The law of large numbers states that as the sample size increases, the sample mean of random variables will approach the theoretical mean. A coin is often used to demonstrate this concept. Adequate sample size is needed to ensure representative results, determined by a formula that considers the desired confidence interval.
The law of large numbers is a statistical theorem that postulates that the sample mean of random variables will approach the theoretical mean as the number of random variables increases. In other words, the larger a statistical sample, the more likely you are to get results that are more accurate than the total picture. Smaller sample numbers tend to skew the result more easily, although they can also be quite accurate.
A coin is a good example that can be used to show the law of large numbers. It is often used in beginner statistics courses to demonstrate how effective this law can be. Most coins have two sides, heads and tails. If the coin is tossed, logic would say that there is an equal chance of the coin landing heads or tails. Of course, this depends on the balance of the coin, its magnetic properties, and other factors, but it’s generally true.
If a coin is tossed only a few times, the results may not indicate that there is an equal chance of heads and tails. For example, flipping a coin four times results in three heads and a tails. It could also produce four heads and no tails. This is a statistical anomaly.
However, the law of large numbers states that as the sample increases, those results will most likely be in line with the true representation of possibilities. If a coin is tossed 200 times, there is a good chance that the number of times it is flipped on heads and tails will be close to 100 each. However, the law or the large numbers don’t predict that it will be exactly 100 each, just that they will probably be more representative of the true range of possibilities than a smaller average.
The law of large numbers demonstrates why an adequate sample is needed. Statistics are used because there is not enough time, or it is impractical, to use the entire population as a sample. However, a population sample means that there will be representative members of the population who are not counted. To ensure that the sample reflects the total population, an adequate number of random variables is required.
Determining how much of a sample is needed usually depends on a number of factors, the main one being the confidence interval. For example, a statistical confidence interval is the level of certainty that the population will fall within certain parameters. Setting a 95 percent confidence interval would mean that there is a reasonable certainty that 95 percent of the population will fall within those parameters. The sample needed for given confidence intervals is determined by a formula that takes into account the number in the population and the desired confidence interval.
While the law of large numbers is a simple concept, the theorems and formulas that help justify it can be quite complex. Simply put, the law or large numbers are the best explanation for why larger samples are better than smaller ones. No one can guarantee with certainty that a statistical sampling will be completely accurate, but this law helps prevent many inaccurate results.
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