Range (statistics)

In arithmetic, the range of a set of data is the difference between the largest and smallest values.

However, in descriptive statistics, this concept of range has a more complex meaning. The range is the size of the smallest interval which contains all the data and provides an indication of statistical dispersion. It is measured in the same units as the data. Since it only depends on two of the observations, it is most useful in representing the dispersion of small data sets.

Independent identically distributed continuous random variables

For n independent and identically distributed continuous random variables X₁, X₂, ..., X_n with cumulative distribution function G(x) and probability density function g(x) the range of the X_i is the range of a sample of size n from a population with distribution function G(x).

Distribution

The range has cumulative distribution function

Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express G(x + t) by G(x), and that the numerical integration is lengthy and tiresome."

Range (music)

In music, the range of a musical instrument is the distance from the lowest to the highest pitch it can play. For a singing voice, the equivalent is vocal range. The range of a musical part is the distance between its lowest and highest note.

The terms sounding range, written range, designated range, duration range and dynamic range have specific meanings.

The sounding range refers to the pitches produced by an instrument, while the written range refers to the compass (span) of notes written in the sheet music, where the part is sometimes transposed for convenience. A piccolo, for example, typically has a sounding range one octave higher than its written range. The designated range is the set of notes the player should or can achieve while playing. All instruments have a designated range, and all pitched instruments have a playing range. Timbre, dynamics, and duration ranges are interrelated and one may achieve registral range at the expense of timbre. The designated range is thus the range in which a player is expected to have comfortable control of all aspects.

Range of a projectile

In physics, assuming a flat Earth with a uniform gravity field, and no air resistance, a projectile launched with specific initial conditions will have a predictable range.

The following applies for ranges which are small compared to the size of the Earth. For longer ranges see sub-orbital spaceflight. The maximum horizontal distance traveled by the projectile

When neglecting air resistance, the range of a projectile will be

If (y₀) is taken to be zero, meaning the object is being launched on flat ground, the range of the projectile will then simplify to

Ideal projectile motion

Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. This assumption simplifies the mathematics greatly, and is a close approximation of actual projectile motion in cases where the distances travelled are small. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance.

Gas

Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or compound molecules made from a variety of atoms (e.g. carbon dioxide). A gas mixture would contain a variety of pure gases much like the air. What distinguishes a gas from liquids and solids is the vast separation of the individual gas particles. This separation usually makes a colorless gas invisible to the human observer. The interaction of gas particles in the presence of electric and gravitational fields are considered negligible as indicated by the constant velocity vectors in the image. One type of commonly known gas is steam.

The gaseous state of matter is found between the liquid and plasma states, the latter of which provides the upper temperature boundary for gases. Bounding the lower end of the temperature scale lie degenerative quantum gases which are gaining increasing attention. High-density atomic gases super cooled to incredibly low temperatures are classified by their statistical behavior as either a Bose gas or a Fermi gas. For a comprehensive listing of these exotic states of matter see list of states of matter.

Gas (musician)

Gas is a music project of Wolfgang Voigt (born 1961), a Cologne, Germany-based electronic musician. Voigt cites his youthful LSD experiences in the Königsforst, a German forest situated near his hometown of Köln, as the inspiration behind his work under the name Gas. He has claimed that the intention of the project is to "bring the forest to the disco, or vice-versa".

Aliases and projects

Voigt is known for his numerous, nearly inexhaustible list of one-off projects and aliases. Of these, his best known is arguably Gas, a project that saw the marriage of ambient music and 4/4 techno.

Other names under which Voigt has released music include, but are not limited to, All, Auftrieb, Brom, C.K. Decker, Centrifugal Force, Crocker, Dextro NRG, Dieter Gorny, Digital, Dom, Doppel, Filter, Freiland, Fuchsbau, Gelb, Grungerman, Kafkatrax, Love Inc., M:I:5, Mike Ink, Mint, Panthel, Popacid, Riss, RX7, Split Inc., Strass, Studio 1, Tal, Vinyl Countdown, W.V., Wassermann, and X-Lvis.

Characteristics

Gas (album)

Gas is the debut album by Wolfgang Voigt's Gas project. It is the second release under the Gas name, preceded only by the Modern EP. It was released on 29 November, 1996 on the Mille Plateaux label. Like all Gas albums, the tracks are untitled. Along with Modern, it is unusual in that the artwork does not share the unifying forest theme common to all other Gas releases.

Track listing