![]() ![]() In this way, to calculate the geometric mean, we can:ġ. ![]() It turns out that the product of n numbers is equal to the sum of their logarithms, so the logarithm of the geometric mean will be equal to the arithmetic mean of the sum of logarithms. This is so because 2 is the number to which the base (10) must be raised to to obtain the original number (100). The base 10 logarithm of 100 is equal to 2: Let’s look at an example to understand it better. The simplest way, in my humble opinion, to say what a logarithm is, is to say that the logarithm of a number indicates the exponent to which a number (the base) must be raised to obtain the original number. Let’s see it.Īlthough it may seem strange, Napier invented logarithms in the 17th century to facilitate mathematical calculations. And it turns out that the geometric mean is closely related to logarithms. To finish this post, we are going to see one of those unexpected relationships that the magic of mathematics offers to us from time to time. It is now clear, 1000 seems to us a more accurate measure of central tendency for this distribution. I clarify, just in case, that raising to 0.2 (1/5) is the same as doing the fifth root. We can use the R program to calculate it: This makes it ideal for averaging geometric progression data, such as ratios, compound interest in economics, and, as in our case, bacterial growths in microbiology.įor example, if a company has grown 15%, 22%, 14%, 18% and 12% in the last 5 years, the correct thing to do would be to calculate the geometric mean to give a measure of average growth for the 5 years. The geometric mean is often used when the values of the distribution change multiplicatively, and not additively. The geometric mean of a series of n numbers is the nth root of the roblema of all the numbers.įor those who know how to appreciate the beauty of mathematical language, what has been said above can be expressed with the following formula: Specifically, we need to resort to the geometric mean. To calculate a more accurate measure of central tendency we need something more robust against the presence of outliers. The problem is that we have an extreme value, 100,000, and the arithmetic mean is irresistibly drawn towards it. The least thoughtful of you will rush to calculate the arithmetic mean, which is 20,440 colonies per dish.Īnd so I would ask you: do you really think that 20,440 is a good measure of central tendency for this distribution of values? If you think about it, you will see that it is not. Now I ask you: what is the average growth with this culture medium? We are going to seed a series of plates with the bugs and, to compare the plates with the different enrichment media, we are going to calculate the mean of the bugs that grow in each series.įor example, we have five plates that we have enriched with a certain medium and we count the colonies that we obtain in each one: 1,000, 100, 100,000, 100 and 1,000. Let us imagine that we are in our laboratory looking for a way to enrich a culture medium to more easily detect the presence of Fildulastrum fildulastrii, the causal germ of that terrible disease that is fildulastrosis. And, to do so, we are going to assume that we have become microbiologists. Today we are going to talk about one of them, the geometric mean. To put it a little more technically, it is not very robust in the presence of outliers.įor these cases, it may be more convenient to use other measures of central tendency that are more robust. However, the mean has a small flaw: it is easily bias by the presence of extreme values in the data distribution. We already know that the arithmetic mean is the most famous of the measures of central tendency of a distribution, so it should come as no surprise that it is involved in so many statistical and mathematical procedures. This makes it ideal for averaging geometric progression data, such as ratios, compound interest in economics, or bacterial growths in microbiology. The geometric mean is used when the values of the data distribution change multiplicatively, and not additively. ![]()
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