1.4. Levels of Measurement

The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level) [Illowsky and Dean, 2023]:

1.4.1. Nominal Scale Level

The nominal scale represents the most basic level of measurement. Data classified on this scale are purely qualitative and are grouped based on categories or names without any numerical significance. For example, when considering the city of Calgary, one could use the nominal scale to categorize neighborhoods into groups such as downtown, suburbs, or industrial areas. Other nominal data might include the predominant colors in Calgary’s cityscape or the names of major rivers like the Bow River and Elbow River. This scale is also used for labeling different types of public transportation, such as buses and trains, or for collecting survey responses on binary questions like yes/no support for a city project.

Other Examples:

• Colors: Identifying the predominant colors in Calgary’s cityscape.

• Names: Listing the names of Calgary’s major rivers (e.g., Bow River, Elbow River).

• Labels: Categorizing different types of public transportation (e.g., buses, trains).

• Favorite Foods: Surveying Calgarians about their favorite local dishes (e.g., beef stew, ginger beef).

• Yes or No Responses: Collecting data on whether residents support a specific city project (e.g., new bike lanes).

1.4.2. Ordinal Scale Level

The ordinal scale is particularly useful when the order of data points is important, but the exact differences between them are not known or are irrelevant. For instance, in a city like Calgary, residents might be asked to rank their preference for potential city events from most to least anticipated. The resulting list could include events such as the Calgary Stampede, the Calgary International Film Festival, and various sports games, among others. Similarly, one could rank local coffee shops based on customer satisfaction, with ratings such as ‘best’, ‘better’, ‘average’, and ‘below average’. These rankings provide a sequence that reflects preferences or perceptions, but they do not quantify the distance between each rank. Another example could be the level of concern citizens have regarding different city issues, ranked from ‘most concerned’ to ‘least concerned’. This could cover topics such as traffic congestion, pollution, or public safety. While these examples show a clear order, they do not allow us to measure how much more one rank is valued over another, which is a key characteristic of the ordinal scale level of measurement.

1.4.3. Interval Scale Level

The interval scale is a type of quantitative measurement that ranks items and measures the distance between them. It features equal intervals between values, allowing for arithmetic operations like addition and subtraction. For example, temperature measurements in Calgary, whether in Celsius or Fahrenheit, are interval data. The 10-degree difference between 20°C and 10°C is equivalent to that between 30°C and 20°C. However, interval scales lack a true zero point, making ratios meaningless. It’s incorrect to say 20°C is twice as hot as 10°C because the zero point isn’t absolute.

Additional Examples:

• Calendar Years: Years can be consistently measured against each other. However, the year 2000 isn’t ‘twice as late’ as the year 1000 due to the absence of an absolute zero year.

• SAT Scores: SAT test scores are interval data with equal scoring intervals, making differences measurable. Yet, a score of 1200 isn’t ‘twice as good’ as a score of 600.

• Time of Day: Time, as indicated by clocks, is an interval scale with consistent hourly intervals. But midnight doesn’t represent a ‘lack of time,’ so it’s not a true zero point.

1.4.4. Ratio Scale Level

The ratio scale possesses all the characteristics of the interval scale, with the addition of a true zero point, which allows for meaningful comparisons of ratios. This is the highest level of measurement and is fully quantitative. In Calgary, if we consider the scores of students on a statistics final exam, we can not only order these scores but also meaningfully say that a score of 80 is twice as high as a score of 40. The zero point on this scale means the absence of the quantity being measured, which allows for the calculation of ratios and differences.

Additional Examples:

• Weight Measurements: In any gym in Calgary, the weight of dumbbells is measured on a ratio scale. If one dumbbell weighs 10 kg and another weighs 20 kg, it’s accurate to say that the latter is twice as heavy as the former because weight has a true zero point.

• Distance: The distance between two locations in Calgary, measured in kilometers or miles, is a ratio scale. If one park is 10 km from a starting point and another is 20 km away, the second park is indeed twice as far.

• Volume of Liquid: The volume of liquid in a bottle, measured in liters, is another example. A 2-liter bottle contains twice the volume of liquid as a 1-liter bottle, demonstrating the meaningfulness of ratios.

• Bank Account Balance: Money in a bank account is measured on a ratio scale. If one account has $200 and another has $400, the second account has twice the amount of money.

Table 1.1 outlines the characteristics of the four levels of measurement in statistics:

Table 1.1 Characteristics of the Four Levels of Measurement in Statistics

Level of Measurement	Put data in categories	Arrange data in order	Subtract data values	Determine if one data value is a multiple of another
Nominal	Yes	No	No	No
Ordinal	Yes	Yes	No	No
Interval	Yes	Yes	Yes	No
Ratio	Yes	Yes	Yes	Yes

In Table 1.1:

1. Level of Measurement:

   • This column categorizes data into four levels: nominal, ordinal, interval, and ratio. Each level represents a different type of measurement scale used for statistical data.

2. Put Data in Categories:

   • This column indicates whether data at a given level can be classified into distinct categories.

   • For example:

     • Nominal: Yes. Data can be categorized but not ordered.

     • Ordinal: Yes. Data can be categorized and ranked.

     • Interval: Yes. Data can be categorized with meaningful intervals between values.

     • Ratio: Yes. Data can be categorized with a true zero point and meaningful ratios.

3. Arrange Data in Order:

   • This column specifies whether data can be ordered or ranked.

   • For example:

     • Nominal: No. Data cannot be ordered.

     • Ordinal: Yes. Data can be ranked in a meaningful order.

     • Interval: Yes. Data can be ordered with meaningful intervals.

     • Ratio: Yes. Data can be ordered with a true zero point and meaningful ratios.

4. Subtract Data Values:

   • This column indicates whether subtraction between data values yields meaningful results.

   • For example:

     • Nominal: No. Subtraction is not meaningful.

     • Ordinal: No. Differences between ranks are not meaningful.

     • Interval: Yes. Differences between values are meaningful.

     • Ratio: Yes. Differences between values are meaningful with a true zero point.

5. Determine if One Data Value Is a Multiple of Another:

   • This column specifies whether it is meaningful to calculate ratios between data values.

   • For example:

     • Nominal: No. Ratios are not meaningful.

     • Ordinal: No. Ratios are not meaningful.

     • Interval: No. Ratios are not meaningful because there is no true zero point.

     • Ratio: Yes. Ratios are meaningful with a true zero point, allowing for comparisons such as “twice as much”.