# Scales of Measurement

The first rule of a scientific investigation is to report honestly and as accurately as possible what a scientist saw and under what conditions. A scientist describes what has been seen and the conditions and procedures followed. The need to provide others an opportunity to corroborate those findings makes this a high-priority requirement. Towards actuality, defining the observational parameters is the initial step in measuring a certain occurrence.

## Scales of Measurement

Measurement is making observations and writing down the data gathered as part of the research. The relationship between the values that are allocated to the characteristics, emotions, or views of a variable is referred to as the level of measurement. For instance, there are various qualities for the variable "if the taste of fast food is good," including very good, good, neither good nor bad, bad, and very awful. We can give the five qualities, in order, the values 1, 2, 3, 4, and 5, to analyze this variable's results. The level of measurement describes the relationship between these five variables. Here, the numbers serve as shorter substitutes for the longer text phrases.

## Nominal or Categorical Scale

The simplest, most basic, and weakest type of measurement is when we can replace real objects with symbols or numbers (without understanding their numerical meanings). In other words, we only describe or categorize things, people, or even features using these symbols or numbers. At the most basic level, a scientist needs to develop a classification system that will allow all recorded occurrences to fit into it. For ease of identification, we assign each category of event or item a name, a number, or a symbol.

Then, a nominal or classificatory scale comprises these symbols or numbers. The scale's categories must be mutually exclusive (each observation may only be classified under one category), exhaustive (there must be enough categories to classify every observation), and unordered. Typically, the categories that make up a nominal scale are called characteristics. As a result, there are only two kinds of sex for mammals: male and female. In a nominal scale, the scaling operation entails dividing a given class into a number of mutually incompatible subclasses. Any subclass member must be equivalent in the scaled property or feature. Equivalence is the sole connection utilized in this scale.

### Statistical Tests for Nominal Scale

The only descriptive statistics that can be used are those that would not be influenced or altered by such interchange since the symbols or labels assigned to each category are arbitrary and can be modified without changing the scale's fundamental information. Crude mode, proportion, and frequency are what they are. However, the data on a nominal scale can be used to test a hypothesis about how occurrences are distributed among the classes. For this, it is possible to employ the chi-square test, the contingency coefficient, and a few more tests based on the binomial expansion.

## Ordinal Scale

The ordinal scale allows the researcher to group people, things, or survey responses according to a certain trait they share. For instance, there are occasions when objects in one class on a nominal scale are not just distinct from those in another class on the same scale but also have some relationship with one another. The members of one class typically possess more of a certain quality or characteristic than members of other classes. Such a connection is frequently indicated with the symbol carat (>), which stands for "more than."

All relationships between classes, including "preferred to," "more than," "greater than," "higher than," etc., are expressed with the symbol >. The ordinal numbers express the relative position or magnitude of the characters about other characteristics. The rank of a category is determined by how many categories come before it in terms of the quantity of the feature being compared, not by how many classes come after it. The discrepancies in ordinal numbers do not indicate the precise variations in the percentage of a characteristic the objects possess.

### Statistical Tests for Ordinal Scale

The best way to determine the central tendency of scores on an ordinal scale is to use the median. For such data, quartile deviation is undoubtedly the best way to gauge dispersion. Numerous non-parametric tests, such as the runs test, sign test, median test, Mann-Whitney U-test, etc., can be used to test a hypothesis with scores on an ordinal scale. The terms "order statistics" and "ranking statistics" frequently describe these tests. Rankings of two sets of observations on the same group of people can be used to calculate interrelations. For these circumstances, Spearman's Rank Difference or Kendall Rank Correlation coefficients are suitable.

## Interval Scale

When ranking qualities on an interval scale, numerically equal distances on the scale correspond to similar distances in the characteristic being measured, and an interval scale provides for comparison of the distance or difference between qualities while still containing all the data of an ordinal scale.

Although there is the lowest endpoint, the zero point, the ratio of any two periods denoted by real numbers is independent of the unit of measurement. This results in a ratio of 1:5, which has no unit, between two intervals of 32 cm and 40 cm and 100 cm and 140 cm. The ratio between the two intervals remains the same if a constant, such as 10 cm, is added to each interval point, resulting in new intervals of 42 cm - 50 cm and 110 cm - 150 cm, respectively.

When analyzing differences between two or more qualities, interval measurement should be done with appropriate caution. When the origin, zero, for both scales is the same, and the measurement units are the same, comparisons are meaningful. Interval scales are used when measuring temperature using a thermometer, time from a chosen beginning point, and altitude relative to mean sea level.

Having all the characteristics of a nominal scale (equivalence relation), an ordinal scale (greater than or transitivity relation), and an ordered metric scale, an interval scale is also a metric scale (transitively related concerning the distance between classes). The ratio of any two intervals can also be specified using this scale. Using units that measure equal distance intervals, interval scale can place things or occurrences into a continuum. The scale's zeroth position is picked at random.

### Statistical Tests for Interval Scale

Even though the numbers linked with an object's position may change according to a regular system, the interval scale preserves both the ordering of objects and the relative differences between them. If the information allows for a linear transformation, a set of observations will be scalable by interval size.

In other words, a set of real numbers is said to be in an interval scale if the equation y = a + bx, where a and b are two positive constants, fulfills the set of real numbers. Data that follow an interval scale can be subjected to all typical parametric tests, including arithmetic mean, median, standard deviation, product-moment correlation, etc. For statistical significance, non-parametric tests like Z, t, and F can also be used on interval scale data.

## Ratio Scale

The ratio scale offers the most accurate measurement since it satisfies all interval scale requirements and an additional, crucial one: it has an invariant or absolute zero. The mathematics operations take on a new dimension because of this invariant zero point. The numbers associated with scale points can also be written as ratios independent of the unit of measurement, much like the ratio of intervals between two classes.

Most frequently, the ratio scale is used in the physical sciences. Regardless of whether two objects are weighed in pounds or kilograms, the ratio remains the same. The same holds for how long two objects are or how long it takes two people to finish a particular task. Suppose the four relations of I equivalence (ii) larger than (iii) the known ratio of any two intervals and (iv) the known ratio of any two real values connected with any two locations on the scale can be operationally attained. In that case, a measurement is said to be in ratio scale.

### Statistical Tests for Ratio Scale

The ratios between two numbers and intervals maintain all the information contained in the scale because the values in a ratio scale are real numbers with a true zero (no upper limit) and only the unit of measurement is arbitrary. This is true even if these true numbers are multiplied by a true positive constant. When a ratio scale is employed, any statistical test, parametric or non-parametric, can be applied. Statistical tools like the geometric mean and coefficient of variation, which need to know the true scores, can be applied to data using ratio scales.

## Criteria for Judging the Measuring Instruments

A measurement must also meet several requirements. The following list of the most crucial factors to consider while assessing a measurement tool.

• Unidimensionality − For a ruler, a scale should only measure one feature at a time, such as length rather than temperature.

• Linearity − A scale must adhere to the straight-line concept to be considered linear. It is necessary to create a scoring system based on movable units. An inch is an inch, whether at one end of a ruler or the other. However, this interchangeability cannot be guaranteed for altitude scales. In these circumstances, ranking is preferred.

• Validity − The capacity of a scale to measure what it is intended to measure is referred to here.

• Reliability − Consistency has this quality. A scale should produce reliable results.

• Accuracy and Precision − A tool should provide a precise and accurate measurement of the thing we are trying to gauge.

• Simplicity − A scale should be as simple as feasible; otherwise, it may become unnecessarily complex, expensive, or even useless.

• Practicability − This covers a wide range of issues, including affordability, practicality, and interpretability. Usually, a trade-off must be made between the "perfect" instrument and what the budget will allow. The benefit received must be equal to the expense incurred.

## Conclusion

The four levels are nominal, ordinal, interval, and ratio measurements. These scales make up a hierarchy, with the nominal scale of measurement having significantly fewer statistical applications than scales higher. Nominal scales provide data on categories; ordinal scales provide sequences, magnitudes between points on the scale are revealed by interval scales, and ratio scales explain the order and absolute distance between any two points on the scale.

Updated on: 13-Feb-2023

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