Where does magnitude collide?

The magnitude of a star or other celestial object is a number on a logarithmic scale that grades how bright it appears, arranged by an old convention so that the brightest objects carry the smallest numbers, a ranking that descends from Hipparchus. Pogson's 1856 Magnitudes of Thirty-six of the Minor Planets fixed the modern scale by defining a difference of five magnitudes as a factor of exactly one hundred in brightness, making each step about 2.512 times. Apparent magnitude records how bright an object looks from Earth, while absolute magnitude records how bright it would look at a fixed standard distance. This referent is a graded number for brightness on the sky, and it is not the length of a vector, the energy of an earthquake, or the size of a measured effect.[1]

Unit

A number on the magnitude scale, where five steps is a factor of one hundred in brightness

Recovery means

Not applicable

Measured by

Read from the light received through calibrated photometry in a defined band, then placed on the scale against standard stars

The magnitude of an earthquake is a number on a logarithmic scale that grades the size of the event, standing in for the energy released as a fault slips, so that each whole step up means a large multiplication of energy. Richter introduced a local magnitude from recorded ground motion in 1935, and Hanks and Kanamori's 1979 A moment magnitude scale tied the modern moment magnitude to the physical seismic moment of the rupture, giving a measure that does not level off for the largest quakes. This referent is a graded number for the strength of a ground-shaking event, and although it too runs on a logarithmic scale, it is unrelated to the brightness of a star, the length of a vector, or the size of a statistical effect.[2][3]

Unit

A number on the magnitude scale, where each whole step is a large jump in released energy

Recovery means

Not applicable

Measured by

Local magnitude read from the largest ground motion on a seismogram; moment magnitude computed from the seismic moment of the rupture

In mathematics the magnitude of a quantity is its size taken apart from its direction or its sign: the magnitude of a number is its absolute value, and the magnitude of a vector is its length, the non-negative number given by its norm. Axler's 2015 Linear Algebra Done Right develops this as the norm on an inner product space, where the magnitude of a vector is the square root of the vector's inner product with itself, generalising the ordinary length of an arrow. The related phrase order of magnitude compares two quantities by roughly how many powers of ten separate them. This referent is an abstract measure of size for a mathematical object and carries no physical content: it is neither a brightness, nor an earthquake's energy, nor a statistical effect.[4]

Unit

A non-negative number giving the length of a vector or the absolute value of a number

Recovery means

Not applicable

Measured by

Computed from the object itself, as an absolute value for a number or as a norm, the root of the inner product, for a vector

In empirical research the magnitude of an effect, its effect size, is a number saying how large a difference or a relationship is, kept separate from whether that difference is statistically significant. Cohen's 1992 A power primer catalogued standard measures such as the standardised mean difference d and the correlation r, with rough labels for small, medium, and large, so that researchers could weigh how big an effect is rather than only whether it differs from zero. A tiny effect can be highly significant in a large sample, and a large effect can miss significance in a small one, which is why the magnitude is reported alongside the p value. This referent is the estimated size of an effect in data, and it has nothing to do with a star's brightness, a quake's energy, or a vector's length.[5]

Unit

A standardised number for how large an effect is, such as Cohen's d or a correlation r

Recovery means

Not applicable

Measured by

Estimated from sample data as a standardised mean difference, a correlation, or a similar ratio, apart from the sample size