Statistische Auffälligkeiten

Wann eine Angabe für ihre Art ungewöhnlich ist: ein robuster Wert auf Logarithmen gegen eine Familie oder eine Gruppe gleichartiger Waffen, die beurteilten Angaben und die ausgelassenen, und was ein Datensatz zu seiner Erklärung angeben kann.

Der robuste Wert

Each figure is compared with a group of records that should measure alike. The distance is taken between logarithms, so a barrel twice its group's median and one half of it are the same distance off, and it is counted in spreads of the group itself.

z = \frac{| \ln x - \tilde{m} |}{\max (1.4826 \cdot MAD, \ln 1.1)}

\tilde{m} = med (\ln x_{i}), MAD = med (| \ln x_{i} - \tilde{m} |)

z > 3.5 \land \max (\frac{x}{\tilde{x}}, \frac{\tilde{x}}{x}) \geq 1.5

wobei

$x$: the record's figure
$x_{i}$: the figure of every record in the group, the record's own included
$\tilde{m}$: the median of the group's logarithms
$\tilde{x}$: the group's median in the figure's own unit
$MAD$: the median absolute deviation; times 1.4826 it reads as a standard deviation would on a normal distribution
$\ln 1.1$: the least spread a group is given, 0.0953 on the logarithmic scale: a tenth either way

A figure is unusual when its score passes 3.5 and it is at least 1.5 times the median, or the median over 1.5 at most. The score alone would raise a 12% difference in a family whose members agree to the millimetre; the ratio alone would raise every carbine in a family of rifles. Together they ask for a figure far off in its own group's terms and far off in plain terms. A cut of 3.5 is the one conventionally used with a score built on the median absolute deviation (Iglewicz and Hoaglin, 1993).

Warum der Median und nicht der Mittelwert

A mean and a standard deviation are moved by the figure being judged. One barrel typed in inches drags a family's mean towards itself and widens its standard deviation until the error looks ordinary. A median does not move, and a median absolute deviation barely does, so the figure under test cannot hide itself.
Length, weight and energy vary by proportion rather than by amount: 50 mm is a large difference on a pistol and nothing on a machine gun. On logarithms a group's spread is a proportion, so one rule reads a derringer and an anti-materiel rifle alike, and doubling and halving are the same distance.
A group whose members share one figure has a spread of zero, and any difference would score as infinite. The spread is never taken as less than a tenth either way.

Die Gruppen

Family: every record that traces back, through the gun it is filed as a variant of, to the same original. A family needs 4 records with the figure before its median is used. Velocity and energy belong to the cartridge rather than the gun, so within a family they are compared only among the members that share one.
Cohort: every gun of the same category, the same primary cartridge and the same action. A cohort needs 8 records with the figure. It is the wider comparison, for guns with no family or a family too small.
A figure unusual against both its family and its cohort is one finding, under the family, the closer comparison. A figure another check already questions (a variant far from its family, a figure in the wrong unit) is left to that check, so one figure is one task.

Die beurteilten Angaben

Which figures are judged, and against which group. A figure that differs between models for good reasons is compared only within a family, where the models share one design.

Figure	Family	Cohort	Why
Barrel length	Yes	Yes	Set by the design and the role; like guns agree closely.
Overall length	Yes	Yes	Follows the barrel and the stock; like guns agree closely.
Empty weight	Yes	Yes	Follows size and material; like guns agree within a few tenths.
Muzzle velocity	Yes	Yes	Follows the cartridge and the barrel; guns firing one cartridge agree.
Muzzle energy	Yes	Yes	Follows the velocity and the bullet; guns firing one cartridge agree.
Magazine capacity	Yes	No	Across a cohort, magazines of 5, 10, 20 and 30 rounds are all standard, and the comparison raised hundreds of right figures. Within a family, one design's magazine is a standard.
Trigger pull	Yes	No	Across a cohort, match and service triggers differ by design, and single and double action pulls are both quoted. Within a family the design is one.
Rate of fire	Yes	No	Across a cohort, a cyclic rate and a practical rate are both quoted, and a mount changes it. Within a family the mechanism is one.
Effective range	No	No	Makers quote it by different conventions (a point target or an area target, with or without optics), so a difference says nothing about the figure.
Year introduced	No	No	Nothing to compare with: a family spans decades by design, and a cohort holds guns of every era.
Bullet weight	No	No	A cartridge's figure rather than a gun's, and too few records carry one to form groups.
Cartridge dimensions	No	No	Already held to arithmetic on each cartridge (a neck against its bullet, a case against its length), which proves an error where a comparison could only suggest one.

Lauf gegen Länge

A barrel's share of its gun's length is set by the layout: about half on a pistol, two thirds on a rifle, more on a bullpup. A barrel and a length may each look ordinary and still disagree with each other, which says one of them belongs to another configuration: a length with the stock folded, a barrel measured with its muzzle device, a length from another variant.

p = \frac{b}{L}, z_{p} = \frac{| p - med (p_{i}) |}{\max (1.4826 \cdot MAD, 0.03)} > 4

The share is scored as it stands, not on logarithms, within each category and action with 8 or more records, and its spread is never taken as less than 0.03 of a gun's length. Past a score of 4, stricter than for a single figure because a share carries the errors of two, it is unusual. A barrel at least as long as its gun is impossible rather than unusual: the record rules raise it, and it is not scored here. Bullpups and integrally suppressed guns are left out of the groups and not judged, because their shares are unusual by design.

Was ein Datensatz erklärt

An unusual figure is not raised when the record itself explains it. Each reason is read from the record's own data (its features, its variant type, the words of its name) and never assumed.

A bullpup, marked as one among its features or in its name, is short for its kind by design: a short overall length is not raised, and its barrel against its length is not judged.
A gun built around its suppressor (an integral suppressor among its features, a suppressed variant type, or a name such as De Lisle, VSS, AS Val, Welrod or MP5SD) is slow on purpose, long for its barrel, and often has a short ported barrel inside the can: its velocity, its energy, a long overall length and a short barrel are not raised.
A shortened variant, marked by its variant type (compact, subcompact, carbine, short rifle, cavalry carbine, barrel length) or its name (Compact, Carbine, SBR, Kurz, Commando, PDW, Mini, Micro, or a rifle named a Pistol), is not raised for a barrel, a length or a weight below its group. Only below: a compact with a barrel longer than its family's is still raised.
A lengthened variant, marked as a marksman, sniper, target or heavy barrel model by its type or its name (Long, HBAR, DMR, Match, Target), is not raised for a barrel, a length or a weight above its group.
An air gun is not raised on velocity or energy: one model is sold at each country's legal power limit, 7.5 J in one market and 16 J or unrestricted in another.
A figure a person has verified against a source is never raised, however unusual, and a finding a reviewer confirmed as right is not raised again.

Zwei Angaben zugleich ungewöhnlich

A record unusual on two related figures in the same direction (two of its barrel, length and weight both short or both long, or its velocity and energy both high or both low) is not raised on those figures. Two independent typing errors that happen to point the same way are rare; a gun that really is shorter, or really fires a hotter load, is common, and its figures move together because they describe one real difference. The exception is a record whose barrel and length also disagree with each other: then one of the pair is probably wrong after all, and both are raised.

Eine Familie durchgerechnet

The Carcano Modello 91 family, from its records: the long rifle that roots it and the guns filed as its variants, 8 barrels in all. The rifle's own barrel is 780 mm.

Record	Barrel, mm	Distance from the median, logarithmic
Carcano Modello 91	780	0.550
Modello 91/38 Short Rifle	535	0.173
M91/41	692	0.430
M91 Cavalry Carbine	450	0.000
M91 TS	450	0.000
Moschetto con Tromboncino Mod. 28	450	0.000
Moschetto Balilla Mod. 1891	367	0.204
Moschetto TS	450	0.000

The median of the logarithms is the logarithm of 450 mm. The distances from it, in the table above, have a median of 0.087; times 1.4826 that is a spread of 0.128, above the floor of a tenth, so it is used as it is.

$z = \frac{| \ln 780 - \ln 450 |}{1.4826 \times 0.087} = \frac{0.55}{0.128} = 4.29$

Barrels this family's median and spread allow

287 mm705 mmim Datensatz: 780 mm

Answer: 780 mm is past 705 mm, a score of 4.29 at 1.73 times the median, so the barrel is unusual for its family and is raised for review.

Unusual, and right. The Modello 91 is the full-length infantry rifle, and most of the guns filed under it are carbines with barrels of about 450 mm. A reviewer reads the figure on a source, confirms it, and the check does not raise it again. That is the check working as intended: it points at figures worth a look, and a person decides.

Wie eine Angabe beurteilt wird

Table 1. How a figure is judged. Rows are read in order; the first that applies decides.
#	Wenn	Urteil	Was es bedeutet
1	A verified figure	Ausgenommen	Set aside: never raised, however unusual.
2	A figure the record explains	Ausgenommen	Set aside: unusual by design (a bullpup, a suppressed gun, a shortened or lengthened variant, an air gun).
3	Two related figures unusual together	Ausgenommen	Set aside when the barrel and length agree with each other: a genuinely different gun.
4	$z > 3.5 \land \max (\frac{x}{\tilde{x}}, \frac{\tilde{x}}{x}) \geq 1.5$	Prüfen	Unusual: a low priority review. Read the figure on a source, then correct it or confirm it.
5	$z_{p} > 4$	Prüfen	A barrel an unusual share of its gun's length: one of the two figures may belong to another configuration.
6	$b \geq L$	Unmöglich	A barrel at least as long as the gun: raised by the record rules, not here.
7	Anything else	Stimmt	Ordinary for its kind.

Prüfungen, die diese Methode verwenden

Findings this method raised

Every task the 3 checks resting on this method put on the review queue, open and closed, with where each one stands.

Reading the findings…

Task	Record	Check	Status	Opened	Closed