In calculus, the mean is an average of several numbers. In fact, the mean is the simplest type of mathematical ‘averages’, therefore, it is a small wonder that even those of us who do not perfectly remember school math lessons know how to use it for different practical purposes. Furthermore, the concept of the ‘mean’ appertains to those mathematical ideas, which properties are intuitively understandable for humans, such as fundamental properties of simple geometric figures: triangles, pyramids, squares, spheres, etc. For example, one does not have to study a great amount of sophisticated mathematical textbooks, to realize that all triangles can be described as a geometric figure with three vertices. Obviously, one should not be aware of the very fact of the existence of the mean definition math to use this concept. The human mind always works with various categories of averages; in fact, it is one of its characteristic peculiarities, which distinguishes it from less complicated mental constructions. It is not designed to multiply six-digit numbers or perform mental computations of differential equations. However, we perfectly cope with various tasks that require the use of averages. Thereby, we surely can state that the concept of the ‘mean definition math’ can be easily comprehended by students, regardless of their academic level, due to the simple fact that our brain is constructed to perform diverse types of computations aimed at determination of average values. However, as well as other mathematical conceptions that may seem quite comprehensible, the mean is fraught with different intricate mathematical nuances, which become obvious only after closer study of different methods targeted at the determination of the mean and various interconnections between the mean and other terms that denote various types of ‘averages’. In truth, one will inevitably perform a gross mistake considering that it is possible to eschew studying of these details if she desires to master her cognitive skills in the sphere of modern statistics. Furthermore, the profound knowledge about different types of ‘averages’, such as the mean definition math, and mathematical techniques that allow determining them will be doubtlessly useful for virtually all people who need to improve their practical mathematical skills. For example, one needs to perform a series of calculations, each of which is connected with an assignment of determination of the average value, just to buy ingredients for an ordinary salad or Christmas presents. Obviously, much more complicated tasks, such as filing taxes or budget planning, also require direct acquaintance with the main statistical rules that permit to determine the average value. Thereby, one can easily eschew various problems with all these tasks just by improving one’s statistical skills and studying the basics of statistics, such as the mean definition math and related terms. Let us study these terms in more detail in order to obtain an overall understanding of the averages in statistics.
In fact, the mean definition math, which is ubiquitously used in colloquial language, does not completely reflects the meaning of this term from the point of view of modern statistics. A common mean definition math defines the mean as the result of dividing the sum of the numbers in the number series by their total number. Nevertheless, in statistics, this definition is not very usable and scientifically precise because it significantly simplifies the concept of the measures of central tendency, one of which is the arithmetic mean. Thereby, in order to understand all the differences between the definition of the arithmetic mean given by the mean definition math and other measures of so-called ‘averages’, such as median, mode and/or range, let us study the mathematical characteristics of these measures together with comprehensible arithmetical methods designed to find them. Here is a concise list of these measures, which also includes a few practical examples, that can be quite useful for those who desire to obtain deep and versatile knowledge about the basics of modern statistics:
In order to gain confidence that all information about diverse elementary measures of central tendency is completely comprehended by the student, it is recommended to perform a series of tests aimed at determination of these measures. Here is an elementary sample of these tests. Seven numbers are given: four, six, eleven, nineteen, two, ten, and four. Our assignment is to find the main measures of central tendency. According to the previous samples, we can easily find the arithmetic mean in accordance with the mean definition math. The total of all numbers in the series is fifty-six. The number of the values in the series is seven. The arithmetic mean is eight. The mode is four (it is the only number that is can be found twice in the series). With an eye to obtaining the median, we have to place all the values in correct order, from smaller to greater values. As a result, we will obtain the series: two, four, four, six, ten, eleven and nineteen. Therefore, the median is six. Finally, the range can be found as the difference between two and nineteen. Thereby, the range is seventeen.