Level or Scale of MeasureEdit
Mainstream statistics recognises four levels or scales of measure. These are
- Ratio (combined with Interval as Scale in SPSS)
These are in order from most name-like to most number-like. Each level has its own characteristics and association with a set of permissible statistical procedures. Below, the level will be characterised and associated with one or more measures of central tendency, viz., mode, mean, and median.
The Nominal level of measure is used for categorical data, where each value has each been assigned to a discrete category. For instance, eye color of participants in a study might be nominally (from Latin nomen for name) categorised into groups
The only procedure for quantitative analysis of these data is counting, to discover frequencies of occurrence. That is, how many individuals are assigned to each category. The categories are often coded numerically (i.e., assigned a unique number) and named (using the values attribute of the variable).
The only measure of central tendency for nominal data is the mode, which is the most frequently occurring category. Note that there is no guarantee that a sample will produce a unique modal value.
The Ordinal level of measure is used for data which form discrete categories and can be naturally ranked on some scale. This ranking is a weak ordering of the data in that two values may share the same rank: the relative rank of a and b is
- a < b or
- a > b or
- a = b
An example of ordinal data is income grouping. While income might be treated as a scalar variable, it is often useful to create categories out of income ranges. For example, the following groupings have been suggested by the Office of National Statistics
|Income Range Floor||Income Range Ceiling||Possible Band Code|
|Lowest in data||£5,199||1|
(I might add a ninth band to deal with all cases with income over £51,999)
As with nominal data, these groupings might each be represented by a numeric code.
The central tendency in ordinal data may be represented by the mode (defined above) and by the median, the value that divides the data into equal halves. This is the middle value when the cases are odd-numbered. Else, the median is usually taken to be the arithmetic mean (see below) of the two middle values.
The differences between the rank levels of this scale cannot be measured or compared: while we know that, of ordinal data points a, b, and c, a<b, b<c, and a<c, we do not know if the distance ab is equal to the distance bc.
Interval data values can be ordered and the distances between them compared. However, the zero point of interval data is arbitrary. An oft quoted example is the measure of temperature on the Celsius scale. Here, the freezing point of water is arbitrarily assigned the value zero and the boiling point of water is arbitrarily assigned the value 100. While 50° is indicated half way between these two marks on the scale, it is not coherently half the boiling point of water. You cannot, for example, logically modify a recipe by halving the indicated oven temperature and doubling the cooking time.
Ratio data is all of: ordered, of comparable distance (successive, integral points on the scale are equally spaced), and on a scale with a true zero point.
For example, consider the measurement of height in meters: some objects have no elevation, which naturally maps to a height of zero meters.
The values can also form ratios, such that any value can be expressed as a ratio of other values. If we find, for example, three people of heights 1.5m, 1.75m and 2m, we can express any one of these as a multiple of any other.
The central tendency in scale data can be indicated by the mode, the median and the arithmetic mean.
The first two are discussed above.
The arithmetic mean is the sum of all the data values divided by the number of data points. In other words, it is what is commonly referred to as the average.
- Stevens, S. S.. [http://www.ons.gov.uk/about-statistics/harmonisation/secondary-concepts-and-questions/S4.pdf Harmonised Concepts and Questions for Social Data Sources: Secondary Standards]. http://www.ons.gov.uk/about-statistics/harmonisation/secondary-concepts-and-questions/S4.pdf.