Measurement is the process of assigning numbers to quantities (variables). The process is so familiar that perhaps we often overlook its fundamental characteristics. A single measure of some attribute (for example, weight) of sample is called statistic. These attributes have inherent properties too that are similar to numbers that we assign to them during measurement.

When we assign numbers to attributes (i.e., during measurement), we can do so poorly, in which case the properties of the numbers to not correspond to the properties of the attributes.

In such a case, we achieve only a “low level of measurement” (in other words, low accuracy). Remember that in the earlier module we have seen that the term accuracy refers to the absolute difference between measurement and real value.

On the other hand, if the properties of our assigned numbers correspond properly to those of the assigned attributes, we achieve a high level of measurement (that is, high accuracy).

American statistician Stanley Smith Stevens is credited with introducing various levels of measurements. Stevens (1946) said: “All measurements in science are conducted using four different types of scales nominal, ordinal, interval and ratio”.

These levels are arranged in ascending order of increasing accuracy. That is, nominal level is lowest in accuracy, while ratio level is highest in accuracy. For the ensuing discussion, the following example is used. Six athletes try out for a sprinter’s position in CUPB Biologists’ Race.

They all run a 100-meter dash, and are timed by several coaches each using a different stopwatch (U through Z). Only the stopwatch U captures the true time, stopwatches V through Z are erroneous, but at different levels of measurement. Readings obtained after the sprint is given in Table.

Nominal level of measurement

Nominal scale captures only equivalence (same or different) and set membership. These sets are commonly called categories, or labels. Consider the results of sprint competition, Table 1. Watch V is virtually useless, but it has captured a basic property of the running times.

Namely, two values given by the watch are the same if and only if two actual times are the same. For example, participants Shatakshi and Tejaswini took same time in the race (13s), and as per the readings of stopwatch V, this basic property remains same (20s each). By looking at the results from stopwatch V, it is cogent to conclude that ‘Shatakshi and Tejaswini took same time in the race’.

This attribute is called equivalency. We can conclude that watch V has achieved only a nominal level of measurement. Variables assessed on a nominal scale are called categorical variables. Examples include first names, gender, race, religion, nationality, taxonomic ranks, parts of speech, expired vs non expired goods, patient vs. healthy, rock types etc.

Correlating two nominal categories is very difficult, because any relationships that occur are usually deemed to be spurious, and thus unimportant. For example, trying to figure out how many people from Assam have first names starting with the letter ‘A’ would be a fairly arbitrary, random exercise.

Ordinal level of measurement

Ordinal scale captures rank-ordering attribute, in addition to all attributes captured by nominal level. Consider the results of sprint competition, Table 1. Ascending order of time taken by the participants as revealed by the true time are (respective ranks in parentheses): Navjot (1), Surbhi (2), Sayyed (3), Shatakshi and Tejaswini (4 each), and Shweta (5).

Besides capturing the same-difference property of nominal level, stopwatches W and X have captured the correct ordering of race outcome. We say that the stopwatches W and X have achieved an ordinal level of measurement. Rank-ordering data simply puts the data on an ordinal scale. Examples at this level of measurement include IQ Scores, Academic Scores (marks), Percentiles and so on. Rank ordering (ordinal measurement) is possible with a number of subjective measurement surveys.

For example, a questionnaire survey for the public perception of evolution in India included the participants to choose an appropriate response ‘completely agree’, ‘mostly agree’, ‘mostly disagree’, ‘completely disagree’ when measuring their agreement to the statement “men evolved from earlier animals”.

Interval level of measurement

Ordinal scale captures relative spacing attribute, in addition to all attributes captured by nominal and ordinal levels. In other words, measurements in the interval level are in correct proportion. Consider the results of sprint competition, Table 1. Values in column Y (readings captured by stopwatch Y) can be obtained from that in column U (readings captured by stopwatch U, the true time) by using a simple formula (which is a ‘linear’ equation):

Y=2U+1

For example, the first reading, 21 (time taken by Surbhi) can be obtained by plugging into the above formula: Y= (2x 10)+1= 21

A hallmark of interval scale is ‘relative spacing’ or proportion. Ratios of differences of values in interval level remain same with that of the true time. For example, consider values at column Y (Shatakshi-Surbhi)/(Sayyed-Surbhi)

= (27–21)/(23–21)

= 6/2

= 3

Let’s do same operation for values in column U:

= (13–10)/(11–10)

= 3/1

= 3

Note that the relative spacing, 3, remains same between measurement at interval level and that of the true time. In other words, looking at the stopwatch Y it is cogent to conclude that the difference in time (Shatakshi-Surbhi) is three times (3, the relative spacing) that of (Sayyed-Surbhi).

In interval measurements, difference between two values are meaningful. However, note that the absolute spacing is different; only relative spacing is same. When the numbers capture same-difference, have the correct order, and have the correct relative interval spacing, we say they have achieved an interval level of measurement.

One example of measurement at interval level is Celsius scale. The unit in Celsius (i.e., 1 °C) is defined as “1/100 of the temperature difference between the freezing and boiling points of water under a pressure of 1 atmosphere”. Note that this definition is centered on the concept of relative spacing discussed above.

As the level is interval, absolute spacing would be wrong. For example, the statement “20°C is twice as hot as 10°C” is fallacious. However, it is cogent to conclude: “the difference between a temperature of 50 °C and 40 °C is the same difference as between 20 °C and 10 °C.”

Also, note that a reading of ‘0’ does not correspond to the absence of the attribute that is being measured at this level. For example, 0°C does not mean absence of any temperature.

Other examples of measurements that fall in interval scale include Temperature in Fahrenheit scale, Percentage, Date-when measured from an arbitrary epoch (such as AD), Location in Cartesian coordinates, Geographical location in Lat-Long, Elevation in ASL, Direction measured in degrees from true or magnetic south and so on.

Ratio level of measurement

Ratio scale captures ratio attribute, in addition to all attributes captured by nominal, ordinal and interval levels. Consider the results of sprint competition, Table. Values in column Z (readings captured by stopwatch Z) can be obtained from that in column U (readings captured by stopwatch U, the true time) by using a simple formula:

Z=2U

The ratio of measurements remains same between Z and U. For example, (Sayyed/Shatakshi) in column Z (22/26) is same as that of column U (11/13). Looking at column Z, it is cogent to conclude “Shweta took twice as long as Surbhi”.

Also note that zero point of stopwatch Z is correct; Navjot took no time at all (or 0 s) in both stopwatch U and stopwatch Z. Therefore, ratio scale has an ‘absolute zero’ (a point where none of the quality being measured exists). For example, a ‘zero degree’ in Kelvin scale of temperature measurement (which is in ratio level) means no temperature at all.

Most measurements in the sciences and engineering is done on ratio scales. In science, measurement is routinely defined as the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind. Some examples of measurements in ratio scale include: Concentration of a chemical, “Photosynthetic Photon Flux Density”, Temperature in Kelvin, mass, length, time, plane angle, energy, pressure and electric charge.

A simple heuristics while deciding whether a measurement is at interval or ratio scale is to ask two questions: Does zero make sense? Are twice or half, valid operators?

Also, note that a few measurements in science do not fall into any of these four levels. Examples include pH scale and logIC50 values, both are expressed in log scale. However, when these values are expressed on concentration scale (H+ ion concentration for pH or IC50 values), measurements fall into ratio level.

Summary

• Measurement is the process of assigning numbers to quantities (variables).
• Most of the measurements in science are conducted using four different types of scales; nominal, ordinal, interval and ratio.
• Nominal level captures equivalency and set membership attributes. For example, My income is the same as yours or different.
• Ordinal level captures rank-ordering attribute. For example, if our incomes are different, mine is greater or less than yours is.
• Interval level captures relative difference attribute. For example, the difference between my income and yours might be, say, twice as great as the difference between my income and the professor’s.
• Ratio scale captures ratio and zero point attributes. For example, my brother’s income is about 10 times what mine is.