Mean
The term "mean" in mathematics typically refers to the average or central value of a set of numbers, making it a crucial concept in statistics, where it is often used to describe measures of central tendency. The most recognized type of mean is the arithmetic mean, which is calculated by adding all the values in a dataset and dividing by the number of values. For instance, for a dataset like 7, 12, and 26, the arithmetic mean would provide a simple average.
Beyond the arithmetic mean, other forms include the weighted mean, which assigns different weights to values, and the geometric mean, useful for datasets that involve exponential growth. The harmonic mean, another variation, focuses on the reciprocals of the values. While the mean is an effective measure of central tendency, it can be misleading in datasets with extreme values, prompting the use of additional statistics like variance and standard deviation, which assess the spread of data around the mean. Understanding these concepts can provide a foundational grasp of how data can be interpreted and analyzed in various fields.
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Mean
Mean is a mathematical term that most often refers to the average or central value of a series of numbers. A mean is used in the science of statistics where it is a common measure of central tendency, a single numerical figure that represents the center of an entire distribution of data. The most widely used form of mean is called the arithmetic mean, which is a simple numerical average. When the term mean is used alone, it is usually referring to the arithmetic mean. There are several other forms of means, the most common of which are the weighted mean, geometric mean, and harmonic mean.
Background
Statistics is a mathematical science that collects and analyzes numerical data to determine the probability that the data represents a larger whole. The origins of statistics can be traced to the work of sixteenth-century Italian mathematician Gerolamo Cardano who wrote a paper about the probable statistical outcomes for the roll of three dice. A century later, French mathematicians Pierre de Fermat and Blaise Pascal developed advanced mathematical theories on the probability of games of chance. While their methods differed, their outcomes were correct and their work later became the basis for the modern science of statistics. As the method developed, scientists were able to predict social tendencies and some future trends with a reliable degree of probability.
One of the primary goals of statistics is to examine the distribution of data to determine measures of central tendency. These are single data values that identify a common or central position within a set of data. Three of the most widely used ways to gauge central tendency are mean, median, and mode. Median refers to the middle value in a set of data arranged in order from smallest to largest. For example, in this list of numbers—3, 23, 33, 34, 38, 54, 76, 81, 84—the median would be 38 because four of the numbers come before it and four after. In a list with an even amount of numbers—3, 23, 33, 34, 38, 54, 76, 81, 84, 90—then the median would be the average between the two middle numbers. In this case 46, which is the average of 38 and 54. Mode is the number that occurs most frequently in a data set. In a data set of 81, 24, 44, 65, 44, 29, 26, 44, 31, 24, the mode would be 44 because it shows up three times in the list.
Overview
The most common way to determine central tendency is the mean, specifically the arithmetic mean. This type of mean is simply the average value of a series of numbers. The mean value for this list—7, 12, 26, 32, 36, 58, 75, 81, 84, 88—would be 49.9, the result of the sum of the data (499) divided by the amount of values in the set (10). When the data values are known, the formula for this is written as X=(∑x)/n.
The Greek uppercase letter sigma (Σ) refers to the "sum of" the data points represented by x, and the number of values in the data set is represented by n.
The arithmetic mean is generally considered a good method of determining central tendency. The mean also determines several other important statistical measures, such as variance and standard deviation. Variance measures the distance each value in a data set is from the mean, while standard deviation plots the dispersion of the data set from the mean. In other words, the further the data points are from the mean, the higher the standard deviation will be. A problem with arithmetic mean is that it does not work well if the data set contains values that are unusually high or low. For example, using the mean value of test scores to determine how well a class has learned the material would provide misleading results if two people did not take the test and scored zeros.
A weighted mean is similar to an arithmetic mean except that certain data values are given more emphasis, or "weight," than others are. For example, a teacher may use a weighted mean to make one exam more important toward a student's final grade. If a course is graded on the scores of four tests, and the final exam is worth 40 percent of the grade, then the first three scores would be worth 20 percent each. So scores of 80, 85, 75, and 90 on the final would be weighted this way: 80 x .20 = 16; 85 x .20 = 17; 75 x .20 = 15; 90 x .40 = 36. The weighted values are then added up to reach a final grade of 84.
A geometric mean is calculated on a logarithmic scale, or a scale that uses values that do not increase in a linear manner, such as counting from 1 to 10. Logarithmic values increase in an exponential manner, a proportional manner determined according to a numerical power, such as a square root. Its technical definition is the nth root of the product of n values. Therefore, whatever the number of values in a data set, that will be the root power. To find the geometric mean of 2, 6, 8, the numbers must first be multiplied (2 x 6 x 8 = 96). The geometric mean is the cubed root of 96, or about 4.58. The cubed root is a number multiplied by itself three times (4.58 x 4.58 x 4.58 = 96.07).
In technical terms, a harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. A reciprocal is simply a 1 divided by a particular number; so the reciprocal of 4 would be 1/4. The harmonic mean of a set of values would be the number of values in the set divided by the sum of the reciprocals in the set. A data set of 1, 5, 8, 10 would yield reciprocals of 1/1, 1/5, 1/8, and 1/10. Added together, they equal 1.425. Since there are four values, then harmonic mean is calculated by 4 divided by 1.425, or 2.807.
Bibliography
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