Statistical Analysis for Education and Psychology Researchers

numbers in measuring phenomena is dependent upon properties that apply to the
phenomena. It is not essential that a measurement scale has all the properties of numbers,
but the more properties that apply to a measurement scale, the more useful that scale
because of the more precise interpretation of meaning.
Similar to the way in which qualitative observations are subdivided, quantitative
observations are often subdivided into two types, discrete and continuous measurements.
Discrete measurements are those for which possible values are distinct and clearly
separated from one another: for example, the school variable ‘number of pupils in sixth
form’. It is impossible to have 43.5 pupils. Discrete measurements are usually counts
which must comprise of positive whole numbers called integers.
Continuous measurements are those which can, at least in theory, assume a
continuous and uninterrupted range of values. Examples include the variables ‘distance
from home to school in kilometres’ and ‘the average point score of pupils aged 16, 17 or
18 entered for 2 or more GCE A-levels or AS equivalent’. This latter example is in fact
one of a number of criteria used in the compilation of national school league tables.

2.2 Properties of Measurement Scales

In many texts measurement scales are referred to and four scales of measurement are
generally distinguished: nominal; ordinal; interval; ratio. Nominal and ordinal scales of
measurement equate to categorical classification. Interval and ratio are true scales of
continuous measurement

Nominal Scale

The nominal scale is used to label and hence classify observations into categories. The
procedure adopted for forming categories should be: well defined, mutually exclusive and
exhaustive. Numbers should not be used mathematically. Frequency counts may,
however, be compiled for different categories. When frequency counts on two or more
qualitative variable are tabulated the frequency table produced is called a contingency
table. Data in the contingency table may be treated statistically, for example, by use of
the Chi square test (see Chapter 6).

Ordinal Scale

The ordinal scale incorporates the properties of the nominal scale, labelling and
classification, and in addition introduces the meaning of order—either through ordered
categories or individual ranks. Numbers are used as labels in ordinal scales and do not
indicate amount or quantity. It should not, therefore, be assumed that intervals between
numbers are equal. For example, if a teacher places pupils in rank order in terms of their
maths achievement scores, from the pupil with the highest score to the pupil with the
lowest score, then it cannot be assumed that the difference in maths ability between
pupils ranked 2nd and 3rd is the same as that between the pupils ranked 1st and 2nd.
Moreover, it cannot be supposed that the pupil ranked 1st in the class has three times as
much maths ability as the 3rd ranked pupil. Statistical tests that are based on ordinal type

Statistical analysis for education and psychology researchers 22