ESC 350: MEASUREMENT
Humans have always observed nature and natural processes with curiosity. Indeed, it was the act
of collecting and compiling observations that enabled humans to interpret nature and its processes.
Geoscientists observe conditions of the atmosphere, lithosphere, and/or hydrosphere to learn about the
Earth System processes. Observations, in fact, are made by counting things, ranking objects, or
measuring various attributes of objects. Each of these activities produces data.
In physical sciences, measurement is often associated with the act of comparing the
dimension/attribute of an object to a known standard, referred to as the measurement unit. In its broadest
sense, though, the term measurement is defined as the assignment of names or numerals to objects or
events according to certain rules. When the term measurement is used in the broad sense, it becomes
necessary to distinguish between measured quantities having different properties. Since the late 1940s, it
has become customary to arrange measured quantities (or data) into four different levels or scales,
nominal, ordinal, interval, and ratio.
Nominal Scale
This, the simplest form of measurement, associates objects or events with categories, groups, or
even numbers. Examples include a geologist categorizing a rock as igneous, metamorphic, or
sedimentary; and a meteorologist describing present weather condition as rainy, snowy, windy, cloudy, or
sunny. Even when numbers are involved, as is the case with telephone and social security numbers of
individuals, more often than not, they do not signify any quantitative attribute. A larger telephone number,
for example, does not connote any kind of quantitative superiority. Given the non-quantitative nature of this
type of measurement, very little computations can be done with the data reported on the nominal scale.
Ordinal Scale
Instead of simply grouping events or objects, if they can be ranked, more useful information can be
conveyed using the resulting data. The name ordinal scale is used when observations are ranked (or
ordered). Ranking rainfall events as light, moderate, or heavy is an obvious example. Likewise, minerals
are often ranked based on their hardness. Invariably, therefore, ordinal scale data imply that one value is
greater, larger, bigger, or better than the other. It is easy to assign numbers (1, 2, 3, etc.) to the ranks.
Hence, measurements belonging to the ordinal scale lend themselves to certain types of computations.
However, the scope of the arithmetic operations is often limited, as the ordinal scale does not indicate the
actual amount by which measurements are separated from each other.
Interval Scale
When the amount by which ranked measurements differ from each other can be quantified, the
resulting data assume true quantitative character. Such data belong to the interval scale. A well-defined
measurement scheme (or unit of measurement) is essential to collect data on the interval scale. Although
measurement schemes often involve a zero point, they are chosen as a matter of convention or
convenience. Therefore, an absolute zero need not be part of such a measurement scheme. Interval
variables, however, have equal distance between each value. When Kelvin, Celsius, or Fahrenheit is be
used to measure temperature, equal intervals connote that observations differ from each other by the exact
same amount. The difference between 90 and 100 is the same as the difference between 32 and 42. Like
temperature, interval scale measurements collected using a particular unit can easily be transferred to
another unit. Other examples of measurements using the interval scale are all around us. The altitude of a
place, time of occurrence of an event, angular distance of an object from a fixed point are familiar
examples.
Ratio Scale
There are measurement scenarios that involve an absolute zero. The zero in the Kelvin scale is a
good example, as is the zero in weight measurement. As per our present understanding, 0 K or the
absolute zero, is the lowest temperature possible. When the measurement scheme involves an absolute
zero, interval scale can be converted into the even more useful ratio scale. Measurements expressed as
ratios can be subjected to operations for determining equality, rank-order, equality of intervals, and equality
of ratios. Ratios, by extension percentages as well, enable measurements to be expressed using
languages such as an object weighs twice as much as another object or an object is 70% warmer than
another object.
Recently, some scientists have suggested that nominal, ordinal, interval, and ratio typologies can
be misleading. Alternate taxonomies for measurement levels, including the following one, have also been
suggested by some.
Names
Grades or ordered labels (e.g., Freshmen, Sophomore, Junior, Senior)
Ranks (starting with rank 1, data are arranged in an increasing or decreasing order)
Counted Fractions (Bounded by 0 and 1)
Counts (non-negative numbers)
Amounts (non-negative real numbers)
Balances (unbounded negative and positive values)
While the concept of measurement scale is important, it may not be as fundamental to data
analysis as once thought. It is counter productive to constrain the scope of data analysis based entirely on
measurement scales. Instead, what is important is to guide one s data analysis by what one hopes to learn
from the observations.
Significant Figures
In physical sciences, the term measurement in a restricted sense, to represent the act of
comparing the dimension/attribute of an object to a known standard. By dimension/attribute, we mean
some intrinsic property of the object, such as its length, mass, or density. Such measurements a numerical
part pertaining to the quantity or amount involved and the measurement unit. When measurements are
made with instruments made by humans, some uncertainty is inevitable. Factors contributing to this
uncertainty include limitations of the measuring device, the skill-level of the person making the
measurement, or both.
No matter what the measurement device, each digit in the numerical part of the measurement
assumes a specific meaning. In this context, the concept of the significant digit is important. A ruler is the
most familiar example of a measuring device. Suppose that in the following figure the whole numbers
represent centimeters and the arrow represents the end of an object that has its beginning exactly at the
zero mark.
Since the end of the object is located between 2.8 and 2.9 centimeters, we may report its length as 2.82
cm. The numerical part of the measurement has 3 significant digits. While we are certain about the validity
of the first two digits, the same cannot be said about the third digit. Even if the skill of the person measuring
the length of the object is not in doubt, it is not prudent to say that the measured length is indeed the true
value. Even though estimating the numerical value associated with the third digit cause error to enter into
our measurement, that digit is accepted as a significant digit. And, only one estimated digit is accepted as
significant.
As we multiply or divide observations, the number of significant figures in the final answer should
not exceed the lowest number of significant figures involved in the operation. While adding or subtracting
numbers, the number of decimal points in the result should match the smallest number of decimal points of
any value involved in the operation. After adding/subtracting numbers, report the answer by rounding it to
the least number of places in the decimal portion of any number in the problem.
The issue of significant figures may get little confusing when an observation is reported as 6500,
as it may be difficult to discern whether the observation has two, three, or four significant figures. However,
when the same number is written in scientific notation as 6.5 × 103, it is clear that there are just two
significant figures; three if the number is written as 6.50 × 103.
Measurement/Experimental Errors
By definition, error is the difference between a measured value and the true value. While this
definition provides a useful concept, the true value is seldom known, hence, it is of no use for quantifying
error. Errors fall into two categories: random and systematic. Often, repeated measurements of the same
quantity yield different values, because observations deviate from the true values due to measurement
error. Such measurement errors are called random errors; because, when measurements are made with
properly calibrated devices by skilled professionals, it is reasonable to expect that there will be just as
many uder estimates as there are over estimates. Further, the occurrence of under and over estimates and
their magnitudes are likely to be devoid of a systematic pattern. When individual observations are in error
by a fixed amount on a consistent basis, another kind error is to blame. It is referred to as systematic error.
Accuracy and Precision
By definition, accurate means capable of providing a correct reading or measurement. If a measurement
correctly produces the attribute of an object, such as its length, mass, or weight, then that measurement is
considered accurate. In other words, accuracy implies closeness of a measurement to the true value.
Precise, on the other hand, implies the exactness of an amount or the exactness with which an action is
executed. If repeated measurements produce the exact same outcome each time, such measurements are
considered precise. Observations with small random errors have higher precision; when systematic errors
are small, observations have higher accuracy. When both types of errors are small, observations will have
high precision and accuracy.