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Do you really understand average? What is the mysterious difference between arithmetic mean and geometric mean?
In everyday language, an average is a single number or value that best represents a set of data. The average most often considered to be the most representative of a list of numbers is the arithmetic mean, which is the sum of all the numbers divided by the number of numbers. For example, the arithmetic mean of the numbers 2, 3, 4, 7, and 9 is 5. Depending on the context, the most representative statistic may be some other measure of central tendency, such as the median or the geometric mean.
In some cases, such as the average of personal income, the median is often used because this prevents the income of a few wealthy people from dragging up the overall arithmetic mean.
One of the universal properties of averages is that if all the numbers in a set of numbers are the same, then their average will also be equal to that number. This property is shared by all types of averages. Another general property is monotonicity: if two sets of numbers, A and B, are the same length, and each number in A is at least as large as the corresponding number in B, then the mean of A will be at least as large as that of B.
In addition, all averages satisfy the linear homogeneity property: if a group of numbers are multiplied by the same positive number, then their averages will change in the same proportion. For some types of weighted averages, such as the weighted arithmetic mean or the weighted geometric mean, the items in the list of numbers are given different weights before the average is calculated. Most average types are permutation insensitive, meaning that all numbers are treated equally when computing their average, regardless of their position in the list.
Differences between arithmetic mean, geometric mean and harmonic mean
The arithmetic mean, geometric mean, and harmonic mean are collectively called the Pythagorean mean. In addition to these means, the mode and median are also often used to estimate the central tendency.
The mode is the most common number in a list, while the median is the number in the middle after sorting the numbers.
For example, in the list of numbers 1, 2, 2, 3, 3, 3, 4, the pattern is 3, while the sorted list 1, 3, 7, 13 has the arithmetic sum of 3 and 7. On average, that is 5.
Different Types of Averaging
Although other types of averages such as τ-th quantiles are not necessarily averages, they can be viewed as solutions to optimization problems. More complex averages include triple means, triple medians, and standardized mean.
In finance, the average percentage return is a special type of averaging that is essentially an application of the geometric mean. When the returns are annual, the metric is called the compound annual growth rate (CAGR). For example, if you experienced a -10% return on your investment in the first year and a +60% return in the second year, you can find the CAGR by solving the equation for the total return.
Moving Average
Moving averages are a common tool in analyzing data and are used to smooth out time series, such as daily stock market prices or years of temperature. People usually choose a value for n and then create a new series by taking the arithmetic average of the first n values and then moving on to the next position to create a smoothed data series.
The simple form of moving average is to take the arithmetic mean, but more complex forms involve weighted averaging to enhance or dampen different cyclical behaviors.
Conclusion
Understanding the different types of averages and where they apply is the cornerstone of mastering data analysis. Are people fully aware of which average they are using when analyzing and interpreting data?
