3 Measurement Scales

Dr. Harmanpreet Singh Kapoor

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   Learning Objectives

  • Introduction.
  • Number’s Property
  • Classification of the Scales
  • Comparison
  • Summary
  • Suggested Readings

    1. Learning Objectives

 

In this module, an attempt has been made to give a brief and relevant information about the topic with examples. This module helps to understand measurement scales and its types and further related information. Questions with answers are included to give an in-depth knowledge of the topic.

 

2. Introduction

 

S. S. Stevens (1906-1973) used different scales of measurement to describe the data. These scales are used for different purposes based on the data and different features that it possess. A proper scale that is used to measure a variable has a great impact on the statistical methods or technique that can be used to analyze it for further inference. So generally, scientist want to use that measurement scale on a variable that explains the features of the data in exact manner so that one can apply the statistical tool appropriately. Researcher defined four different measures known as scales of measurement. These are

 

(a)  nominal (b) ordinal (c) interval (d) ratio. Ratio scale has the features that other scales have. Thus one can say that each of these scales of measurement contains more features in increasing order and therefore each succeeding scale of measurement is used for more sophisticated statistical analysis than the previous one. Before discussing about scales there are some important properties of numbers that a person must be aware of. These properties will help the person to differentiate between scales in a better way.

    2.1 Number’s properties

 

Numbers are used for the presentation of the data. It is essential for a person to understand its four properties and these properties determine how numbers can be used. These are

 

(a)  Identity: Numbers are used for identification only. For example, football player’s T-Shirt number is used for labeling purpose. One can judge the performance of the player with the T-shirt number. Another example is, different trains have different numbers. These numbers help you to find out the right train for your destination. Even in bike racing game different bikes are labelled with different numbers to differentiate with one another. This will help you to know the performance of which no. bike and whose bike is it. Here for example bike no. 1 is not numerically different from bike no. 2. 

 

(b)  Magnitude: The second important property of a number is called magnitude. Numbers are used for identity purpose as discussed in the previous section and its value represent an amount of something. Also larger number indicate more of this quality then it is called magnitude property of number. Two coins are better than one coin, four men are more than three men etc. 

 

(c)  Equal Interval: Some property of numbers are such that if it indicates amount or magnitude in such a way that the difference between numerical values has same meaning in the scale from all ranges. For example, the interval difference between 90 and 80 marks is same as the interval difference between the 70 and 60 marks. This number property allows some numerical calculations like adding and subtracting on the numbers. 

 

(d)  True Zero

 

This property of number allows one to apply all the mathematical operations on the numbers that can be interpreted as well. There are some numbers that have a value of zero that really means something. Having a value zero does not mean the absence of the characteristics of the study in the number. For example zero degree temperature in a room does not mean the absence of temperature as zero degree temperature has its own meaning. However, on a length, weight scale, number of claims etc are the value in which zero means really zero.

 

These four properties are used by different scales and based on these properties one can easily differentiate between different scales.

 

2.2 Classification of the scales

 

Classification of the scales is based on the data under study. Different measurement of scales are used for the extracting the information available in the data. Data are of two types: (a) Quantitative data (b) Qualitative data. As we already discussed about the types of data in the previous module. So one can visit the material from previous module about different types of data and variable. We are going to use these term here again to give more information about different scales of measurement. As in the previous section, we discussed four different measurement of scales. The first one is nominal scale. It is used for qualitative data.

 

(a) Nominal Scale

 

A nominal scale of measurement is used for different categories like color, sex, nations, states, rivers etc. Nominal scale is a scale that categorize things into sections that are different in term of qualitative features. For example, the categorization of the school teachers in terms of sex, religion etc. are covered under nominal data. While filling any survey form or form of applying visa/ school admission/ college admission/ passport etc., we use different code to represent data such as sex, religion etc. These are also examples of nominal data. As nominal scale assign number to different categories, it does not have any quantitative significance or value. For example, for analysis purpose if we assign 1 digit to male or 2 digit to female. As sex can be described through quantitative figures but digit has no quantitative significance here. It is used for further statistical analysis like diagrammatic representation of the data, counting of the number of observation of particular category. For example, in a survey of sex of school teachers it is found that among total 200 teachers 150 teachers belong to male category and 50 belong to female category. Hence nominal scale is used for grouping of different categories. These observations can be processed further for analysis purpose.

 

Hence Nominal data are considered as that form of data that cannot be ordered or have more than two categories. For example, color of hair (black, brown, blonde etc.), marital status (married, unmarried, divorced, separated), nature of disaster (fire, theft, accident, earthquake, etc.). In these examples, on can observe that there are more than two categories as well as these categories are unordered. It means that one cannot compare black color with brown color and comment about it. Also one cannot compare categories of marital status and nature of disaster with each other.

 

The data values are assigned numbers for keeping record in a computer data file. For example, different types of blood groups A+, A-, AB, O+, O- are recorded as 1,2…… and similarly different types of accidents are recorded as 0 or 1, 1 or 2 etc. Here we can assign different numbers to categories as number has no quantitative significance it is just used for the representation purpose. There is no order associated with the data and arithmetic operations cannot be performed. Hence, we cannot add values of different categories as mathematical it seems to be relevant but actually it is not relevant. For example, if we assign A+ as 1 value and A- as 2 value. If we add 1+2 it is equal to 3 but in actual we cannot club A+ and A- group persons as they have different characteristics.

 

Next we will discuss about ordinal scale. This scale is used in those cases where data can be categorized into different categories like nominal scale but also these categories can be ranked based on the characteristics of the data.

 

(b) Ordinal Scale

 

In ordinal scale, we are more concerned with the ordering of observations based on characteristics that it possess. It is also known as a scale that can order the data according to the relationship among the values of the characteristics of the data.

 

For example, while categorizing the designation of the employees in a bank, we can categorize them as Manager, Assistant Manager, Clerical Staff etc. Here we can rank the employees based on their designation and assign numerical values to them that is Manager-1, Assistant Manager-2, Clerical Staff-3 etc. These codes are used for further analysis but here the values have different meaning as used for nominal scale. In nominal scale, sex categories male and female are not comparable so the value 1 assign to male and 2 to female are not comparable but in ordinal scale minimum digit can be assigned to highest order category and maximum digit is assigned to lowest order category or minimum digit can be assigned to lowest order category and maximum digit can be assigned to highest order category. It depends on the experimenter that determine the sequence of assignment of numerical values to different categories.

 

As ordinal scale is applied to qualitative data in which observations are ranked in some order, but difference between data values cannot be determined or are meaningless. For example, if we order data in terms of designation and assign 1 value to Manager, 2 value to Assistant Manager. We cannot use arithmetic operations on these data values as in nominal scale. Here difference of the numerical values has no meaning and cannot be interpreted. For example, if we take difference 1-2 the mathematics value is -1 but here negative value of one has no quantitative significance. Thus one should keep these things in mind while applying ordinal scale to the data.

 

Ordinal data consist of observation that can be ordered in terms of their characteristics. For example, tidiness among students have categories as messy, fairly, tidy or very neat, build of body has categories like fat, medium sized or thin, agreement has categories like strongly agree, agree, neither agree nor disagree, disagree, strongly disagree. Now one can order observation based on these categories in term of high or low level of cleanliness. Similarly, responses on body type and agreement can also be ordered based on categories available.

 

Hence ordinal scale can be used for all cases where there are ordering in categories like height among students in a school, monthly salary of teachers/lecturers/ professors, marks obtained by students in an examination, agreement, level of cleanliness etc.

 

Now we will discuss third scale that is interval scale. As we already discussed that both nominal and ordinal scales are used for qualitative data. Due to this reason, the numerical values that we assigned to data have not qualitative significance but in the interval scale the values can be compared in terms of magnitude only but not in relative purpose. We will discuss this in detail in the following:

    (c) Interval Scale

 

In an interval scale, data can be arranged in some order and difference between the consecutive units have some quantitative significance. The interval level of measurement results from counting and measuring but not for comparing in terms of ratio. Hence in interval scale data ordering of the units and difference between the units can be calculated. As the zero values is arbitrarily chosen for interval data and does not imply an absence of the characteristics being measured. Thus ratio of the units cannot be evaluated for interval data. For example, IQ (Intelligent Quotient) score represent interval data. If one person A has IQ score as 80 and other person B has IQ score as 40. A has higher IQ than B and IQ scores can be arranged in an order. A IQ score is 40 points higher than B IQ score, that is IQ scores can be ordered as well as difference between IQ scores are meaningful. However, we cannot conclude that A’s IQ is 2 times than B’s IQ. As zero IQ score does not indicate a lack of intelligence among the person. Hence one cannot conclude the units based on ratios in interval data.

 

Hence in an interval scale of measurement is based on observations that are ordered as well are of equal length and zero value has it importance. Addition or subtraction is possible on this scale, but no multiplication or division.

 

There are few examples of interval scale like test scores represent interval level data, temperature also represent level data etc.

 

(d) Ratio Scale

 

Ratio scale is a scale in which the observations are ordered and are of equal interval length with true zero point. So that one can perform basic mathematics like addition, subtraction, multiplication and division (except by zero) on the values. The ratio level of measurement results from counting or measuring. Ratio scale data can be arranged in order. Also differences, ratios of observations can be calculated and interpreted. Ratio level data has an absolute zero and a value of zero indicates a complete absence of the characteristic of interest. For example, length is a ratio variable. There is a true zero number as zero length means absence of length characteristics. Similarly, weight and speed are also ratio variable. Hence ratio scale is interpretable and consists of true zero number.

 

For example, the fat consumed per day by Indian is an example of ratio scale data. As youngster use to consume more fat than old aged persons. Hence consumption of fat is comparable among youngster and old aged persons in terms of category, order, equal interval and relative. As different mathematical operations can be performed on the data. Hence the data belong to ratio scale.

 

Another example, in a study of emergency call to ambulance in a particular state in a country with having a sample size of approx. 10000 in a month. The call are received on daily basis and number of calls received is ratio scale data. Calls on a particular day can be compared with any other day in terms of ratio. Also no call means absence of characteristics.

 

3 Comparison between different scales

 

In the following section, we will provide you the summary of the measurement scales. In this section, features of the measurement scales are provided in points that will help the person to remember the features of different scales. It is also easy for a person to compare the features of different scales.

 

(a) Nominal Scale

 

It is a level of measurement that is lowest among all scales and is mostly used with the data that are qualitative in nature rather than quantitative. Some of the important features of nominal scale are mentioned below:

  • In nominal scale, the data is divided into different groups or categories.
  • These groups or categories are assigned units i.e numbers and objects are determined through the category or group to which they belong.
  • In a nominal scale, one can assign different names to the category to which they belong and for analysis purpose numerical values are used to represent these names.
  • It is important to note that because the units of a nominal scale are categories, hence no arithmetic operations can be applied between the units of a nominal scale.
  • Also there is no quantitative relationship between the categories.
  • A basic features of the nominal scale is that of equivalence.
  • Every member of a given category or group are same on the basis of classification.
  • Experimenter often used nominal scale for categorizing the data into groups and also to count number of observation in a particular group and draw inference based on these values.
  • These frequency value also help one in comparing the number of items within each group or category.
  • A nominal scale does not possess any mathematical features like equal interval, absolute zero point etc. It allows categorization of objects into mutually exclusive categories.

   (b) Ordinal Scale

 

Ordinal Scale is the next higher scale after nominal scale. It is also used for qualitative data but it has some other features than nominal scale like categorizing data with order or ranking. There are some features of ordinal scale that are given below:

  • It possesses a low level of mathematical property like one cannot apply arithmetic operation on the data values.
  • One can rank the category based on whether they possess more, less or the same amount of object being measured.
  • Although with ordinal scale one can compare the objects in terms of “better than”, “equal to” or “less than” but it says nothing about the magnitude of difference between adjacent units on the scale.
  • It does not have the property of equal interval between consecutive units.
  • It does not tell about the absolute level of the units for e.g. all units belong to same higher category or lower category. Thus one cannot give any statement about the data in absolute term.

    (c) Interval Scale

 

It is higher scale than both nominal and ordinal scale. It is used for the qualitative data but one can apply arithmetic operation like multiplication and division on the data having interval scale. Some of the features of interval scale are mentioned below:

  • The interval scale possesses the properties of magnitude and equal interval between consecutive units but does not have an absolute zero point. There are equal amount of the variable being measured between adjacent units on the scale.
  • The interval scale possesses the properties of the ordinal scale and nominal scale.
  • In an interval scale, equal differences between the numbers on the scale represent equal differences in the magnitude of the variable.
  • It follows that greater differences between the numbers on the scale represent greater difference in the magnitude of the variable being measured.
  • Also smaller difference between the numbers on the scale represent the smaller difference in the magnitude of the variable being measured.

   The difference between the interval and ordinal scale is that in ordinal we can order the objects based on the characteristics but difference between the numbers has no quantitative significance but in interval scale one can order the observation and also take interval difference that has quantitative significance.

 

(d) Ratio Scale

 

Ratio Scale is the highest level of all measurement scales. It has all the properties all nominal, ordinal and interval scale and other features that differentiate this scale from other scales. These are:

  • Ratio scale holds the property of absolute zero.
  • It means that a ratio sale data can be compared on the basis of ratio. Without absolute zero property it is not possible to calculate ratio of the data and give interpretation
  • In ratio scale, equal differences between the numbers on the scale represent equal differences in the magnitude of the variable.
  • Any object with zero value means complete absence of the characteristics in the object.
  • In interval scale, one cannot compute ratio due to unavailability of property of absolute zero. For example, one cannot compute the ratio of different temperature values in a room as it is calculated on an interval scale but in ratio scale one can do this.

    In the following section, some self-check multiple choice questions are given for practice purpose. This will help the reader to test one knowledge about different measure of scales. If you are unable to answer with confidence then read the definitions again from the above section and answer. It is possible that one may feel other options as suitable answer but this is not the case here.

  1. Summary

   In this module, we try to give an in-depth knowledge of different measure of scales. Different scales are discussed with examples that help the reader to understand the topic in an easy manner. In the previous section, different scales are compared in term of their feature and properties that they possess. As we are aware of the importance of this topic in the statistics. Question and answer part will help you to check understanding of this topic.

  1. Suggested Readings

Agresti, A. and B. Finlay, Statistical Methods for the Social Science, 3rd Edition, Prentice Hall, 1997.

 

Daniel, W. W. and C. L. Cross, C. L., Biostatistics: A Foundation for Analysis in the Health Sciences, 10th Edition, John Wiley & Sons, 2013.

 

Hogg, R. V., J. Mckean and A. Craig, Introduction to Mathematical Statistics, Macmillan Pub. Co. Inc., 1978.

 

Meyer, P. L., Introductory Probability and Statistical Applications, Oxford & IBH Pub, 1975.

 

Triola, M. F., Elementary Statistics, 13th  Edition, Pearson, 2017.

 

Weiss, N. A., Introductory Statistics, 10th Edition, Pearson, 2017.

you can view video on Measurement Scales

 

One can refer to the following links for further understanding of the statistics terms.

 

http://biostat.mc.vanderbilt.edu/wiki/pub/Main/ClinStat/glossary.pdf

 

http://www.stats.gla.ac.uk/steps/glossary/alphabet.html

 

http://www.reading.ac.uk/ssc/resources/Docs/Statistical_Glossary.pdf

 

https://stats.oecd.org/glossary/

 

http://www.statsoft.com/Textbook/Statistics-Glossary

 

https://www.stat.berkeley.edu/~stark/SticiGui/Text/gloss.htm

 

https://stats.oecd.org/glossary/alpha.asp?Let=A