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.
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk, as an estimate of the true MSE.
Forecasting is the process of making predictions based on past and present data. Later these can be compared (resolved) against what happens. For example, a company might estimate their revenue in the next year, then compare it against the actual results creating a variance actual analysis. Prediction is a similar but more general term. Forecasting might refer to specific formal statistical methods employing time series, cross-sectional or longitudinal data, or alternatively to less formal judgmental methods or the process of prediction and resolution itself. Usage can vary between areas of application: for example, in hydrology the terms "forecast" and "forecasting" are sometimes reserved for estimates of values at certain specific future times, while the term "prediction" is used for more general estimates, such as the number of times floods will occur over a long period.
In the military science of ballistics, circular error probable (CEP) is a measure of a weapon system's precision. It is defined as the radius of a circle, centered on the mean, whose perimeter is expected to enclose the landing points of 50% of the rounds; said otherwise, it is the median error radius. That is, if a given munitions design has a CEP of 100 m, when 100 munitions are targeted at the same point, 50 will fall within a circle with a radius of 100 m around their average impact point.
The average absolute deviation (AAD) of a data set is the average of the absolute deviations from a central point. It is a summary statistic of statistical dispersion or variability. In the general form, the central point can be a mean, median, mode, or the result of any other measure of central tendency or any reference value related to the given data set. AAD includes the mean absolute deviation and the median absolute deviation.
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value". The error of an observation is the deviation of the observed value from the true value of a quantity of interest. The residual is the difference between the observed value and the estimated value of the quantity of interest. The distinction is most important in regression analysis, where the concepts are sometimes called the regression errors and regression residuals and where they lead to the concept of studentized residuals. In econometrics, "errors" are also called disturbances.
The standard error (SE) of a statistic is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM). The standard error is a key ingredient in producing confidence intervals.
In probability theory and statistics, the coefficient of variation (CV), also known as Normalized Root-Mean-Square Deviation (NRMSD), Percent RMS, and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation to the mean , and often expressed as a percentage ("%RSD"). The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R, by economists and investors in economic models, and in psychology/neuroscience.
The mean absolute percentage error (MAPE), also known as mean absolute percentage deviation (MAPD), is a measure of prediction accuracy of a forecasting method in statistics. It usually expresses the accuracy as a ratio defined by the formula:
The mean absolute difference (univariate) is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean absolute difference, which is the mean absolute difference divided by the arithmetic mean, and equal to twice the Gini coefficient. The mean absolute difference is also known as the absolute mean difference and the Gini mean difference (GMD). The mean absolute difference is sometimes denoted by Δ or as MD.
In statistics, mean absolute error (MAE) is a measure of errors between paired observations expressing the same phenomenon. Examples of Y versus X include comparisons of predicted versus observed, subsequent time versus initial time, and one technique of measurement versus an alternative technique of measurement. MAE is calculated as the sum of absolute errors divided by the sample size:
The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is either one of two closely related and frequently used measures of the differences between true or predicted values on the one hand and observed values or an estimator on the other.
In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference. The magnitude of the value indicates the size of the difference.
In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a standard or reference or starting value. The comparison is expressed as a ratio and is a unitless number. By multiplying these ratios by 100 they can be expressed as percentages so the terms percentage change, percent(age) difference, or relative percentage difference are also commonly used. The terms "change" and "difference" are used interchangeably.
Demand forecasting refers to the process of predicting the quantity of goods and services that will be demanded by consumers at a future point in time. More specifically, the methods of demand forecasting entail using predictive analytics to estimate customer demand in consideration of key economic conditions. This is an important tool in optimizing business profitability through efficient supply chain management. Demand forecasting methods are divided into two major categories, qualitative and quantitative methods. Qualitative methods are based on expert opinion and information gathered from the field. This method is mostly used in situations when there is minimal data available for analysis such as when a business or product has recently been introduced to the market. Quantitative methods, however, use available data, and analytical tools in order to produce predictions. Demand forecasting may be used in resource allocation, inventory management, assessing future capacity requirements, or making decisions on whether to enter a new market.
Symmetric mean absolute percentage error is an accuracy measure based on percentage errors. It is usually defined as follows:
In statistics, the mean signed difference (MSD), also known as mean signed deviation and mean signed error, is a sample statistic that summarises how well a set of estimates match the quantities that they are supposed to estimate. It is one of a number of statistics that can be used to assess an estimation procedure, and it would often be used in conjunction with a sample version of the mean square error.
In statistics and management science, a tracking signal monitors any forecasts that have been made in comparison with actuals, and warns when there are unexpected departures of the outcomes from the forecasts. Forecasts can relate to sales, inventory, or anything pertaining to an organization's future demand.
In statistics, the mean absolute scaled error (MASE) is a measure of the accuracy of forecasts. It is the mean absolute error of the forecast values, divided by the mean absolute error of the in-sample one-step naive forecast. It was proposed in 2005 by statistician Rob J. Hyndman and Professor of Decision Sciences Anne B. Koehler, who described it as a "generally applicable measurement of forecast accuracy without the problems seen in the other measurements." The mean absolute scaled error has favorable properties when compared to other methods for calculating forecast errors, such as root-mean-square-deviation, and is therefore recommended for determining comparative accuracy of forecasts.
Mean directional accuracy (MDA), also known as mean direction accuracy, is a measure of prediction accuracy of a forecasting method in statistics. It compares the forecast direction to the actual realized direction. It is defined by the following formula:
