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:

- The mean is the simplest mathematical average of a set of two or more numbers. Of course, this mean definition math defines only the simple arithmetic mean. In other words, the mean can be defined as the total of all the numbers in the given number series, which is divided by the number of numbers in the series. In addition, to the arithmetic mean, there also exists a group of measures, which are also referred as means, such as the harmonic mean and the geometric mean. These measures are also known as the Pythagorean means, due to the scientist who discovered them first. Unfortunately, these measures are a little bit more complicated than elementary statistical measures, therefore, if one wants to know more about them, she should examine mathematical textbooks designed for universities and specialized schools. In our case, we can use the mean definition math, which defines the simple arithmetic mean. Let us study its characteristics by performing a simple virtual experiment. Five numbers are given: five, three, eleven, seventeen and forty-nine. How to find the mean of this set of numbers? According to the mean definition math, all we have to do is to add up all the numbers in the set and then divide them by five. After a series of elementary computations, we will obtain the desired result: the arithmetic mean is seventeen. Therefore, the mean can be easily determined, if we know the number of objects in the set and the value of each given object.
- The median is the middle value in the set of given numbers. In order to find the median one has to list the given numbers in numerical order in the direction of increasing values. The median is the middle number. Let us consider the previous example. Now we have to list all the given numbers in numerical order. As a result, we will obtain the following set of numbers: three, five, eleven, seventeen and forty-nine. Thereby, the median is eleven. However, this is the simplest example that can be used for demonstration of the characteristics of the mean definition math. In fact, what should we do, if we have two middle values? In this case, the median is the halfway between these numbers. For example, in the set of numbers, which includes five, nine, ten, twelve, twenty-three and sixty-eight, the median is the halfway between ten and twelve. Thereby, the median is eleven.
- The mode is the number, which is listed in the given set of numbers more often than any other number. Again, we shall use the simple set of numbers that comprises ten, eighteen, thirty-three, eighteen, five, twenty-seven, forty-two and fifty-three. According to our definition of the mode, we have to find the number which occurs the most, as well as we acted trying to find the mean in accordance with the mean definition math. In our case, the mode is eighteen. Obviously, if the obtained list of numbers includes no repeated values, then the set has no mode.
- The range is a simple mathematical average that can be described as the difference between the smallest value in the given set of numbers and the largest value in the same set. In truth, the definition of the range may be even more understandable than the mean definition math. In the previous virtual experiment, we have worked with the numbers: forty-two, ten, eighteen, twenty-seven, five, thirty-three, eighteen and fifty-three. The largest value is fifty-three, whereas the smallest is five. In order to find the range of the received set of numbers, we have to find the difference between five and fifty-three. 53 – 5 = 48. Thereby, the range of this set of values is forty-eight.

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.

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