will be with plus or minus 10 cm of the average height. For women (with a mean of 1.622 meters) this means that 95% of all women will be between 1.522 and 1.722 cm in height. Statisticians are able to determine the likely error for any given sample size. The graph below shows the effect of sample size for estimates of average height. For a sample consisting of just one person, 95% of the time, that person will be within about 10 cm of the true average. If your sample consists of two randomly selected individuals, 95% of the time, the average of these two individuals will be within about 6.6 centimeters of the true average. With a sample of five individuals, 95% of the time, the average of this sample will be within about 2.5 centimeters of the population average. With a sample of twenty people, 95% of the time, the sample average will be within about half a centimeter of the population mean. With a sample of 100 people, 95% of the time, the sample average will be less than 25 millimeters of the population mean.
Note that these values assume that the sampling is truly random. It assumes that any given person in the world is as likely to be sampled as any other person.
The Law of Large Numbers
The law of large numbers simply states that as the sample size increases, the sample is more likely to provide a more accurate estimate of the true population value.
A fair coin should be equally likely to appear heads or tails. That is, it should appear heads 50% of the time. If you flip a fair coin just once, it will either come up heads 100% of the time or 0% of the time. If you flip a fair coin twice, it will either come up heads 100% of the time, 0% of the time, or 50% of the time. In fact, it will come up heads 50% twice as often as either 100% or 0%. The greater the number of times you flip a coin, the greater the likelihood that the number of times it appears heads will approach the fair value of 50%. That is, the more times you sample the coin’s behavior, the greater the likelihood that the sample value will approach the true population value.
At this point, we can introduce some useful terminology.
The average (or mean) for the population is represented by the lower-case Greek letter mu (μ).
The average or mean for the sample is represented by the value x with a horizontal bar placed on top (x̄). When spoken, this symbol is referred to as “x-bar.” Sometimes the population mean is represented by the uppercase letter “M.”
The number of items in a sample is typically represented by the italicized uppercase letter N. The italicized lowercase letter n is sometimes used to identify the number of cases in some sample. For example, twenty-five people were recruited for an experiment (N=25); each participant responded to 10 musical works resulting in 250 data points (n=250).
In statistical calculations, we are always interested in the number of ways that values are free to vary. This concept is referred to as the degrees of freedom (abbreviated df). Typically, the degrees of freedom are related to the number of observations made. So as the sample size increases, normally the degrees of freedom also increase.