Midhinge

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In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.

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The midhinge is related to the interquartile range (IQR), the difference of the third and first quartiles (i.e. ), which is a measure of statistical dispersion. The two are complementary in sense that if one knows the midhinge and the IQR, one can find the first and third quartiles.

The use of the term "hinge" for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis in the late 1970s, [1] and "midhinge" is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the trimean (), which originated in the same context and equals the average of the median () and the midhinge.

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Interquartile range measure of statistical dispersion

In descriptive statistics, the interquartile range (IQR), also called the midspread, middle 50%, or H‑spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q3 − Q1. In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale.

Median middle quantile of a data set or probability distribution

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In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Skewness measure of the asymmetry of random variables

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The interquartile mean (IQM) is a statistical measure of central tendency based on the truncated mean of the interquartile range. The IQM is very similar to the scoring method used in sports that are evaluated by a panel of judges: discard the lowest and the highest scores; calculate the mean value of the remaining scores.

Box plot method for graphically depicting groups of numerical data through their quartiles

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  1. the sample minimum
  2. the lower quartile or first quartile
  3. the median
  4. the upper quartile or third quartile
  5. the sample maximum

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Important note: in some papers, MAE Mean absolute error is often abbreviated as MAD .

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In statistics, the mid-range or mid-extreme of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set, defined as:

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values predicted by a model or an estimator and the values observed. The RMSD represents the square root of the second sample moment of the differences between predicted values and observed values or the quadratic mean of these differences. These deviations are called residuals when the calculations are performed over the data sample that was used for estimation and are called errors when computed out-of-sample. The RMSD serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. RMSD is a measure of accuracy, to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent.

In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles:

The sample mean or empirical mean and the sample covariance are statistics computed from a collection of data on one or more random variables. The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken.

In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.

L-estimator

In statistics, an L-estimator is an estimator which is an L-statistic – a linear combination of order statistics of the measurements. This can be as little as a single point, as in the median, or as many as all points, as in the mean.

In statistics, a trimmed estimator is an estimator derived from another estimator by excluding some of the extreme values, a process called truncation. This is generally done to obtain a more robust statistic, and the extreme values are considered outliers. Trimmed estimators also often have higher efficiency for mixture distributions and heavy-tailed distributions than the corresponding untrimmed estimator, at the cost of lower efficiency for other distributions, such as the normal distribution.

In statistics, a robust measure of scale is a robust statistic that quantifies the statistical dispersion in a set of numerical data. The most common such statistics are the interquartile range (IQR) and the median absolute deviation (MAD). These are contrasted with conventional measures of scale, such as sample variance or sample standard deviation, which are non-robust, meaning greatly influenced by outliers.

References

  1. Tukey, J. W. (1977) Exploratory Data Analysis, Addison-Wesley. ISBN   0-201-07616-0