Mode (statistics) in the context of "Averages"

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is$${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}$$

The parameter ⁠${\displaystyle \mu }$⁠ is the mean or expectation of the distribution (and also its median and mode), while the parameter ${\textstyle \sigma ^{2}}$ is the variance. The standard deviation of the distribution is ⁠${\displaystyle \sigma }$⁠ (sigma). A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words. While the absolute likelihood for a continuous random variable to take on any particular value is zero, given there is an infinite set of possible values to begin with. Therefore, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.

More precisely, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of a continuous variable's PDF over that range, where the integral is the nonnegative area under the density function between the lowest and greatest values of the range. The PDF is nonnegative everywhere, and the area under the entire curve is equal to one, such that the probability of the random variable falling within the set of possible values is 100%.

A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features from a collection of information, while descriptive statistics (in the mass noun sense) is the process of using and analysing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics) by its aim to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. This generally means that descriptive statistics, unlike inferential statistics, is not developed on the basis of probability theory, and are frequently nonparametric statistics. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example, in papers reporting on human subjects, typically a table is included giving the overall sample size, sample sizes in important subgroups (e.g., for each treatment or exposure group), and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, the proportion of subjects with related co-morbidities, etc.

Some measures that are commonly used to describe a data set are measures of central tendency and measures of variability or dispersion. Measures of central tendency include the mean, median and mode, while measures of variability include the standard deviation (or variance), the minimum and maximum values of the variables, kurtosis and skewness.

The evolution of biological complexity is one important outcome of the process of evolution. Evolution has produced some remarkably complex organisms – although the actual level of complexity is very hard to define or measure accurately in biology, with properties such as gene content, the number of cell types or morphology all proposed as possible metrics.

Many biologists used to believe that evolution was progressive (orthogenesis) and had a direction that led towards so-called "higher organisms", despite a lack of evidence for this viewpoint. This idea of "progression" introduced the terms "high animals" and "low animals" in evolution. Many now regard this as misleading, with natural selection having no intrinsic direction and that organisms selected for either increased or decreased complexity in response to local environmental conditions. Although there has been an increase in the maximum level of complexity over the history of life, there has always been a large majority of small and simple organisms and the most common level of complexity appears to have remained relatively constant.

An average of a collection or group is a value that is most central or most common in some sense, and represents its overall position.

In mathematics, especially in colloquial usage, it most commonly refers to the arithmetic mean, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5. In situations where the data is skewed or has outliers, and it is desired to focus on the main part of the group rather than the long tail, "average" often instead refers to the median; for example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people. In certain real-world scenarios, such computing the average speed from multiple measurements taken over the same distance, the average used is the harmonic mean. In situations where a histogram or probability density function is being referenced, the "average" could instead refer to the mode. Other statistics that can be used as an average include the mid-range and geometric mean, but they would rarely, if ever, be colloquially referred to as "the average".

In statistics, a multimodal distribution is a probability distribution with more than one mode (i.e., more than one local peak of the distribution). These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal.

Otodus megalodon (/ˈmɛɡələdɒn/ MEG-əl-ə-don; meaning "big tooth"), commonly known as megalodon, is an extinct species of giant mackerel shark that lived approximately 23 to 3.6 million years ago (Mya), from the Early Miocene to the Early Pliocene epochs. This prehistoric fish was formerly thought to be a member of the family Lamnidae and a close relative of the great white shark (Carcharodon carcharias), but has been reclassified into the extinct family Otodontidae, which diverged from the great white shark during the Early Cretaceous.

While regarded as one of the largest and most powerful predators to have ever lived, megalodon is only known from fragmentary remains, and its appearance and maximum size are uncertain. Scientists have argued whether its body form was more stocky or elongated than the modern lamniform sharks. Maximum body length estimates between 14.2 and 24.3 metres (47 and 80 ft) based on various analyses have been proposed, though the modal lengths for individuals of all ontogenetic stages from juveniles to adults are estimated at 10.5 meters (34 ft). Their teeth were thick and robust, built for grabbing prey and breaking bone, and their large jaws could exert a bite force of up to 108,500 to 182,200 newtons (24,390 to 40,960 lbf).

Size is an important aspect of dinosaur paleontology, of interest to both the general public and professional scientists. Dinosaurs show some of the most extreme variations in size of any land animal group, ranging from tiny hummingbirds, which can weigh as little as two grams, to the extinct titanosaurs, such as Argentinosaurus and Bruhathkayosaurus which could weigh as much as 50–130 t (55–143 short tons).

The latest evidence suggests that dinosaurs' average size varied through the Triassic, early Jurassic, late Jurassic and Cretaceous periods, and dinosaurs probably only became widespread during the early or mid Jurassic. Predatory theropod dinosaurs, which occupied most terrestrial carnivore niches during the Mesozoic, most often fall into the 100–1,000 kg (220–2,200 lb) category when sorted by estimated weight into categories based on order of magnitude, whereas recent predatory carnivoran mammals peak in the range of 10–100 kg (22–220 lb). The mode of Mesozoic dinosaur body masses is between one and ten metric tonnes. This contrasts sharply with the size of Cenozoic mammals, estimated by the National Museum of Natural History as about 2 to 5 kg (4.4 to 11.0 lb).

In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.

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 (both abbreviated as MAD).