What Is the Definition of a Mathematical Average

In mathematics and statistics, average refers to the sum of a group of values divided by n, where n is the number of values in the group. An average is also known as a mean. Like the median and the mode, the average is a measure of central tendency, meaning it reflects a typical value in a given set. Averages are used quite regularly to determine final grades over a term or semester. Averages are also used as measures of performance. For example, batting averages express how frequently a baseball player hits when they are up to bat. Gas mileage expresses how far a vehicle will typically travel on a gallon of fuel. In its most colloquial sense, average refers to whatever is considered common or typical. Mathematical Average A mathematical average is calculated by taking the sum of a group of values and dividing it by the number of values in the group. It is also known as an arithmetic mean. (Other means, such as geometric and harmonic means, are calculated using the product and reciprocals of the values rather than the sum.) With a small set of values, calculating the average takes only a few simple steps. For example, let us imagine we want to find the average age among a group of five people. Their respective ages are 12, 22, 24, 27, and 35. First, we add up these values to find their sum: 12 22 24 27 35 120 Then we take this sum and divide it by the number of values (5): 120 Ã · 5 24 The result, 24, is the average age of the five individuals. Mean, Median, and Mode The average, or mean, is not the only measure of central tendency, though it is one of the most common. The other common measures are the median and the mode. The median is the middle value in a given set, or the value that separates the higher half from the lower half. In the example above, the median age among the five individuals is 24, the value that falls between the higher half (27, 35) and the lower half (12, 22). In the case of this data set, the median and the mean are the same, but that is not always the case. For example, if the youngest individual in the group were 7 instead of the 12, the average age would be 23. However, the median would still be 24. For statisticians, the median can be a very useful measure, especially when a data set contains outliers, or values that greatly differ from the other values in the set. In the example above, all of the individuals are within 25 years of each other. But what if that were not the case? What if the oldest person were 85 instead of 35? That outlier would bring the average age up to 34, a value greater than 80 percent of the values in the set. Because of this outlier, the mathematical average is no longer a good representation of the ages in the group. The median of 24 is a much better measure. The mode is the most frequent value in a data set, or the one that is most likely to appear in a statistical sample. In the example above, there is no mode since each individual value is unique. In a larger sample of people, though, there would likely be multiple individuals of the same age, and the most common age would be the mode. Weighted Average In an ordinary average, each value in a given data set is treated equally. In other words, each value contributes as much as the others to the final average. In a weighted average, however, some values have a greater effect on the final average than others. For example, imagine a stock portfolio made up of three different stocks: Stock A, Stock B, and Stock C. Over the last year, Stock As value grew 10 percent, Stock Bs value grew 15 percent, and Stock Cs value grew 25 percent. We can calculate the average percent growth by adding up these values and dividing them by three. But that would only tell us the overall growth of the portfolio if the owner held equal amounts of Stock A, Stock B, and Stock C. Most portfolios, of course, contain a mix of different stocks, some making up a larger percentages of the portfolio than others. To find the overall growth of the portfolio, then, we need to calculate a weighted average based on how much of each stock is held in the portfolio. For the sake of example, well say that Stock A makes up 20 percent of the portfolio, Stock B makes up 10 percent, and Stock C makes up 70 percent. We weight each growth value by multiplying it by its percentage of the portfolio: Stock A 10 percent growth x 20 percent of portfolio 200Stock B 15 percent growth x 10 percent of portfolio 150Stock C 25 percent growth x 70 percent of portfolio 1750 Then we add up these weighted values and divide them by the sum of the portfolio percentage values: (200 150 1750) Ã · (20 10 70) 21 The result, 21 percent, represents the overall growth of the portfolio. Note that it is higher than the average of the three growth values alone—16.67—which makes sense given that the highest performing stock also makes up the lions share of the portfolio.