Measures are the processes of quantifying the length, size, capacity or quantity of substances. It’s essential in every day life.
The word’measure’ comes from the Greek’metron’, meaning limited proportion. It’s used in many disciplines including physics, chemistry, biology and engineering.
Measures are a set of basic concepts in mathematics and probability theory. These include length, area, volume, mass and other common notions.
Depending on its definition, measures can be generalized to assume negative values (as in electrical charge) or they can be treated as a unit in their own right. They are foundational to many analytical concepts in math and physics, including integration theory and probability theory.
Typically, the definition of a measure requires that the measure of a set is a countable additive function in a non-negative real or complex number. However, some authors replace the requirement that a set has an empty measure with an equivalent requirement that it contains a point of a finitely additive measure.
Measures are the processes of quantifying the length, size, capacity or quantity of substances. They are vital to our day-to-day lives.
In mathematics, measures are a natural concept derived from the ideas of length, area, probability and so on. In the mathematical development of this concept, measure is additive, that is, it assigns a non-negative number to subsets of a set following the mathematical nature of these concepts.
This is achieved in a natural way by requiring that the measure of two disjoint sets should be zero, and that the measure of an empty set should be also 0. Another generalization is a finitely additive measure, known as a content.
Professionals who work with data often depend on the accuracy of measurements. This is especially true for scientists, who use accurate measurements to make determinations and test theories.
Accuracy refers to the degree of closeness of a measured value to a standard or true value. Precision, on the other hand, describes the degree to which repeated measurements under identical conditions show the same results.
The two terms are usually described separately but they can be used together in colloquial language, although their meanings are not rigorously exchangeable. ISO 5725-1 defines accuracy as the closeness of the average of measurement results to an actual (true) value and precision as the closeness of agreement between the individual results.
Variability is the extent to which data points diverge from their average value and/or the degree to which these data points differ from each other. It is an important measure in statistical and financial analysis.
Variables can be significant factors in investment returns, especially for risky investments. Professional investors equate a high variability of returns to a greater degree of risk.
There are three main measures of variability. They are standard deviation, variance, and range. The standard deviation is the average squared difference of each data point from the mean.
Uncertainty is the lack of absolute certainty about a quantity. The uncertainty can be measured in terms of a number like the standard error, or it can be expressed in terms of a probability distribution.
A measurement result is not always precise enough, and scientists have to account for the uncertainties that come with every observation they make. The resulting range of possible values is often expressed in bars called error bars or confidence intervals.
This process, which is known as error propagation, allows scientists to determine the uncertainty range of a measurement. It works by adding the various experimental uncertainties together and calculating their combined effect.