Quantile in the context of "Linear regression"

The median of a set of numbers is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as the “middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of the center. Median income, for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in robust statistics.

Median is a 2-quantile; it is the value that partitions a set into two equal parts.

In descriptive statistics, a decile is any of the nine values that divide the sorted data into ten equal parts, so that each part represents 1/10 of the sample or population. A decile is one possible form of a quantile; others include the quartile and percentile. A decile rank arranges the data in order from lowest to highest and is done on a scale of one to ten where each successive number corresponds to an increase of 10 percentage points.

In statistics, quartiles are a type of quantiles which divide the number of data points into four parts, or quarters, of more-or-less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are a form of order statistic. The three quartiles, resulting in four data divisions, are as follows:

Along with the minimum and maximum of the data (which are also quartiles), the three quartiles described above provide a five-number summary of the data. This summary is important in statistics because it provides information about both the center and the spread of the data. Knowing the lower and upper quartile provides information on how big the spread is and if the dataset is skewed toward one side. Since quartiles divide the number of data points evenly, the range is generally not the same between adjacent quartiles (i.e. usually (Q₃ - Q₂) ≠ (Q₂ - Q₁)). Interquartile range (IQR) is defined as the difference between the 75th and 25th percentiles or Q₃ - Q₁. While the maximum and minimum also show the spread of the data, the upper and lower quartiles can provide more detailed information on the location of specific data points, the presence of outliers in the data, and the difference in spread between the middle 50% of the data and the outer data points.

Income inequality has fluctuated considerably in the United States since measurements began around 1915, moving in an arc between peaks in the 1920s and 2000s, with a lower level of inequality from approximately 1950-1980 (a period named the Great Compression), followed by increasing inequality, in what has been coined as the great divergence.

The U.S. has the highest level of income inequality among its (post-industrialized) peers. When measured for all households, U.S. income inequality is comparable to other developed countries before taxes and transfers, but is among the highest after taxes and transfers, meaning the U.S. shifts relatively less income from higher income households to lower income households. In 2016, average market income was $15,600 for the lowest quintile and $280,300 for the highest quintile. The degree of inequality accelerated within the top quintile, with the top 1% at $1.8 million, approximately 30 times the $59,300 income of the middle quintile.

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable is a simple linear regression; a model with two or more explanatory variables is a multiple linear regression. This term is distinct from multivariate linear regression, which predicts multiple correlated dependent variables rather than a single dependent variable.

In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.