Geometric mean in the context of "Averages"

In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk/ arr-ith-MET-ik), arithmetic average, or just the mean or average is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such as geometric and harmonic.

Arithmetic means are also frequently used in economics, anthropology, history, and almost every other academic field to some extent. For example, per capita income is the arithmetic average of the income of a nation's population.

An average of a collection or group is a value that is most central or most common in some sense, and represents its overall position.

In mathematics, especially in colloquial usage, it most commonly refers to the arithmetic mean, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5. In situations where the data is skewed or has outliers, and it is desired to focus on the main part of the group rather than the long tail, "average" often instead refers to the median; for example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people. In certain real-world scenarios, such computing the average speed from multiple measurements taken over the same distance, the average used is the harmonic mean. In situations where a histogram or probability density function is being referenced, the "average" could instead refer to the mode. Other statistics that can be used as an average include the mid-range and geometric mean, but they would rarely, if ever, be colloquially referred to as "the average".

In probability theory and statistics, covariance is a measure of the joint variability of two random variables.

The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. One feature of covariance is that it has units of measurement and the magnitude of the covariance is affected by said units. This means changing the units (e.g., from meters to millimeters) changes the covariance value proportionally, making it difficult to assess the strength of the relationship from the covariance alone; In some situations, it is desirable to compare the strength of the joint association between different pairs of random variables that do not necessarily have the same units. In those situations, we use the correlation coefficient, which normalizes the covariance by dividing by the geometric mean of the total variances (i.e., the product of the standard deviations) for the two random variables to get a result between -1 and 1 and makes the units irrelevant.

This article provides a list of data to rank the largest refugee crises in modern history by the event(s) that caused them. Only those events that resulted in the creation of at least one million refugees—not including internally displaced persons—are listed below.

For events for which estimates vary, the geometric mean of the lowest and highest estimates is calculated in order to rank them. As the dates for some events are disputed, the provided data only covers the time since or between the listed years.

Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable. [There is also a method for predicting the conditional geometric mean of the response variable, .] Quantile regression is an extension of linear regression used when the conditions of linear regression are not met .It was introduced by Roger Koenker in 1978. As a complementary and extended approach to the least squares method, quantile regression addresses the limitations of least squares method in the presence of heteroscedasticity and ensures the robustness of quantile regression through its robustness to outliers, which compensates for the weakness of least squares method in dealing with outlier data.