Fraction (mathematics)

Another kind of fraction is the percentage (Latin per centum meaning "per hundred", represented by the symbol %), in which the implied denominator is always 100. Thus, 51% means 51/100. Percentages greater than 100 or less than zero are treated in the same way, e.g. 311% equals 311/100, and −27% equals −27/100. The related concept of permille or parts per thousand (ppt) has an implied denominator of 1000, while the more general parts-per notation, as in 75 parts per million (ppm), means that the proportion is 75/1,000,000. Whether common fractions or decimal fractions are used is often a matter of taste and context.


The word mathematics comes from Ancient Greek μάθημα (máthēma), meaning "that which is learnt", "what one gets to know", hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), ars mathematica, meant "the mathematical art".

Absolute difference

ordinary distance metric
The absolute difference is used to define other quantities including the relative difference, the L 1 norm used in taxicab geometry, and graceful labelings in graph theory. When it is desirable to avoid the absolute value function – for example because it is expensive to compute, or because its derivative is not continuous – it can sometimes be eliminated by the identity This follows since |x − y| 2 = (x − y) 2 and squaring is monotonic on the nonnegative reals. * Absolute deviation |x − y| ≥ 0, since absolute value is always non-negative. |x − y| = 0 if and only if x = y. |x − y| = |y − x| (symmetry or commutativity).


ratiosproportionratio analysis
It is usual either to reduce terms to the lowest common denominator, or to express them in parts per hundred (percent). If a mixture contains substances A, B, C and D in the ratio 5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total and multiply by 100, we have converted to percentages: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25:45:20:10).

Percentage point

pppercentage-pointpercentage points
Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation). Percentage 1 part in 100. Per mille 1 part in 1,000. Basis point 1 part in 10,000. Parts-per notation. Baker percentage. Percent point function.

Errors and residuals

residualserror termerror
Likewise, the sum of absolute errors (SAE) refers to the sum of the absolute values of the residuals, which is minimized in the least absolute deviations approach to regression. The difference between the height of each man in the sample and the unobservable population mean is a statistical error, whereas. The difference between the height of each man in the sample and the observable sample mean is a residual. Absolute deviation. Consensus forecasts. Error detection and correction. Explained sum of squares. Innovation (signal processing). Lack-of-fit sum of squares. Margin of error. Mean absolute error. Observational error. Propagation of error. Probable error. Regression dilution.

Approximation error

relative errorabsolute errorerror
One commonly distinguishes between the relative error and the absolute error. Given some value v and its approximation v approx, the absolute error is where the vertical bars denote the absolute value. If v \ne 0, the relative error is and the percent error is In words, the absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100. These definitions can be extended to the case when v and are n-dimensional vectors, by replacing the absolute value with an n-norm.


Changes in percentages can be reported in at least two forms, percentage change and percentage point change. Percentage change represents the relative change between the two quantities as a percentage, while percentage point change is simply the number obtained by subtracting the two percentages. As an example, suppose that 30% of widgets made in a factory are defective. Six months later, 20% of widgets are defective. The percentage change is −33 1⁄3%, while the percentage point change is −10 percentage points. The method of complements is a technique used to subtract one number from another using only addition of positive numbers.

Dimensionless quantity

dimensionlessdimensionless numberdimensionless quantities
Other common proportions are percentages % (= 0.01), ‰ (= 0.001) and angle units such as radians, degrees (°= undefined⁄180) and grads(= undefined⁄200). In statistics the coefficient of variation is the ratio of the standard deviation to the mean and is used to measure the dispersion in the data. The Buckingham theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (e. g., pressure and volume are linked by Boyle's Law – they are inversely proportional).


number systemnumericalnumeric
Two different fractions may correspond to the same rational number; for example 1⁄2 and 2⁄4 are equal, that is: In general, : if and only if If the absolute value of m is greater than n (supposed to be positive), then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. For example −7 can be written −7⁄1. The symbol for the rational numbers is Q (for quotient), also written \mathbb{Q}. The symbol for the real numbers is R, also written as \mathbb{R}.


skewedskewskewed distribution
This means that often samples from a symmetric distribution (like the uniform distribution) have a large quantile-based skewness, just by chance. Bowley's measure of skewness is γ(u) evaluated at u = 3/4. Kelley's measure of skewness uses u = 0.1. Groeneveld & Meeden have suggested, as an alternative measure of skewness, where μ is the mean, ν is the median, |…| is the absolute value, and E is the expectation operator. This is closely related in form to Pearson's second skewness coefficient. Use of L-moments in place of moments provides a measure of skewness known as the L-skewness. A value of skewness equal to zero does not imply that the probability distribution is symmetric.

Bias of an estimator

unbiasedunbiased estimatorbias
Any minimum-variance mean-unbiased estimator minimizes the risk (expected loss) with respect to the squared-error loss function (among mean-unbiased estimators), as observed by Gauss. A minimum-average absolute deviation median-unbiased estimator minimizes the risk with respect to the absolute loss function (among median-unbiased estimators), as observed by Laplace. Other loss functions are used in statistics, particularly in robust statistics. As stated above, for univariate parameters, median-unbiased estimators remain median-unbiased under transformations that preserve order (or reverse order).

Statistical dispersion

Median absolute deviation (MAD). Average absolute deviation (or simply called average deviation). Distance standard deviation. Coefficient of variation. Quartile coefficient of dispersion. Relative mean difference, equal to twice the Gini coefficient. Entropy: While the entropy of a discrete variable is location-invariant and scale-independent, and therefore not a measure of dispersion in the above sense, the entropy of a continuous variable is location invariant and additive in scale: If Hz is the entropy of continuous variable z and y=ax+b, then Hy=Hx+log(a). Variance (the square of the standard deviation) – location-invariant but not linear in scale.

Coefficient of variation

CVrelative standard deviationcoefficients of variation
It is often expressed as a percentage, and is defined as the ratio of the standard deviation \ \sigma to the mean \ \mu (or its absolute value, | \mu |). 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. In addition, CV is utilized by economists and investors in economic models and in determining the volatility of a security.

International System of Units

SISI unitsSI unit
A temperature difference of one kelvin is the same as one degree Celsius: 1/100 of the temperature differential between the freezing and boiling points of water at sea level; the absolute temperature in kelvins is the temperature in degrees Celsius plus about 273; human body temperature is about 37 °C or 310 K. A 60 W incandescent light bulb consumes 0.5 amperes at 120 V (US mains voltage) and about 0.26 amperes at 230 V (European mains voltage). The value of a quantity is written as a number followed by a space (representing a multiplication sign) and a unit symbol; e.g., 2.21 kg, 7.3 m 2, 22 K.

Average absolute deviation

mean absolute deviationmean deviationMAD
L ∞ norm statistics: the mid-range minimizes the maximum absolute deviation. trimmed L ∞ norm statistics: for example, the midhinge (average of first and third quartiles) which minimizes the median absolute deviation of the whole distribution, also minimizes the maximum absolute deviation of the distribution after the top and bottom 25% have been trimmed off.. Deviation (statistics). Mean absolute error. Errors and residuals in statistics. Least absolute deviations. Loss function. Mean absolute percentage error. Mean difference. Mean squared error. Median absolute deviation. Squared deviations.

Mean absolute difference

MDMean differenceaverage absolute difference
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 (not to be confused with the absolute value of the mean signed difference) and the Gini mean difference (GMD). The mean absolute difference is sometimes denoted by Δ or as MD.

List of mathematical symbols by subject

Interpretation: A short textual description of the meaning of the formula in the previous column. Article: The Wikipedia article that discusses the meaning (semantics) of the symbol. LaTeX: The LaTeX command that creates the icon. Characters from the ASCII character set can be used directly, with a few exceptions (pound sign #, backslash \, braces {}, and percent sign %). High-and low-position is indicated via the characters ^ and _ and is not explicitly specified. HTML: The icon in HTML, if it is defined as a named mark. Non-named characters can be indicated in the form can &#xnnnn by specifying the Unicode code point of the next column.

Fourier transform

Fouriercontinuous Fourier transformuncertainty principle
Dirichlet-Dini theorem), the value of at is taken to be the arithmetic mean of the left and right limits, and provided that the integrals are taken in the sense of Cauchy principal values. L 2 versions of these inversion formulas are also available. The Fourier transform can be defined in any arbitrary number of dimensions. As with the one-dimensional case, there are many conventions. For an integrable function f (x), this article takes the definition: where x and are -dimensional vectors, and x · ξ is the dot product of the vectors.

Symmetric derivative

symmetric difference quotientsymmetric differentiability
A well-known counterexample is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. Neither Rolle's theorem nor the mean value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.

Correlation and dependence

The population correlation coefficient \rho_{X,Y} between two random variables X and Y with expected values \mu_X and \mu_Y and standard deviations \sigma_X and \sigma_Y is defined as where is the expected value operator, means covariance, and is a widely used alternative notation for the correlation coefficient. The Pearson correlation is defined only if both of the standard deviations are finite and nonzero. The correlation coefficient is symmetric:. It is a corollary of the Cauchy–Schwarz inequality that the correlation cannot exceed 1 in absolute value.

Ramp function

rampramp depolarisation
Possible definitions are: * A piecewise function: * The max function: * The mean of an independent variable and its absolute value (a straight line with unity gradient and its modulus): max(a,b) a = x and b = 0 * The Heaviside step function multiplied by a straight line with unity gradient: * The convolution of the Heaviside step function with itself: * The integral of the Heaviside step function: * Macaulay brackets: The ramp function has numerous applications in engineering, e.g., in the theory of DSP. In finance, the payoff of a call option is a ramp (shifted by strike price).


Mathematically, this corresponds to the absolute value function. Full-wave rectification converts both polarities of the input waveform to pulsating DC (direct current), and yields a higher average output voltage. Two diodes and a center tapped transformer, or four diodes in a bridge configuration and any AC source (including a transformer without center tap), are needed. Single semiconductor diodes, double diodes with common cathode or common anode, and four-diode bridges, are manufactured as single components.

Price elasticity of demand

price elasticityinelasticinelastic demand
More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price. Price elasticities are almost always negative, although analysts tend to ignore the sign even though this can lead to ambiguity. Only goods which do not conform to the law of demand, such as Veblen and Giffen goods, have a positive PED. In general, the demand for a good is said to be inelastic (or relatively inelastic) when the PED is less than one (in absolute value): that is, changes in price have a relatively small effect on the quantity of the good demanded. The demand for a good is said to be elastic (or relatively elastic) when its PED is greater than one.


left and right derivativederivative from the rightleft
One can also define a symmetric derivative, which equals the arithmetic mean of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not. If a real-valued, differentiable function f, defined on an interval I of the real line, has zero derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity and one-sided differentiability of f. The version for right differentiable functions is given below, the version for left differentiable functions is analogous.