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Chapter 1
Measuring Scales
1.1 Statistics and Its Applications
In the realm of theoretical and pure sciences, such as mathematics and physics, the
proof of a theorem or principle leads to an unequivocal conclusion. For instance, in a
right triangle, the square of the hypotenuse is always equal to the sum of the squares
of the other two sides. In physics, the velocity of an object is determined by dividing
the object’s displacement by the time taken. These events in the pure sciences always
occur with 100% certainty and are not subject to occasional or partial occurrences.
If even a single case were to be found where such a relationship does not hold true,
it would challenge the validity of the principle or theorem.
However, in the biological sciences, events do not exhibit such absolute certainty.
For example, despite the emphasis on handwashing, maintaining social distance,
and wearing masks to prevent Covid-19, there are instances where individuals who
follow these guidelines still contract the virus. Conversely, there are individuals who
do not adhere to the health protocols but do not contract the coronavirus.
When evaluating the efficacy of a new Covid-19 vaccine, people vary in their
response to the vaccine and their exposure to the disease. This means that some
unvaccinated individuals may not contract Covid-19, while some vaccinated
individuals might. In such cases, if the percentage of cases is significantly lower
among the vaccinated group compared to the unvaccinated, can we conclude that
the vaccine was not effective? Or could it be attributed to chance, or even
differences between the two groups in terms of social class, age, gender, or
occupation?
Research has demonstrated that smoking and alcohol consumption pose health
risks and can lead to heart disease and lung cancer. However, one can find elderly
individuals who have been long-time smokers and drinkers but show no symptoms
of these diseases and maintain good physical health. Conversely, there are young
individuals who have never used alcohol or tobacco but s till suffer from heart disease
or lung cancer. Moreover, many drugs used to treat a specific disease are effective
for the majority of patients, but there are cases where these drugs do not work.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
S. H. Saneii and H. Doosti, Practical Biostatistics for Medical and Health Sciences,
https://doi.org/10.1007/978-981-97-3083-4_1
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2 1 Measuring Scales
In the biological sciences, observations often vary from one situation to another,
and exact replication of results is not always possible. These variations arise due to
genuine and inherent differences or due to sampling variability. It is the
responsibility of statisticians to identify these differences and determine the extent
to which variations can be attributed to chance and how much is related to inherent
differences in the studied phenomena.
Statistics encompasses the scientific processes of planning, collecting, organizing,
summarizing, presenting, analyzing, and interpreting data, and it finds application
in various disciplines and fields. In the realm of medical and biological sciences,
statistics serves numerous purposes, including:
•
Assessing the accuracy of measurements.
•
Comparing different methods and techniques of measurement.
•
Evaluating the performance of diagnostic tests.
•
Establishing normal clinical and laboratory values.
•
Estimating disease prognosis.
•
Monitoring patients’ progress and response to treatment.
•
Determining the appropriate sample size for research studies.
•
Determining the incidence, prevalence, and mortality rates of diseases.
These applications highlight the vital role of statistics in guiding
decision-making, drawing meaningful conclusions, and gaining valuable insights in
the medical and biological sciences. By employing statistical techniques,
researchers and practitioners can make informed decisions, improve healthcare
outcomes, and advance scientific knowledge.
1.2 Type of Data
Observation refers to the act of noting a fact or event, often involving the
measurement of a quantity using appropriate instruments, as defined by Merriam
Webster. Not all observations are of the same kind and possess varying
characteristics. Each phenomenon and observation differs in nature from others,
leading to distinct approaches in measuring and recording their properties. Such
diverse nature of observations necessitates different analytical methods.
Observations can be recorded through measurement, counting, sorting, or simply
naming. In general, observations are broadly categorized into two groups:
“categorical or qualitative” and “numerical or quantitative”. These categories serve
as fundamental distinctions in understanding and analyzing different types of data.
Categorical or Qualitative Data
Categorical or qualitative observations encompass observations that cannot be
quantitatively measured or assigned numerical values. Instead, they are classified
based on their kind or similarity to one another. These observations can be
represented using codes or letters, which serve as symbols rather than actual
quantities or values. Examples of categorical data include pain levels in patients
(e.g., no pain, slight, mild, moderate, severe), the gender of patients (male, female),
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1.2 Type of Data 3
the types of diseases (e.g., cardiac, cancer, infectious, physical), and blood types
(A,B,O,AB).
The number of levels or categories assigned to each variable in categorical data
is limited and can be predetermined. When a variable has only two possible
responses, such as male or female for gender, it is referred to as a dichotomous
variable. On the other hand, if a variable has more than two possible responses, it is
called a polytomous variable. For example, the gender of individuals is a
dichotomous variable, while the opinions of individuals about a health program,
with responses ranging from “very poor” to “very good”, represent a polytomous
variable.
Qualitative observations can further be classified into the following two groups.
Nominal Data
Some observations cannot be subjected to a meaningful scale of measurement,
making it impossible to compare the answers in terms of better or worse, bigger or
smaller. In these cases, the only approach is to assign distinct names or categories
to the various possible answers or modes of the variable. Consequently, the data is
primarily distinguished by their separate categories. Examples of such nominal
data include the gender of a baby (boy, girl), blood types (A, B, O, and AB), marital
status (single, married, divorced, widowed), and colors of objects (green, blue,
yellow, black, red, etc.).
In nominal data, the focus is on identifying the distinct categories rather than
establishing any inherent order or hierarchy. These categories are not amenable to
numerical comparison or quantitative analysis. Instead, they serve as labels or
identifiers for different groups or characteristics within the dataset.
Ordinal Data
Certain types of data can be arranged in a specific order, either from small to large
or from bad to good (and vice versa). In this type of data, the focus is on the relative
positions and the order of items, without making any assumptions about the precise
differences between two states or stages. This type of measurement, based on
rankings, is known as ordinal scales.
For instance, the severity of a disease can be categorized into “mild, moderate,
severe” as a way of indicating its intensity. However, on an ordinal scale, it is not
possible to determine if the difference between a severity level of 3 and 4 is the same
as that between 7 and 8. Furthermore, two babies with the same Apgar score of 6
can still exhibit differences between them.
Similarly, let’s consider a customer satisfaction survey where respondents rate
their experience on a scale of “very dissatisfied, dissatisfied, neutral, satisfied, very
satisfied”. While we can determine that “very satisfied” is ranked higher than
“satisfied”, we cannot conclude that the difference between “neutral” and
“satisfied” is the same as the difference between “dissatisfied” and “neutral”. The
distances between these levels are not consistently measurable in terms of the
underlying variable being measured.
However, as scientific understanding progresses, new scales may be developed
to transform qualitative data into measurable and quantitative data. For instance, in
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4 1 Measuring Scales
the past, intelligence was often categorized as “very clever, clever, moderate”, etc.
But now, intelligence levels can be measured and expressed numerically through
intelligence quotient (IQ) scores.
Tip
If the ordinal scale has many levels (e.g., summed score of a questionnaire),
the data may have the statistical properties of interval scale.
As new measurement techniques and tools emerge, some qualitative data can be
translated into quantitative data, enabling a more precise and numerical
representation.
Numerical or Quantitative Data
Quantitative or numerical data refers to observations that can be measured or
counted and represented by numbers. Examples of quantitative data include the
number of family members, the prevalence of a specific disease, body temperature,
height, age, weight, white blood cell (WBC) count per unit of blood, and blood
pressure. Numerical measurements provide a higher level of precision and
consistency compared to qualitative data.
For instance, when we compare weights, we can confidently state that 500 g is
exactly double the weight of 250 g. The difference between 200 and 500 g carries
the same significance as the difference between 600 and 900 g. It is important to note
that not all quantitative data are alike, and they can be categorized into two distinct
groups: discrete and continuous.
Discrete Variables
Discrete data refers to variables that can only take specific, distinct values and cannot
assume any values in between. These values can be listed, even if the list is extensive
or infinite. Examples of discrete variables include family size, number of decayed
teeth, number of health personnel, number of white blood cells (WBC) in a unit of
blood, eyeglass size, and shoe size.
Count data, which are characterized by specific values such as 1, 2, 3, and so
on, are part of the discrete scale. There are no conceivable values between two
successive values, and they exhibit a distinct separation or “step” in their continuity.
For instance, family size can only be expressed as whole numbers such as 3, 4, 5, and
so forth, whereas stating the family size as 3.36 persons would not be meaningful.
However, certain articles may provide such numbers as an average of family sizes
in a population. These numbers represent estimates of the mean or average based
on observations of family sizes, obtained by dividing the sum of family sizes by the
total number of families. Typically, these average values are not whole numbers.
Similarly, eyeglass lenses (e.g., 1.00, 1.25, 1.50, 1.75, 2.00, 2.25, and 2.50) and
shoe sizes (e.g., 37.5, 38, 38.5, and 39) are examples of discrete data. Each value is
distinct, with no other values existing between them, creating a noticeable gap.
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1.2 Type of Data 5
Age is also considered a discrete variable since it is measured and expressed in
separate units such as months and years, without decimal units of time. To illustrate
this concept, imagine that a person’s age corresponds to the number of candles lit on
their most recent birthday cake (assuming everyone celebrates their birthday).
Tips
Age is also a discrete variable because it is expressed in separate units like
month and year, with no decimal unit of time.
Age is equal to the number of candles lit at their last birthday party.
Continuous Variable
If an observation has the potential to take on any conceivable value, it is considered
to have a continuous scale. This implies that there can exist additional values
between any two successive values, even at infinitesimally small intervals. In other
words, the range of possible values for a continuous variable is not restricted to a
finite list. Concrete examples of continuous data include length, volume, and
weight. When someone states their weight as 75 kg, it does not necessarily mean
that their exact weight is precisely 75,000 g. Rather, the actual weight could be
slightly higher or lower if measured with greater precision. In settings such as
pharmaceutical factories or jewelry workshops, goods and raw materials are often
weighed to several decimal places for enhanced accuracy. This allows for the
conceptualization of a different value between any two consecutive measurements
with greater precision. The following illustration demonstrates the distinction
between discrete and continuous scales.
Unlike discrete data, continuous data does not have distinct separations or gaps
between values. It allows for infinite possibilities within a given range or interval.
For example, between any two heights, there can be an infinite number of values.
It is important to note that continuous data is not limited to physical measurements.
Time can also be considered a continuous variable, where it can be measured with
precision down to the fraction of a second.
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6 1 Measuring Scales
Tips
Continuous data must be measured, but not counted.
Next value for any particular value of discrete data is known, but not for
continuous data.
In summary, continuous data encompasses variables that can assume any value
within a specified range or interval, allowing for decimal precision and infinite
possibilities between two points.
In the realm of quantitative data, there are two subcategories: interval and ratio
scales. The distinction between these scales lies in the arbitrariness or substantive
difference of their origin (i.e., the zero value). Nevertheless, in most cases, the
methods employed for their analysis remain the same. Software programs like
SPSS commonly classify both nominal and ordinal observations as either nominal
or ordinal scales. However, the statistical analysis approach remains consistent for
both types, and SPSS treats them as scales.
Interval Data
Observations that have a conventional and consistent origin, where the absence of
something does not imply “nothing” or “lack of”, are categorized as interval data.
In this case, negative values are also possible, as the starting point is arbitrary and
does not represent an absence. For instance, the freezing point of water serves as the
origin (zero point) for temperature, indicating the absence of a specific temperature
level. Thus, a temperature of −12° signifies being 12° below zero (the starting point).
The fundamental principle of the interval scale is that “equal intervals” correspond
to “equal variations”. However, there is no fixed relationship between the ratios of
the observed quantities. For example, if the temperatures of areas A and B are 40
and 10 °C, respectively, it does not imply that the temperature of area A is four times
that of area B. It simply means that the temperature of area B is 30 °C higher and
warmer than that of area A.
Another illustration of an interval scale is the calendar date. Typically, the birth
year of a prominent figure or the occurrence of a significant event is considered
the reference point for each country’s calendar date. For instance, Islamic countries
often designate the migration of Prophet Muhammad as the starting date (zero), while
European countries use the birth year of Jesus. Suppose event A took place in the
year 2000 AD and event B occurred in the year 500 AD. This does not imply that
event A transpired four times later than event B, but it signifies that event A occurred
1500 years after event B. On an interval scale, the temperature difference between
8° and 11° holds the same meaning as the difference between 4° and 7°. Conversely,
on an ordinal scale, the distinction between ranks 8 and 11 is not equivalent to the
difference between ranks 4 and 7.
In some cases, if an ordinal scale exhibits a large number of levels (e.g., with a
summed score ranging from 0 to 100, unlike a five-level rating on an opinion sheet),
the data may possess statistical properties similar to interval scales. Summated scales,
where scores are derived by summing responses to multiple questions, are often
treated as interval scales as well.
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1.2 Type of Data 7
For instance, in a questionnaire consisting of 20 ordinal questions (coded from 0 to
4), the total score can range from 0 to 80. In this context, the score of the questionnaire
can be regarded as an interval scale. Similarly, IQ is determined through a set of
multiple-choice questions and is thus considered an interval scale. When comparing
two individuals with IQ scores of 150 and 100, it is incorrect to state that the first
person’s score is 1.5 times higher than the latter’s. The only valid assertion is that
the difference in IQ between the two individuals is 50. This is because the IQ test
lacks an actual zero point; a score of zero does not signify a complete absence of
intelligence but rather reflects the individual’s performance in that specific test. If the
person were evaluated using different criteria and questions, their IQ score would
inevitably change.
When converting the measurement system for interval observations, the value of
zero in the new system no longer aligns with zero in the original system. For example,
if temperature is converted from 0 °C, the corresponding value in Fahrenheit would be
15°. Similarly, when transitioning from the solar calendar to the Gregorian calendar,
the value of solar year zero changes to 624 AD.
The key distinction between the interval scale and the ordinal scale lies in the
fact that the intervals between points on the interval scale are equal, whereas, on the
ordinal scale, the intervals between points are not equal.
Ratio Data
The ratio scale is characterized by having an actual value of zero, which represents
nothingness and absence. On this scale, negative values are not possible. For example,
a value of zero in the ratio scale indicates no weight or complete absence of the
measured attribute. In this scale, an object weighing 150 g is precisely three times
heavier than an object weighing 50 g. If we combine three objects weighing 50 g
each, their total weight will equal that of a single object weighing 150 g. Units
of measurement such as volume, length, weight, distance, area, speed, and time
exemplify the ratio scale.
By defining fundamental mathematical operations and associating them with
different scales and data, we can understand their characteristics as follows:
•
The nominal scale is not associated with any mathematical operations. It does not
allow for mathematical comparisons or operations of any kind.
•
The ordinal scale corresponds to the symbols “=”, “<”, and “>”. It allows for
comparisons between observations, but other mathematical operations such as
addition or subtraction are not applicable.
•
The interval scale corresponds to the symbols “+” and “−”. In addition to
comparisons, it enables us to determine the magnitude of the difference between
values.
•
The ratio scale corresponds to the symbols “×” and “/”. It encompasses
comparisons, measures the difference between values, and provides information
about the magnitude of one value relative to another in terms of multiplication
or division (Fig.
1.1).
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8 1 Measuring Scales
Numerical
)Quantitative(
Categorical
)Qualitative(
Observation
Polytomous
Dichotomous
Discrete
Continuous
Interval
Ratio
Ordinal
Nominal
Fig. 1.1 Types of observations
When analyzing data with statistical software, the type of input data and its
measurement scale must be specified. In SPSS, these scales must be introduced as
“nominal” (for polytomous and dichotomous data), “ordinal” (for ordinal
variables), and “scale” (for quantitative data including interval and ratio) via the
variable view tab and then the “measure” column.
1.3 Exercises
1. Determining the structure of data
•
A shoe store wants to analyze its sales data to identify the most popular shoe
sizes. What type of data structure and measurement scale would be most
appropriate for this analysis?
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1.3 Exercises 9
•
A vision clinic is conducting a study on the effectiveness of different
eyeglass prescriptions for correcting refractive errors. Which data structure
and measurement scale would be suitable for recording and analyzing the
data?
•
A nutritionist is investigating the relationship between calcium intake and bone
health. What data structure and measurement scale should be used to collect
and analyze data on the amount of calcium available in different nutrition
sources?
•
A pharmaceutical company is conducting a clinical trial to determine the
optimal dosage of a new medication. Which data structure and measurement
scale would be most suitable for recording and analyzing the dosage data?
2. Data Collection Methods
For each item listed below, determine the most appropriate data collection method:
A. Shoes and clothes size.
B. Refractive error (glasses).
C. The amount of calcium available in nutrition.
D. Drug dosage.
E. Treatment duration of a specific disease.
F. Infant weight.
G. The score of a student on biostatistics.
H. The volume of a gallon.
I. Workload of a clerk in a health center.
J. The infant gender.
K. The height of an infant.
L. Blood type.
M.
Race of people.
N. The interval between two deliveries of a mother.
O. Number of decayed teeth.
P. Relative humidity.
Q. Systolic blood pressure.
R. Pulse rate.
S. Education level.
T. The price of toothpaste.
U. The opinions of TV viewers.
V. The amount of lead in the air at gas stations.
W. The amount of chlorine in one cubic meter of water.
X. Amount of doctor’s visits.
Y. Carat of a piece of gold.
Z. Age of a student.
Note: In this exercise, you can suggest appropriate data collection methods based
on the context of each item.
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Chapter 2
Tables and Charts
2.1 Introduction
Presenting data and observed values in the best possible way is another task of
statistics. Without organizing the observations measured and collected, they can
neither be analyzed, nor interpreted, nor used. Therefore, classifying and organizing
observations in the form of frequency distribution tables and graphs is one of the
basic requirements for statistical methods.
2.2 How to Construct a Frequency Table
The construction of one- and multidimensional tables for qualitative data (nominal
or ordinal, dichotomous or polytomous) is straightforward, but for quantitative data
(discrete or continuous), it requires more experience and precision. First, the number
of classes and their limits are determined, then one person reads the raw data and
another person enters tally marks in the appropriate row. The tally marks should be
diagonal or vertical, and the fifth tally mark should always intersect with the first
four tally marks to make counting tally marks easier. After reading all the raw data
and ensuring that the data has been correctly transferred to the table, the number of
strokes in each row is counted and written as the absolute frequency ( f
i
). The index
i indicates the class number.
There are different points of views about the number of classes in a table, which
all are based on experience. The recommended number of classes is between (5–25)
or (8–15) or (10–25) or … I recommend the number of classes between 5 and 15 in
relation to the total number of observations. For qualitative data such as blood types
(A, B, AB, O), gender (male, female), geographical area (urban, rural), the same
number of responses of each qualitative variable can be used.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024
S. H. Saneii and H. Doosti, Practical Biostatistics for Medical and Health Sciences,
https://doi.org/10.1007/978-981-97-3083-4_2
11
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