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53-410: The Palermo Technical Impact Hazard Scale is a logarithmic scale used by astronomers to rate the potential hazard of impact of a near-Earth object (NEO). It combines two types of data — probability of impact and estimated kinetic yield—into a single "hazard" value. A rating of 0 means the hazard is equivalent to the background hazard (defined as the average risk posed by objects of

106-446: A 1946 Science article titled "On the theory of scales of measurement". In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal", "ordinal", "interval", and "ratio", unifying both " qualitative " (which are described by his "nominal" type) and " quantitative " (to a different degree, all the rest of his scales). The concept of scale types later received

159-607: A central moment. The ratio type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit of measurement of the same kind (Michell, 1997, 1999). Most measurement in the physical sciences and engineering is done on ratio scales. Examples include mass , length , duration , plane angle , energy and electric charge . In contrast to interval scales, ratios can be compared using division . Very informally, many ratio scales can be described as specifying "how much" of something (i.e. an amount or magnitude). Ratio scale

212-588: A classification can be viewed as progress. Numbers may be used to represent the variables but the numbers do not have numerical value or relationship: for example, a globally unique identifier . Examples of these classifications include gender, nationality, ethnicity, language, genre, style, biological species, and form. In a university one could also use residence hall or department affiliation as examples. Other concrete examples are Nominal scales were often called qualitative scales, and measurements made on qualitative scales were called qualitative data. However,

265-508: A linear scale where each unit of distance corresponds to the same increment, on a logarithmic scale each unit of length is a multiple of some base value raised to a power, and corresponds to the multiplication of the previous value in the scale by the base value. In common use, logarithmic scales are in base 10 (unless otherwise specified). A logarithmic scale is nonlinear , and as such numbers with equal distance between them such as 1, 2, 3, 4, 5 are not equally spaced. Equally spaced values on

318-399: A logarithmic scale can be helpful when the data: A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic graph paper was a commonly used scientific tool. If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot

371-430: A logarithmic scale have exponents that increment uniformly. Examples of equally spaced values are 10, 100, 1000, 10000, and 100000 (i.e., 10 , 10 , 10 , 10 , 10 ) and 2, 4, 8, 16, and 32 (i.e., 2 , 2 , 2 , 2 , 2 ). Exponential growth curves are often depicted on a logarithmic scale graph . The markings on slide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on

424-402: A nonrule". Hand says, "Basic psychology texts often begin with Stevens's framework and the ideas are ubiquitous. Indeed, the essential soundness of his hierarchy has been established for representational measurement by mathematicians, determining the invariance properties of mappings from empirical systems to real number continua. Certainly the ideas have been revised, extended, and elaborated, but

477-418: A quantitative relation between sensation intensity and stimulus intensity is not merely false but is in fact meaningless unless and until a meaning can be given to the concept of addition as applied to sensation. That is, if Stevens's sone scale genuinely measured the intensity of auditory sensations, then evidence for such sensations as being quantitative attributes needed to be produced. The evidence needed

530-698: A quantity ( physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm. Examples of logarithmic units include units of information and information entropy ( nat , shannon , ban ) and of signal level ( decibel , bel, neper ). Frequency levels or logarithmic frequency quantities have various units are used in electronics ( decade , octave ) and for music pitch intervals ( octave , semitone , cent , etc.). Other logarithmic scale units include

583-462: A range and repeating (like degrees in a circle, clock time, etc.), graded membership categories, and other types of measurement do not fit to Stevens's original work, leading to the introduction of six new levels of measurement, for a total of ten: While some claim that the extended levels of measurement are rarely used outside of academic geography, graded membership is central to fuzzy set theory , while absolute measurements include probabilities and

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636-559: A relative degree of difference between them. Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as "sick" vs. "healthy" when measuring health, "guilty" vs. "not-guilty" when making judgments in courts, "wrong/false" vs. "right/true" when measuring truth value , and, on the other hand, non-dichotomous data consisting of a spectrum of values, such as "completely agree", "mostly agree", "mostly disagree", "completely disagree" when measuring opinion . The ordinal scale places events in order, but there

689-803: A variable on a nominal level). L. L. Thurstone made progress toward developing a justification for obtaining the interval type, based on the law of comparative judgment . A common application of the law is the analytic hierarchy process . Further progress was made by Georg Rasch (1960), who developed the probabilistic Rasch model that provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments. Typologies aside from Stevens's typology have been proposed. For instance, Mosteller and Tukey (1977) and Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998), van den Berg (1991). Mosteller and Tukey noted that

742-404: Is 40th, it cannot be said that Devi's position is four times as good as that of Ganga. Ordinal scales only permit the ranking of items from highest to lowest. Ordinal measures have no absolute values, and the real differences between adjacent ranks may not be equal. All that can be said is that one person is higher or lower on the scale than another, but more precise comparisons cannot be made. Thus,

795-616: Is a classification that describes the nature of information within the values assigned to variables . Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scales, of measurement: nominal , ordinal , interval , and ratio . This framework of distinguishing levels of measurement originated in psychology and has since had a complex history, being adopted and extended in some disciplines and by some scholars, and criticized or rejected by others. Other classifications include those by Mosteller and Tukey , and by Chrisman. Stevens proposed his typology in

848-408: Is doubtful if he understood it himself ... no measurement theorist I know accepts Stevens's broad definition of measurement ... in our view, the only sensible meaning for 'rule' is empirically testable laws about the attribute. A nominal scale consists only of a number of distinct classes or categories, for example: [Cat, Dog, Rabbit]. Unlike the other scales, no kind of relationship between

901-420: Is little prima facie evidence to suggest that such attributes are anything more than ordinal (Cliff, 1996; Cliff & Keats, 2003; Michell, 2008). In particular, IQ scores reflect an ordinal scale, in which all scores are meaningful for comparison only. There is no absolute zero, and a 10-point difference may carry different meanings at different points of the scale. The interval type allows for defining

954-401: Is measured in megatons , and yr is the unit of T divided by one year. For instance, this formula implies that the expected value of the time from now until the next impact greater than 1 megatonne is 33 years, and that when it occurs, there is a 50% chance that it will be above 2.4 megatonnes. This formula is only valid over a certain range of E . However, another paper published in 2002 –

1007-426: Is no attempt to make the intervals of the scale equal in terms of some rule. Rank orders represent ordinal scales and are frequently used in research relating to qualitative phenomena. A student's rank in his graduation class involves the use of an ordinal scale. One has to be very careful in making a statement about scores based on ordinal scales. For instance, if Devi's position in his class is 10th and Ganga's position

1060-572: Is not allowed. The mode is allowed. In 1946, Stevens observed that psychological measurement, such as measurement of opinions, usually operates on ordinal scales; thus means and standard deviations have no validity , but they can be used to get ideas for how to improve operationalization of variables used in questionnaires . Most psychological data collected by psychometric instruments and tests, measuring cognitive and other abilities, are ordinal, although some theoreticians have argued they can be treated as interval or ratio scales. However, there

1113-515: Is often used to express an order of magnitude such as for temperature in Orders of magnitude (temperature) . The geometric mean and the harmonic mean are allowed to measure the central tendency, in addition to the mode, median, and arithmetic mean. The studentized range and the coefficient of variation are allowed to measure statistical dispersion. All statistical measures are allowed because all necessary mathematical operations are defined for

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1166-412: Is referred to as a log–log plot . If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot. A modified log transform can be defined for negative input ( y < 0) to avoid the singularity for zero input ( y = 0), and so produce symmetric log plots: for a constant C =1/ln(10). A logarithmic unit is a unit that can be used to express

1219-404: Is the intellectual handmaiden to Stevens's "operational theory of measurement", which was to become definitive within psychology and the behavioral sciences , despite Michell's characterization as its being quite at odds with measurement in the natural sciences (Michell, 1999). Essentially, the operational theory of measurement was a reaction to the conclusions of a committee established in 1932 by

1272-404: Is the median. A percentile or quartile measure is used for measuring dispersion. Correlations are restricted to various rank order methods. Measures of statistical significance are restricted to the non-parametric methods (R. M. Kothari, 2004). The median , i.e. middle-ranked , item is allowed as the measure of central tendency ; however, the mean (or average) as the measure of central tendency

1325-595: Is used for simpler descriptions in the non-scientific media. As of October 2024, two asteroids have a cumulative Palermo Scale value above −2: (29075) 1950 DA (−0.93) and 101955 Bennu (−1.40). Three have cumulative Palermo Scale values between −2 and −3: 2000 SG 344 (−2.77), 2008 JL 3 (−2.86) and 2010 RF 12 (−2.97). Of those that have a cumulative Palermo Scale value between −3 and −4, five were discovered in 2024: 2024 BY 15 (−3.30), 2024 JW 16 (−3.63), 2024 TK 5 (−3.76), 2024 QL 1 (−3.88) and 2024 TX 13 (−3.95). The scale compares

1378-601: The British Association for the Advancement of Science to investigate the possibility of genuine scientific measurement in the psychological and behavioral sciences. This committee, which became known as the Ferguson committee , published a Final Report (Ferguson, et al., 1940, p. 245) in which Stevens's sone scale (Stevens & Davis, 1938) was an object of criticism: …any law purporting to express

1431-636: The Richter magnitude scale point. In addition, several industrial measures are logarithmic, such as standard values for resistors , the American wire gauge , the Birmingham gauge used for wire and needles, and so on. The two definitions of a decibel are equivalent, because a ratio of power quantities is equal to the square of the corresponding ratio of root-power quantities . Scale (measurement) Level of measurement or scale of measure

1484-414: The coefficient of variation . More subtly, while one can define moments about the origin , only central moments are meaningful, since the choice of origin is arbitrary. One can define standardized moments , since ratios of differences are meaningful, but one cannot define the coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of)

1537-880: The degree of difference between measurements, but not the ratio between measurements. Examples include temperature scales with the Celsius scale , which has two defined points (the freezing and boiling point of water at specific conditions) and then separated into 100 intervals, date when measured from an arbitrary epoch (such as AD), location in Cartesian coordinates, and direction measured in degrees from true or magnetic north. Ratios are not meaningful since 20 °C cannot be said to be "twice as hot" as 10 °C (unlike temperature in kelvins ), nor can multiplication/division be carried out between any two dates directly. However, ratios of differences can be expressed; for example, one difference can be twice another; for example,

1590-605: The broadest sense, is defined as the assignment of numerals to objects and events according to rules (Stevens, 1946, p. 677). Stevens was greatly influenced by the ideas of another Harvard academic, the Nobel laureate physicist Percy Bridgman (1927), whose doctrine of operationalism Stevens used to define measurement. In Stevens's definition, for example, it is the use of a tape measure that defines length (the object of measurement) as being measurable (and so by implication quantitative). Critics of operationalism object that it confuses

1643-408: The classes can be relied upon. Thus measuring with the nominal scale is equivalent to classifying . Nominal measurement may differentiate between items or subjects based only on their names or (meta-)categories and other qualitative classifications they belong to. Thus it has been argued that even dichotomous data relies on a constructivist epistemology . In this case, discovery of an exception to

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1696-504: The energies of the atmospheric impacts that they used in their determination. In 2002 the near-Earth object (89959) 2002 NT 7 reached a positive rating on the scale of 0.18, indicating a higher-than-background threat. The value was subsequently lowered after more measurements were taken. 2002 NT 7 is no longer considered to pose any risk and was removed from the Sentry Risk Table on 1 August 2002. In September 2002,

1749-507: The four levels are not exhaustive and proposed seven instead: For example, percentages (a variation on fractions in the Mosteller–Tukey framework) do not fit well into Stevens's framework: No transformation is fully admissible. Nicholas R. Chrisman introduced an expanded list of levels of measurement to account for various measurements that do not necessarily fit with the traditional notions of levels of measurement. Measurements bound to

1802-409: The highest Palermo rating was that of asteroid (29075) 1950 DA , with a value of 0.17 for a possible collision in the year 2880. By March 2022, the rating had been reduced to −2.0. As of October 2024, it has a rating of −0.93. For a brief period in late December 2004, with an observation arc of 190 days, asteroid 99942 Apophis (then known only by its provisional designation 2004 MN 4 ) held

1855-402: The likelihood of the detected potential impact with the average risk posed by objects of the same size or larger over the years until the date of the potential impact. This average risk from random impacts is known as the background risk. The Palermo Scale value, P , is defined by the equation: where The background impact frequency is defined for this purpose as: where the energy threshold E

1908-439: The mathematical rigour that it lacked at its inception with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b), and R. Duncan Luce (1986, 1987, 2001). As Luce (1997, p. 395) wrote: S. S. Stevens (1946, 1951, 1975) claimed that what counted was having an interval or ratio scale. Subsequent research has given meaning to this assertion, but given his attempts to invoke scale type ideas it

1961-578: The most natural display of numbers in some cultures. The top left graph is linear in the X- and Y-axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y-axis of the bottom left graph, and the Y-axis ranges from 0.1 to 1000. The top right graph uses a log-10 scale for just the X-axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y-axis. Presentation of data on

2014-488: The only non-trivial operations that generically apply to objects of the nominal type. The mode , i.e. the most common item, is allowed as the measure of central tendency for the nominal type. On the other hand, the median , i.e. the middle-ranked item, makes no sense for the nominal type of data since ranking is meaningless for the nominal type. The ordinal type allows for rank order (1st, 2nd, 3rd, etc.) by which data can be sorted but still does not allow for

2067-452: The ordinal type in behavioural science is in fact somewhere between the true ordinal and interval types; although the interval difference between two ordinal ranks is not constant, it is often of the same order of magnitude. For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across the range of an assessment. Thus, some argue that so long as

2120-518: The plausibility and ignorance in Dempster–Shafer theory . Cyclical ratio measurements include angles and times. Counts appear to be ratio measurements, but the scale is not arbitrary and fractional counts are commonly meaningless. Log-interval measurements are commonly displayed in stock market graphics. All these types of measurements are commonly used outside academic geography, and do not fit well to Stevens's original work. The theory of scale types

2173-583: The ratio scale. While Stevens's typology is widely adopted, it is still being challenged by other theoreticians, particularly in the cases of the nominal and ordinal types (Michell, 1986). Duncan (1986), for example, objected to the use of the word measurement in relation to the nominal type and Luce (1997) disagreed with Stevens's definition of measurement. On the other hand, Stevens (1975) said of his own definition of measurement that "the assignment can be any consistent rule. The only rule not allowed would be random assignment, for randomness amounts in effect to

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2226-408: The record for the highest Palermo scale value, with a value of 1.10 for a possible collision in the year 2029. The 1.10 value indicated that a collision with this object was considered to be almost 12.6 times as likely as a random background event: 1 in 37 instead of 1 in 472. With further observation through 2021 there is no risk from Apophis for the next 100+ years. Logarithmic scale Unlike

2279-416: The relations between two objects or events for properties of one of those of objects or events (Moyer, 1981a, b; Rogers, 1989). The Canadian measurement theorist William Rozeboom was an early and trenchant critic of Stevens's theory of scale types. Another issue is that the same variable may be a different scale type depending on how it is measured and on the goals of the analysis. For example, hair color

2332-405: The remarkable thing is his insight given the relatively limited formal apparatus available to him and how many decades have passed since he coined them." The use of the mean as a measure of the central tendency for the ordinal type is still debatable among those who accept Stevens's typology. Many behavioural scientists use the mean for ordinal data anyway. This is often justified on the basis that

2385-487: The rise of qualitative research has made this usage confusing. If numbers are assigned as labels in nominal measurement, they have no specific numerical value or meaning. No form of arithmetic computation (+, −, ×, etc.) may be performed on nominal measures. The nominal level is the lowest measurement level used from a statistical point of view. Equality and other operations that can be defined in terms of equality, such as inequality and set membership , are

2438-474: The same size or larger over the years until the date of the potential impact). A rating of +2 would indicate the hazard is 100 times as great as a random background event. Scale values less than −2 reflect events for which there are no likely consequences, while Palermo Scale values between −2 and 0 indicate situations that merit careful monitoring. A similar but less complex scale is the Torino Scale , which

2491-608: The same year as the paper on which the Palermo scale is based – found a power law with different constants: This formula gives considerably lower rates for a given E . For instance, it gives the rate for bolides of 10 megatonnes or more (like the Tunguska explosion ) as 1 per thousand years, rather than 1 per 210 years as in the Palermo formula. However, the authors give a rather large uncertainty (once in 400 to 1800 years for 10 megatonnes), due in part to uncertainties in determining

2544-622: The scales. The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value: The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value: Some of our senses operate in a logarithmic fashion ( Weber–Fechner law ), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of hearing perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be

2597-584: The ten-degree difference between 15 °C and 25 °C is twice the five-degree difference between 17 °C and 22 °C. Interval type variables are sometimes also called "scaled variables", but the formal mathematical term is an affine space (in this case an affine line ). The mode , median , and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical dispersion include range and standard deviation . Since one can only divide by differences , one cannot define measures that require some ratios, such as

2650-429: The unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on ordinal scale variables. Statistical analysis software such as SPSS requires the user to select the appropriate measurement class for each variable. This ensures that subsequent user errors cannot inadvertently perform meaningless analyses (for example correlation analysis with

2703-403: The use of an ordinal scale implies a statement of "greater than" or "less than" (an equality statement is also acceptable) without our being able to state how much greater or less. The real difference between ranks 1 and 2, for instance, may be more or less than the difference between ranks 5 and 6. Since the numbers of this scale have only a rank meaning, the appropriate measure of central tendency

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2756-527: Was later rendered false by the discovery of the theory of conjoint measurement by Debreu (1960) and independently by Luce & Tukey (1964). However, Stevens's reaction was not to conduct experiments to test for the presence of additive structure in sensations, but instead to render the conclusions of the Ferguson committee null and void by proposing a new theory of measurement: Paraphrasing N. R. Campbell (Final Report, p. 340), we may say that measurement, in

2809-452: Was the presence of additive structure —a concept comprehensively treated by the German mathematician Otto Hölder (Hölder, 1901). Given that the physicist and measurement theorist Norman Robert Campbell dominated the Ferguson committee's deliberations, the committee concluded that measurement in the social sciences was impossible due to the lack of concatenation operations. This conclusion

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