Statistical data, whether qualitative or quantitative, is generated or obtained through some measurement or observational processes. There are 4 levels of measurement.
Each measurement or observation made on any object or variable can be attributed to one of the 4 scales of measurement;
- interval, and
Each character has unique characteristics and implications for the type of statistical procedures that can be used with it.
We elaborate on these measurements.
All qualitative measurements are nominal, regardless of whether the categories are designated by names (red, white, male) or numerals (June 20, Room no. 10, bank account no. student ID, etc.).
In the nominal level of measurement, the categories differ from one another only in names. In other words, one category of a characteristic is not necessarily higher or lower, greater or smaller than the other category.
Sex (male and female), religion (Muslim, Hindu, Christians, etc.) are examples of nominal measurements. The categories are homogeneous and mutually exclusive, with no assumption about the ordered relationships between the categories.
To work with such non-numerical data with statistical tools, we need to impose a numerical scheme on the data.
For example, with gender, 0 might be assigned to males and 1 to female. With religion, the scheme might be to use 1 for Muslim, 2 for Hindu, 3 for Christian, etc.
In each of these cases, the numerical data have been artificially created, but none of the numbers have any numerical meaning. We call such data nominal data because they are numerical in name only.
In the measurement scale, the nominal level of measurement is the lowest or weakest level of measurement, and the resulting data are nominal.
When there is an ordered relationship between the categories, we achieve what we refer to as the ordinal level of measurement.
Unlike the nominal level, here we have the typical relations “higher”, “more than”, “less difficult”, “more prejudiced”, “more feminine”, “less favorable”, “more profitable”, “less costly” and the like.
More specifically, the relationships are expressed in terms of the algebra of inequalities: a is less than b (a < b) or a is greater than b (a > b).
Thus, your university degree (for example, Masters, Bachelor, etc.), job title (for example, manager, deputy manager, accountant), socio-economic status (high, medium-low), academic performance (outstanding, very good, good, poor), monthly frequency of visits of a physician in a clinic (frequently, occasionally, rarely, never), level of agreement on the issue of imposing VAT in food items (strongly agree, agree, disagree, strongly disagree) and the like, all are examples of the ordinal level of measurement.
Note that an ordinal scale is distinguished from a nominal scale by the additional property of order among the categories included on the scale.
You can rate, for example, the level of agreement on the issue of VAT on a 4-point scale of 1 (strongly agree) to 4 (strongly disagree).
Still, such ratings have no real significance in the sense of usual arithmetic operations, but they certainly represent a way to introduce an ordering relation.
The chief properties of the ordinal level of measurement are
- The categories are distinct, mutually exclusive and exhaustive;
- The categories are possible to be ranked or ordered;
- The distance or differences from one category to the other category is not necessarily constant.
The interval level of measurement includes all the properties of the nominal and the ordinal level but an additional property that the difference (interval) between values is known and of constant size.
A thermometer, for example, measures temperature in degrees, which are of the same size at any point in the scale.
The difference between 20° C and 21° C is the same as the difference between 12° C and 13° C. The temperature 12° C, 13° C, 20° C and 21° C can be ranked and the differences between the temperatures can easily be determined.
It is also important to note that 0 is just an arbitrary point on the scale. It does not necessarily represent the absence of heat, just that it is cold. 0 degrees Celsius is 32 degrees on the Fahrenheit scale.
Owing to this, we cannot say that a temperature of 64°F is twice as warm as a temperature of 32°F.
Note that the Celsius equivalence of 32°F (the freezing point of water) is 0°C, while the equivalence of 64°F is 17.8°C. 17.8°C is not twice as warm as 0°C.
The Gregorian calendar is another example of an interval scale: 0 is used to separate BC and AD. It does not mean that there was no time before 0. We refer to the years before 0 as BC and to those after 0 as AD.
Incidentally, 0 is a hypothetical date in the Gregorian calendar because there never was a year 0. The other examples are IQ, calendar time (6 AM, 10 AM, etc.). The interval levels of data have the following properties:
- The data classification are mutually exclusive and exhaustive;
- The data can be meaningfully ranked or ordered;
- The difference between one data-classification to the next is known and constant.
In practice, almost all quantitative data fall under the ratio level of measurement. It has all the ordering and distance properties of the interval level.
Also, a ‘zero-point’ can be meaningfully designated, and thus the ratio between two numbers is also meaningful.
Examples of ratio level of measurement include wages, stock prices, sales values, age weight, and height.
Thus it makes sense to speak of 0 sales when there are no sales in the store. It is also quite meaningful to say a 4-feet tallboy is twice as tall as a 2-feet tallboy. A family with 6 members is twice as large as of a family with 3 members.
In comparing the four levels of measurement, we can conclude that an ordinal measure is a nominal measure, and also, has the ordinality property, an interval measure is an ordinal measure plus it has a unit of measurement. The ratio measure has all the properties of nominal, ordinal, and interval measures, plus it has an absolute or true zero.
The characteristic properties of the four levels of measurement and the way of deciding whether a particular level of measurement qualifies as nominal, ordinal, interval, or ratio; the following flowchart may be used:
- Do the numbers express a quantitative value or order?
If no, then -> nominal level. If yes, then ask:
- Do the differences between the numbers represent equal units of measurement (e.g., 3-2=4-3)?
If no, then -> ordinal level. If yes, then ask:
- Does the measurement have an absolute zero?
If no, then -> interval level. If yes, then -> ratio level.
The accompanying table attempts to compare the various levels of measurement.
Measurement Scales and their Comparison
|Interval||Ordered, equal intervals and arbitrary zero point|
|Ratio||Ordered, equal intervals and true zero point|