Random Variables (discrete and continuous)
Key Words and Terms:
· Scale, ranked
· Scale, ratio
DEFINITION and IMPORTANCE OF SCALES
Random variables are basically attributes or
are numerical descriptions. If they are attributes they are qualitative and if numerical they are quantitative. Scientific
work invariably uses quantitative approaches because they are more precise, accurate, objective, and convenient. However not
everything in the universe is precise and elegant. In is inevitable that the scientists will have to confront qualitative
attributes in real life however much he may try to avoid them. One of the areas of encounter is in communicating scientific
findings to the general public in which qualitative rather than quantitative descriptions are preferred.
QUALITATIVE and QUANTITATIVE SCALES
The qualitative scale is used for attribute or categorical variables. These variables
have no intrinsic numerical value. They arise as a result of classification. Qualitative variables are of three types: nominal,
ordinal, and ranked.
The quantitative scale is used for variables that arise as a result of measurement or
counts. Quantitative variables have an intrinsic numerical value. Quantitative variables are of two types: numerical continuous
& numerical discrete
IMPORTANCE OF SCALES:
All statistics is based on scales. Knowledge of scales is needed for correct choice of
statistical analytic procedures. The computer cannot on its own understand the correct scale to use. It can produce an output
that is meaningless if done using the wrong scale. There are two types of scales, qualitative and quantitative. We also talk
of quantitative and qualitative variables. Qualitative variables were used before but with more growth of science and technology,
quantitative variables have become more popular. Before the wide availability of computers, quantitative work was approached
with some trepidation. Now with wide availability of high-speed computing this fear has virtually disappeared
QUALITATIVE SCALES: nominal, ordinal, ranked
The nominal scale is unordered. Ordering is impossible even if desired because there is
no natural ordering of the categories. The categories of the nominal scale may be (a) 2, dichotomous/either-or e.g.: die/live
(b) 3, trichotomous: healthy, sick, and dead (c) more than 4 poly-chotomous / multi-chotomous: e.g.: race (black, white, Asian).
The distance between categories is not informative. The size of groups or categories is not relevant. The basis of categorization
is subjective and customary. The criteria are not rigorous. Misclassification in the nominal scale is possible and occurs
often. Customary categorization is not taken seriously and is not done with the same rigor as other types of classification. The nominal scale is used in the following: race, ethnicity, diagnosis, etc
There is a natural ordering in the ordinal scales. The order between the groups is pre-determined.
Any number of categories is possible. However too many categories are difficult to appreciate intuitively. The numerical scale
is better where the categories are more than a handful. The distance between categories on the ordinal scale is not uniform.
Size of the category is irrelevant. The categorization is based on some measure
of magnitude that may be determined subjectively or objectively. There is however no information about the absolute magnitude.
We for example can say not say that a PhD degree is at a certain quantifiable level of achievement. All we can do is an internal
comparison in which we say that PhD is above MA and BA degrees. Misclassification in the ordinal scale is likely because it
is often based on subjective criteria. Examples of the ordinal scale are: Stage
of metastatic cancer (I, II, III, IV), performance status of patient (0, 1, 2, 4), and academic degree (PhD, MA, BA)
In a ranked scale observations are arrayed in order of magnitude either ascending or descending.
Arraying is on the basis of an attribute like size, beauty, and expense. The attributes may be quantitative or qualitative.
The categories or ranks are given as 1st, 2nd, 4rd, etc. The distance between ranks is not
considered. The number in each rank is not important. Confusion may occur in case of tied ranks. The general rule is to skip
one rank to the next rank after the tie. Examples of the ranked scale are: ranking in sports competition, public elections,
and class examination performance.
NUMERICAL CONTINUOUS SCALE
DEFINITION and USES:
The numerical continuous scale is a result of measurement of length, weight, speed, volume,
time, or a combination of these. Continuous data cannot take exact values. Responses assume any value including decimals with
no restrictions at all. Any point on the scale is possible; the only limitation is accuracy of measurement. Further mathematical
manipulations are limited by degree of accuracy of the measurements. Readings on this scale are not always perfectly accurate because of the inevitable
rounding off error. The numerical continuous is the most used scale in medicine because it is the most detailed quantification.
It has 2 forms: the interval scale and the ratio scale.
INTERVAL NUMERICAL CONTINUOUS SCALE
The interval scale is a type of continuous scale in which the difference between 2 readings
has a meaning. The magnitude of the difference between 2 readings is the same at all parts of the scale for example the following
have the same magnitude: 20-10, 100-90. The ratio between two readings has no meaning for example a temperature of 100 degrees
Celsius in not twice as much as a temperature of 50 degrees Celsius. Zero on this scale is arbitrary and has no biological
meaning or significance. Temperature and the calendar are examples of interval scales. The interval scale has both positive
and negative values. The positive values are above the arbitrary zero and the negative ones are below it. Examples: temperature
on the Celsius scale, calendar.
Gabriel Daniel Fahrenheit (1688-1736) invented a thermometer named after him in 1714.
The freezing point of salty water was set at 0 degrees. The freezing point of normal water was set at 32 degrees. The boiling
point of normal water was set at 212 degrees. In 1724 Celsius (1701-1744) invented the thermometer named after him with freezing
point of water set at 100 degrees and boiling point set at 0 degrees. A year later the points were reversed such that the
freezing point became 0 degrees and the boiling point became 100 degrees.
RATIO NUMERICAL CONTINUOUS SCALE
The ratio scale is a type of numerical continuous in which zero has a biological significance.
Values on this scale can only be positive; negative valued would be meaningless. Both the difference and the ratio of 2 readings
on this scale have a meaning and can be interpreted. Intervals between 2 readings have the same meaning at different parts
of the scale e.g. 90-80 is the same as 70-60. The ratios however have different magnitudes at different parts of the scale
for example 90/80 is not the same as 70/60. Examples are bodyweight, height, blood pressure, and serum cholesterol
The ratio scale has the distinguishing characteristic of having an absolute zero
NUMERICAL DISCRETE SCALE:
The numerical discrete scale is a result of counting. It is a numerical scale that uses
only whole positive integers. There is no continuum of values. No values are permissible between any two integers. The numerical
discrete scale, unlike the numerical continuous, is exact because it is count of whole numbers. Examples of the numerical
discrete scale are: heart rate, respiratory rate, number of patients in a ward, number of vertebrae. The process of measurement
turns the continuous into the discrete. Discrete numbers, unlike the continuous, are exact.
EXERCISE ON SELECTING THE RIGHT SCALE
Indicate the most appropriate scale for
each of the following as follows: a=nominal, b=ordinal, c=rank, d=numerical continuous, e=numerical discrete
Blood pressure (mmHg)
Serum cholesterol level
Class position in the final examination
Government Civil Service grade/position
Year of study in the medical school (1st, 2nd, 4rd etc)
Number of cancer deaths
Highest certificate of education attained
New cases of coronary heart disease
Athletic competition prizes
Response to treatment by new drug (Complete, partial, none)
Number of bones in the human hand
Number of admissions to a hospital