Absolute Mean Deviation
For a long time, I have not only debated with myself, but with others about standard deviation. To me, the formula makes as little sense as pre-calculus would do to an elementary student. Why would you want to square all the differences between the x values and the mean, and then find an average of those, and then square root the average? When you could just as simply find the difference between x values and the mean, convert them all to positives and then find the average between them? Before I proceed degrading the standard deviation any more, and promoting the absolute mean deviation, let me, take the time to explain the two. The standard deviation is a measure of spread, one of the most common ways in which a dataset can be analyzed. The process of attaining the standard deviation is by summing the squared values of each measurement’s deviation, dividing that value by the total number of measurements and then taking the positive square root of the result. Assume that you have a dataset of ten positive integers with a mean of ten: 13,6,12,10,11,9,10,8,12,9
Standard Deviation
In this instance, the sum of the squared deviations equals 40. The next step is to divide forty by the number of measurements, in this case 10. That leaves us with four, the variance value. The standard deviation is the positive value of the variance square-rooted, which then indicates that the standard deviation for this data set is two. On the other hand, the process of obtaining the absolute mean deviation is much simpler. The first step is to identify the mean, and subtract all of the x values from the mean (the same step was conducted to find the standard deviation). However, instead of squaring the numbers, the absolute mean deviation requires all deviations to be positive using absolute values. After which, an average of the positive deviations are taken yielding in the absolute mean deviation. Once again, assume that you have a dataset of ten positive integers with a mean of ten: 13,6,12,10,11,9,10,8,12,9
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| Mean Absolute Deviation |
The sum of the deviations equals 16. Following that, the sums of the deviations are to be divided by the number of measurements, in this case 10. Hence, the absolute mean deviation is 1.6 for the above-mentioned dataset. Seeing, as it required lot less time and effort to produce the absolute mean deviation, as opposed to the standard deviation; why is the standard deviation still preferred to the absolute mean deviation? The formula, and an example of the absolute mean deviation is herewith shown below:
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absolute-deviation-formula
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The mean deviation is much easier for newer researchers to understand intuitively than the standard deviation, because it is simply the average of the deviations. It has a clear meaning; the absolute mean deviation is the amount by which particular values differ from the mean on average. The standard deviation, however, does not have a clear meaning. In 1914, a British astrophysicist known as Arthur Eddington pointed out that calculating the absolute mean deviation is not only easier, but also more accurate than the standard deviation, especially since the margin for error in his field was greater. He claimed that it worked especially better with empirical data than the standard deviation did. However, Fisher, who believed that the standard deviation was more efficient than the absolute mean deviation under ideal circumstances, formed a counterargument. What Fisher failed to realize, is that it is nearly impossible for a measurement to be entirely error free, and hence the absolute mean deviation would work better in most situations. Moreover, scientists like Fisher have often tried to compare the standard deviation (population) versus standard deviation (sample) with absolute mean deviation (sample) versus the standard deviation (population). Comparisons like that often lead to inaccurate findings. Other social scientists and scientists believe that the absolute value symbols used in the absolute mean deviation formula ensure that it is much harder for the formula to be algebraically manipulated, as opposed to the standard deviation formula.
On the other hand, there are several situations where the absolute mean deviation is preferred, and there are several advantages attached to using the absolute mean deviation. In situations where some of the measurements do contain slight errors, the distribution is not perfectly normal, and when further analysis needs to be conducted, the absolute mean deviation is easier to obtain and easier to understand. This translates into the fact that for the standard deviation to be more effective than the absolute mean deviation, the data would have to be error free, and without any contamination. To begin with, utilizing the absolute mean deviation reduces the potential of errors within the dispersion, because it gives us less of a distorted view of the spread. The standard deviation requires the subtraction of the mean from each x value to be squared, ensuring that each unit of distance from the mean is exponentially (rather than additively) greater. Make no mistake; the act of square rooting the sum of the squares does not effectively and completely eradicate this bias. This serves as the primary reason why (in example 1) the standard deviation is 0.4 units greater than the absolute mean deviation.
The works of Barnett and Lewis discovered that the advantage in efficiency and effectiveness that the standard deviation is dramatically reversed when even an error element as small as 0.2% (2 error points in 1000 observations) is found within the data. In most realistic situations, there is bound to be at least a miniscule percentage of error elements, and as seen in the works of Barnett and Lewis, the advantage of the standard deviation over the absolute mean deviation is immediately reversed when that is the case. In addition, an assumption, which underlies the superiority of the standard deviation, is that it involves working with samples selected from a fixed population. However, that will not always be the case, especially in situations when working with the population, with a non-probability sample, or with a probability sample with considerable amounts of lack of responses. In each of these situations, it is acceptable to calculate the variance of certain values in comparison to the mean; however, forming the population standard deviation is improper and inaccurate. Lastly, the practice of removing outliers has been immensely lopsided with the use of the standard deviation. Because of the large bias and inconsistencies faced with squaring and then square-rooting data in order to produce the SD, the exclusion and deletion of outliers has often occurred, despite there not being a need for it to occur. Regardless of the importance of the data, those in education are often told to remove or ignore valid measurements with large deviations because they negatively influence correlations. However, the fault in that, is that the large nature of the deviations are only exaggerated by the production of the SD, and that perhaps the MD would not result in such large deviations and hence the removal of key pieces of data. Moreover, extreme values (not exaggerated extreme values) are quite essential in a variety of natural and social phenomena, including but not limited to city population growth, income distributions, earthquakes and traffic jams. As a result, in practice and in empirical situations, the absolute mean deviation (MD) should be preferred to the standard deviation (SD) due to the easiness in which the former can be utilized, and the fact that it has been proven more effective and accurate.
Especially since most of the calculations conducted today are formed using formulas and are outsourced to technology, the difficulty in using absolute values is dramatically reduced, and simplified. Given that, the SD and MD nearly perform the same tasks, and in most situations, the MD tends to be more accurate, it would only make practical sense for researchers, scientists and social scientists to take the simpler route in the absolute mean deviation. Complexity could be the cause of failure when performing correlations
