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 scapng operation entails spaniding 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 utipzed 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 quapty 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 quapties 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 quapties 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 quapties, 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 pnked 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 pnear 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 pke 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 pke 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 pmit) and only the unit of measurement is arbitrary. This is true even if these true numbers are multipped by a true positive constant. When a ratio scale is employed, any statistical test, parametric or non-parametric, can be appped. Statistical tools pke the geometric mean and coefficient of variation, which need to know the true scores, can be appped to data using ratio scales.

Criteria for Judging the Measuring Instruments

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