Statistical ‘averages’: mean, median, mode, range, and their use in practice
Naturally, it is an unacceptable logical blunder to consider that different types of statistical averages, such as mean, median, mode and/or range, are not widely used in various spheres of study, which are not based on math and statistics. On the contrary, profound understanding of the main characteristics of these types of averages and methods of their calculation are necessary for any educated person. For example, it can be effective to represent the entire data series by using a single value, which characterizes the ‘average’ value of the entire series when analyzing diverse large data sets. In other words, in order to simplify cumbersome computations, indicate the main properties of the data set and/or determine the main elements of the data set we can determine the so-called central tendency. Therefore, mean, median, mode and range can be defined as the most demonstrative ways to describe it. Obviously, these characteristics are widely used in those fields of study, which require a great number of different operations with large data sets because their statistical value is in direct proportion to the quantity of objects represented in the given set. For example, such statistical ‘averages’ as mean, median, mode allow us to determine the characteristic properties of the series with sufficient precision, whereas the range is representative only in case of large data sets. Let us explain it using a simple example. According to the definition of the range, if the data set is small and the difference between the smallest and the largest values in the series is small than the central tendency of this series (mean, median, mode, etc.) is not representative. Therefore, all these properties of fundamental statistical ‘averages’ make them indispensable for virtually all statistical calculations. For example, statistics as well as IT specialists need to perfectly understand the characteristics of mean, median, mode and range in order perform various statistical measurements, manage systems, plan capacity and balance load, monitor maintenance and troubleshoot issues, etc. Obviously, these different objectives dictate that these specialists calculate mean, median, mode, range and/or diverse types of their combinations, attempting to demonstrate a statistically significant quantity, trend or deviation from the norm of the data set. Of course, determining mean, median, mode and range is only the start, however, they permit us to determine the central tendency of the data set from the very beginning.
Mean, median, mode and range: definitions and ways of finding these statistical ‘averages’
As it was already emphasized, mean, median, mode and range appertain to the group of fundamental mathematical terms, which are used in practically all branches of modern statistics, accounting, financial management, budget planning, various areas of computer science, etc. Obviously, with an eye to achieving a satisfying understanding of these doubtlessly significant mathematical terms, it is recommended to examine not only their main mathematical properties and methods in which they may be used in practically oriented assignments but also different techniques used to find them. This educational operation has several aims. Firstly, by studying the simplest methods according to which these ‘averages’ can be determined, one will inevitably achieve a better understanding of basics of statistics. In fact, this topic may serve as a rich source of different mathematical exercises and practical assignments, thereby, by performing these simple computations one can considerably improve her logical skills. Secondly, this topic supply students with a great amount of objectives, which require not only an ability to create abstract mathematical models but also the ability to relate theoretical information with practical problems. Therefore, its importance for those students who study economics, computer science, accounting, management or architecture can hardly be overestimated. Thirdly, by comparing these ‘averages’ between each other and studying differences between them, one can understand not only a mathematical but also a practical meaning of these conceptions, which is, obviously, important for future specialists. Thus, in order to achieve all these goals let is examine the definitions of mean, median, mode and range, as well as their significant differences. Here is a concise list, which includes also characteristic examples and elementary exercises targeted at simplifying of the educational material:
- The mean is the elementary statistical average of a series of two or more numbers. 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 objects, which constitute the series. It is also should be noted that in addition to the commonly-known arithmetic mean, there also exists a group of measures that are also referred as various types of the mean, such as the geometric mean and the harmonic mean or the so-called Pythagorean means (according to the name of the mathematician who discovered them first). Nevertheless, in this article you will not find information about them. These measurements can be considered as too sophisticated compared with elementary statistical measurements, therefore, if one wished to know more about them, she should examine mathematical textbooks created for use in universities and specialized schools. We shall concentrate our attention on the arithmetic mean and ways of its definition. Thereby, let us study its fundamental characteristics by performing a demonstrative and comprehensible virtual experiment. Seven numbers are given: six, eleven, nine, three, four, eighteen and five. How to find the mean of this series of numbers? In fact, the solution is quite simple because all we have to do is to add up all the numbers in the given series and then divide them by seven. After a series of elementary computations, we will receive the desired result: the arithmetic mean is eight. Therefore, we can postulate that the arithmetic mean can be easily determined if we know the number of objects in the given series, as well as the value of each given object.
- The median is the middle value in the set of given numbers. With an eye to finding the median, one has to list all the given numbers in the series in numerical order (in the direction of increasing values). Thereby, the middle number is the median. Let us repeat the previous example, using the already known series of numbers in order to simplify all needed calculations. Therefore, now we have to list all the given numbers in numerical order. As a result, we will obtain the following set of numbers: three, four, five, six, nine, eleven and eighteen. Therefore, the median is six. 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 series of numbers, which includes twenty-three, twenty-nine, thirty-two, thirty-six, forty-four and forty-eight, the median can be calculated as the halfway between thirty-two and thirty-six. Thereby, the median is thirty-four.
- The range is a simple mathematical average, which can be described as the difference between the largest value and the smallest value in the studied series of numbers. In the previous virtual experiment, we have used the following set of numbers: seventeen, twenty, sixty-nine, forty-eight, eighty-five, twenty-seven, forty-eight, twenty, thirty-two and forty-eight. To make our computations more demonstrative let us list all these numbers in the numerical order from the smallest to the largest values. Here is the result: seventeen, twenty, twenty, twenty-seven, thirty-two, forty-eight, forty-eight, forty-eight, sixty-nine and eighty-five. The smallest value is seventeen and the largest value is eighty-five. In order to find the range of the given series of numbers, we have to find the difference between eighty-five and seventeen. The resulted equation: 85 – 17 = 68. Therefore, the range of this set of values is sixty-eight.
- The mode is the number (object), which is listed in the given set of numbers more often than any other number (objects). Again, we shall examine the elementary series of numbers, which contains twenty, sixty-nine, forty-eight, eighty-five, seventeen, twenty-seven, forty-eight, twenty, thirty-two and forty-eight. According to the given definition of the mode, we have to find the number that occurs the most often in the given series. In this case, the mode is forty-eight. Obviously, if the obtained register of numbers comprises no repeated values, then the given series has no mode.
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