## Level or Scale of MeasureEdit

Mainstream statistics recognises *four* levels or scales of measure. these are

- Nominal
- Ordinal
- Interval
- Ratio

Each level has its own characteristics and is associated with a set of permissible statistical procedures. In what follows, the level will be characterised and associated with measures of central tendency. I will treat *interval* and *ratio* data as varieties of *scale* because this is how they are treated in SPSS but I will try to describe each adequately.

## Nominal DataEdit

The nominal level of measure is used for *categorical* data, where some item observed has been assigned to a discrete category. For instance, eye color of participants in a study might be categorised

- brown
- blue
- green
- other

These categories are named and the data are counted as nominal (from Latin *nomen* for name). The only procedure for quantitative analysis of these data is counting to discover frequencies of occurence of items: how many participants are assigned to each category. The categories are often *coded* numerically and in SPSS/PAWS the numeric codes can be given labels (using the *values* attribute of the variable).

The only measure of central tendency for nominal data is the mode which is the most frequently occuring category. There is no guarantee that a sample will produce a unique modal value.

## Ordinal DataEdit

The ordinal level of measure is used for data which form discrete categories but where these categories can be ranked on some scale. This ranking is a weak ordering of the data in that two items may share the same rank. In general the relationship ranking items *a* and *b* is

*a*<*b*or*a*>*b*or*a*=*b*

An example is income grouping. While income might be treated as a scalar variable, it is often useful to reanalyse raw income data into a less finely grained ranking of income groups. The following groupings have been suggested by the Office of National Statistics for example^{[1]}

Income Range Floor | Income Range Ceiling | Possible Band Code |
---|---|---|

Lowest in data | £5,199 | 1 |

£5,200 | £10,399 | 2 |

£10,400 | £15,599 | 3 |

£15,600 | £20,799 | 4 |

£20,800 | £25,999 | 5 |

£26,000 | £31,199 | 6 |

£31,200 | £36,399 | 7 |

£36,400 | £51,999 | 8 |

(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* and by the *median* which is the value that divides the data into two equal halves. If the observed frequencies of each possible value are ranked, then in an odd numbered ranking, the median is the middle item. In an even numbered ranking, the median is usually taken to be the mean of the two middle values.

The differences between the rank levels of this scale cannot be measured or compared. While we know that **a<b** and **b<c** and that **a<c** we do not know if the distance **ab** is equal to the distance **bc**.

## Scale DataEdit

### Interval DataEdit

Interval data values can be ordered and the distances between them compared. 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 modify a recipe by halving the indicated oven temperature and doubling the cooking time.

### Ratio DataEdit

Ratio data values are ordered and the distances between the points on the scale are equal and can be compared and the scale has a true zero point. For example, consider the measurement of height in meters: some objects have no elevation and a height zero means just that. 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 arithmetic mean is given by the sum of all the data values divide by the number of data points (in other words, what is commonly referred to as the average).

## NotesEdit

- ↑ 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.*