Lecture 6
Measurement and Scaling Techniques
Measurement is defined as a process of associating numbers or symbols to
observations obtained in a research study. The observations could be quantitative or
qualitative.
Most of the analysis can be conducted using quantitative data. For example, mean,
standard deviation, etc. can be computed for quantitative characteristics. Qualitative
characteristics can be counted and cannot be computed. Therefore, the researcher
must have a clear understanding of the type of characteristic or variable before
colleting the data. The observations on qualitative variables may also be assigned
numbers. For example, we can record a person’s marital status as 1, 2, 3 or 4
depending on whether the person is single, married, widowed or divorced. We can
also record “Yes” or “No” answers to a question as “0” and “1”. In this artificial or
nominal way, categorical data (qualitative or descriptive) can be made into
numerical data and if we thus code the various categories, we refer to the numbers
we record as nominal data. Nominal data are numerical in name only, because they
do not share any of the properties of the numbers we deal in ordinary arithmetic. For
instance, if we record marital status as 1, 2, 3, or 4 as stated above, we cannot write
4 > 2 or 3 < 4 and we cannot write 3 – 1 = 4 – 2, 1 + 3 = 4 or 4 ÷ 2 = 2.
In those situations when we cannot do anything except set up inequalities, we refer
to the data as ordinal data. For instance, if one mineral can scratch another, it receives
a higher hardness number and on Mohs’ scale the numbers from 1 to 10 are assigned
respectively to talc, gypsum, calcite, fluorite, apatite, feldspar, quartz, topaz,
sapphire and diamond. With these numbers we can write 5 > 2 or 6 < 9 as apatite is
harder than gypsum and feldspar is softer than sapphire, but we cannot write for
example 10 – 9 = 5 – 4, because the difference in hardness between diamond and
sapphire is actually much greater than that between apatite and fluorite. It would also
be meaningless to say that topaz is twice as hard as fluorite simply because their
respective hardness numbers on Mohs’ scale are 8 and 4. The greater than symbol
(i.e., >) in connection with ordinal data may be used to designate “happier than”
“preferred to” and so on.
When in addition to setting up inequalities we can also form differences, we refer to
the data as interval data. Suppose we are given the following temperature readings
(in degrees Fahrenheit): 58°, 63°, 70°, 95°, 110°, 126° and 135°. In this case, we can
write 100° > 70° or 95° < 135° which simply means that 110° is warmer than 70°
and that 95° is cooler than 135°. We can also write for example 95° – 70° = 135° –
110°, since equal temperature differences are equal in the sense that the same amount
of heat is required to raise the temperature of an object from 70° to 95° or from 110°
to 135°. On the other hand, it would not mean much if we said that 126° is twice as
hot as 63°, even though 126° ÷ 63° = 2. To show the reason, we have only to change
to the centigrade scale, where the first temperature becomes 5/9 (126 – 32) = 52°,
the second temperature becomes 5/9 (63 – 32) = 17° and the first figure is now more
than three times the second. This difficulty arises from the fact that Fahrenheit and
Centigrade scales both have artificial origins (zeros) i.e., the number 0 of neither
scale is indicative of the absence of whatever quantity we are trying to measure.
When in addition to setting up inequalities and forming differences we can also form
quotients (i.e., when we can perform all the customary operations of mathematics),
we refer to such data as ratio data. In this sense, ratio data includes all the usual
measurement (or determinations) of length, height, money amounts, weight, volume,
area, pressures etc.
The above stated distinction between nominal, ordinal, interval and ratio data is
important for the nature of a set of data may suggest the use of particular statistical
techniques. A researcher has to be quite alert about this aspect while measuring
properties of objects or of abstract concepts.
MEASUREMENT SCALES
From what has been stated above, we can write that scales of measurement can be
considered in terms of their mathematical properties. The most widely used
classification of measurement scales are: (a) nominal scale; (b) ordinal scale; (c)
interval scale; and (d) ratio scale.
(a) Nominal scale:
Nominal scale is simply a system of assigning number symbols to events in order to
label them. The usual example of this is the assignment of numbers of basketball
players in order to identify them. Such numbers cannot be considered to be
associated with an ordered scale for their order is of no consequence; the numbers
are just convenient labels for the particular class of events and as such have no
quantitative value. Nominal scales provide convenient ways of keeping track of
people, objects and events. One cannot do much with the numbers involved. For
example, one cannot usefully average the numbers on the back of a group of football
players and come up with a meaningful value. Neither can one usefully compare the
numbers assigned to one group with the numbers assigned to another. The counting
of members in each group is the only possible arithmetic operation when a nominal
scale is employed. Accordingly, we are restricted to use mode as the measure of
central tendency. There is no generally used measure of dispersion for nominal
scales. Chi-square test is the most common test of statistical significance that can be
utilized, and for the measures of correlation, the contingency coefficient can be
worked out. Nominal scale is the least powerful level of measurement. It indicates
no order or distance relationship and has no arithmetic origin. A nominal scale
simply describes differences between things by assigning them to categories.
Nominal data are, thus, counted data. The scale wastes any information that we may
have about varying degrees of attitude, skills, understandings, etc. In spite of all this,
nominal scales are still very useful and are widely used in surveys and other ex-post-
facto research when data are being classified by major sub-groups of the population.
(b) Ordinal scale:
The lowest level of the ordered scale that is commonly used is the ordinal scale. The
ordinal scale places events in order, but there is no attempt to make the intervals of
the scale equal in terms of some rule. Rank orders represent ordinal scales and are
frequently used in research relating to qualitative phenomena. A student’s rank in
his graduation class involves the use of an ordinal scale. One has to be very careful
in making statement about scores based on ordinal scales. For instance, if Rahim’s
position in his class is 10 and Mohan’s position is 40, it cannot be said that Rahim’s
position is four times as good as that of Mohan. The statement would make no sense
at all. Ordinal scales only permit the ranking of items from highest to lowest. Ordinal
measures have no absolute values, and the real differences between adjacent ranks
may not be equal. All that can be said is that one person is higher or lower on the
scale than another, but more precise comparisons cannot be made. Thus, the use of
an ordinal scale implies a statement of ‘greater than’ or ‘less than’ (an equality
statement is also acceptable) without our being able to state how much greater or
less. The real difference between ranks 1 and 2 may be more or less than the
difference between ranks 5 and 6. Since the numbers of this scale have only a rank
meaning, the appropriate measure of central tendency is the median. A percentile or
quartile measure is used for measuring dispersion. Correlations are restricted to
various rank order methods. Measures of statistical significance are restricted to the
non-parametric methods.
(c) Interval scale:
In the case of interval scale, the intervals are adjusted in terms of some rule that has
been established as a basis for making the units equal. The units are equal only in so
far as one accepts the assumptions on which the rule is based. Interval scales can
have an arbitrary zero, but it is not possible to determine for them what may be called
an absolute zero or the unique origin. The primary limitation of the interval scale is
the lack of a true zero; it does not have the capacity to measure the complete absence
of a trait or characteristic. The Fahrenheit scale is an example of an interval scale
and shows similarities in what one can and cannot do with it. One can say that an
increase in temperature from 30° to 40° involves the same increase in temperature
as an increase from 60° to 70°, but one cannot say that the temperature of 60° is
twice as warm as the temperature of 30° because both numbers are dependent on the
fact that the zero on the scale is set arbitrarily at the temperature of the freezing point
of water. The ratio of the two temperatures, 30° and 60°, means nothing because
zero is an arbitrary point. Interval scales provide more powerful measurement than
ordinal scales for interval scale also incorporates the concept of equality of interval.
As such more powerful statistical measures can be used with interval scales. Mean
is the appropriate measure of central tendency, while standard deviation is the most
widely used measure of dispersion. Product moment correlation techniques are
appropriate and the generally used tests for statistical significance are the ‘t’ test and
‘F’ test.
(d) Ratio scale:
Ratio scales have an absolute or true zero of measurement. The term ‘absolute zero’
is not as precise as it was once believed to be. We can conceive of an absolute zero
of length and similarly we can conceive of an absolute zero of time. For example,
the zero point on a centimeter scale indicates the complete absence of length or
height. But an absolute zero of temperature is theoretically unobtainable and it
remains a concept existing only in the scientist’s mind. The number of minor traffic-
rule violations and the number of incorrect letters in a page of type script represent
scores on ratio scales. Both these scales have absolute zeros and as such all minor
traffic violations and all typing errors can be assumed to be equal in significance.
With ratio scales involved one can make statements like “Jyoti’s” typing
performance was twice as good as that of “Reetu.” The ratio involved does have
significance and facilitates a kind of comparison which is not possible in case of an
interval scale. Ratio scale represents the actual amounts of variables. Measures of
physical dimensions such as weight, height, distance, etc. are examples. Generally,
all statistical techniques are usable with ratio scales and all manipulations that one
can carry out with real numbers can also be carried out with ratio scale values.
Multiplication and division can be used with this scale but not with other scales
mentioned above. Geometric and harmonic means can be used as measures of central
tendency and coefficients of variation may also be calculated. Thus, proceeding from
the nominal scale (the least precise type of scale) to ratio scale (the most precise),
relevant information is obtained increasingly. If the nature of the variables permits,
the researcher should use the scale that provides the most precise description.
Researchers in physical sciences have the advantage to describe variables in ratio
scale form but the behavioural sciences are generally limited to describe variables in
interval scale form, a less precise type of measurement.
