1.5 Measurement Scales
| Properties | Valid Math Operations | |
|---|---|---|
| Nominal / Categorical | None | \(=\), \(\neq\), \(\lt\), \(\gt\) |
| Ordinal / Rank | Magnitude | \(=\), \(\neq\), \(\lt\), \(\gt\) |
| Interval | Magnitude | \(=\), \(\neq\), \(\lt\), \(\gt\) |
| Interval | \(=\), \(\neq\), \(\lt\), \(\gt\) | |
| Ratio | Magnitude | \(=\), \(\neq\), \(\lt\), \(\gt\) |
| Interval | \(=\), \(\neq\), \(\lt\), \(\gt\) | |
| Absolute 0 | \(x_{1} / x_{2} = k\) |
Nominal (categorical) scales
The categories are mutually exclusive and exhaustive. The categories have no meaning by themselves – the placement of an observation in a particular category simply indicates that it is different from observations in other categories. It does not imply one category is more or less different.
- Examples: Car model name, eye color
Nominal cannot normally distributed.
Ordinal (rank) scales
Ordinal scales are like nominal scales with additional property of order among the categories included on the scale.
- Examples: social class, percentile ranks, and Olympic medal placing.
With ordinal scales, we can talk about the order of a set of objects, but we cannot say how much bigger one category is to another.
Ordinal cannot normally distributed.
Interval scales (equi-interval scales)
Sometimes called equi-interval scales to distinguish this level of measurement from nominal and ordinal scales.
This scale can be added, subtracted, multiplied, and divided without affecting the relative distances among scores.. The intervals between two scores can be determined by arithmetic manipulation of the scores.
An important characteristic of interval scales is that the zero point is arbitrary, which means that we cannot meaningfully interpret the size of particular score ratios. As a consequence of being measured on equi-units, it is possible for interval data to be normal under asymptotic assumptions. This is just a fancy way of saying that variables measured in interval scales have central tendency that can be measured by mode, median, or mean. Standard deviations for these variables can also be calculated.
Some would argue that centeral tendency for nominal and ordinal data can be calculated via mode. For practical purposes, I think that is valid, but in purist sense, I don’t think central tendency is defined for nominal and ordinal data.
Examples: Temperature scale like Celcius, Farenheit
10 Co vs 20 Co and 20 Co vs 30 Co: the degrees are 10 units apart, but just because \(\frac{20}{10}=2\), it doesn’t mean 20 Co is \(2 \times 10\) Co.
Your local weather report might say today’s weather (20 Co) is twice as hot as yesterday’s weather (10 Co), but that would be wrong because Celcius scale has no concept of ratio! This has to be with the fact that there is no meaningful \(0\) in Celcius. Celcius scale does have 0 to represent freezing point, but this is not a true 0 that represents absence of something, unlike Kelvin scale where 0 is complete absence of molecular motion. Therfore …Not an example: Kelvin
Kelvin is not an interval scale. It is a ratio scale because it has properties of interval scale with absolute zero. If two liquids have 10 Kelvin and 20 Kelvin respectively, that does indeed mean the latter is is twice as the former.
Ratio scales
Ratio scales have all the properties of interval scales with addition of an absolute zero.
- distance and time measures have genuine zero points.
This is what people most generally mean by “numeric” column