>>What we can do with data depends on the kinds of data that we collect. How can we
display it and what kinds of mathematical operations can be performed?
Consider the numbers forty and eighty. The numbers forty and eighty could describe the
weights, in ounces, of two boxes. Or the numbers forty and eighty could describe the football
jersey numbers of a defensive back and tight end, respectively. Naturally, we can perform
arithmetic operations on the weights of the two boxes. Forty times two is eighty. That
is, the box that weighs eighty ounces indeed weighs twice as much as the forty-ounce box.
But that same arithmetic operation, forty times two equals eighty, makes no sense with
the jersey numbers. Those are still numerical values, but we know that mathematical operations
make no sense in this case. If you take “half the speed” or “half the weight” of the
player wearing eighty, you don’t get the player wearing forty. This is a straightforward
example that seems like a no-brainer, but there are other examples where folks often
make this mistake.
There are four scales of measurement that generally describe data that we might collect:
Nominal, Ordinal, Interval, and Ratio.
The nominal scale generally measures categories or classifications. For example, gender is
measured on a nominal scale, and there are two values that the variable “gender”
can take on: female and male. We can’t perform arithmetic operations on these categories:
three times female doesn’t equal two times male. Even when the categories are numerical,
we still can’t perform arithmetic operations. Consider the congressional districts in the
state of Oklahoma. District 1 plus District 2 doesn’t equal District 3, nor is any order
implied: District 1 is not necessarily bigger or more populous than District 2, same with
District 3 and so on. And the same is true with the football jerseys: eighty divided
by two doesn’t equal forty here, nor is a player wearing the jersey “eighty” better
than one wearing “forty,” or vice versa. Even if the jersey numbers represent positions
on the field, those positions are just categories.
The ordinal scale measures categories that have an implied order or rank. A non-numeric
example of an ordinal scale would be the medals awarded during the Olympics: gold, silver,
bronze. These are categories, but there is an order implied, as a gold medal signifies
a better performance relative to a silver medal, and likewise a bronze medal. A numerical
example that’s popular in the Midwest deals with tornado intensity ratings of EF1 to EF5.
The intensity of an EF5 tornado is greater than that of a tornado rated EF1, but an EF5
is not five times as intense. Similarly, the damage resulting from an EF2 and an EF3 tornado
do not equate necessarily to one EF5 tornado. An order is implied, but arithmetic operations
are not possible. A well-known numeric example that is regularly mishandled deals with questionnaires
measured on what is called a Likert scale, the common 1 to 5 or 1 to 10 ordered scale
ranging from “strongly disagree” with the given statement at hand to “strongly
agree.” The distance between 1 and 2 is not necessarily the same as the distance between
2 and 3, nor is an answer of “4” meant to be twice the strength of an answer of “2.”
The interval scale measures numerical, not categorical, data where the distances between
consecutive numbers have meaning. There aren’t many examples of interval scale measures,
but the most commonly discussed one is Fahrenheit temperature. With Fahrenheit temperature numbers,
the temperatures can be ranked, and the amounts of heat between consecutive readings are the
same: the distance between fifty-one degrees and fifty-two degrees is the same as the distance
between seventy-five degrees and seventy-six degrees. Also, the zero point is just a matter
of convention in an interval scale and not a natural or fixed zero point. That is, a
zero measurement doesn’t indicate an absence of value. In the Fahrenheit scale, 00F has
no implied meaning, just another value on the scale, and it doesn’t represent an absence
of temperature. Nor do positive and negative values have any particular meaning either.
The ratio scale is the highest level of measurement, with the same properties as interval but where
ratios are meaningful. Also, the zero point implies the absence of the value we’re interested
in. Most anything we as engineers measure will be on a ratio scale: weight, height,
length, volume, time, distance, and so on.
Broadly speaking, variables fall into two categories: qualitative and quantitative.
Qualitative data tend to describe characteristics that aren’t measured with numbers and are
measured with nominal or ordinal scales. Quantitative variables are those that are measured with
numbers and are measured with an interval or ratio scale. Quantitative data are further
described as being either discrete or continuous. Discrete variables are measured as integers
and tend to represent counts. For example, the number of cantilever bridges in Oklahoma
would be measured with an integer, as there can’t be 8.34 cantilever bridges. Continuous
variables are measured on a continuous number line. For example, the weight that a particular
cantilever bridge can withstand can take on several decimal places and is thus a continuous
variable.
Let’s take a look at a data set from the National Bridge Inventory from 2013, which
contains information describing over 630,000 bridges on public roads in the United States.
Shown is a subset of the more than 23,000 bridges that are found in Oklahoma. Let’s
distinguish the variables (running across the top) from the observations (each row represents
one bridge observation) from the sample of bridges. We generally refer to the actual
values of the variables for each observation as “data” – here, the data are found
in all the intersections of the columns and rows. Some of the variables include a bridge
identification that I created (so it really has no meaning), the county where the bridge
is located, the street or highway that the bridge carries, the object that the bridge
is crossing (like a creek or highway), and its location. All of the variables I just
mentioned are qualitative variables that describe each bridge in some way. The year that the
bridge was built should likely be viewed as a quantitative variable, and likely as a continuous
variable (as discrete variables are usually counts, and years are not really counts).
The span length is the distance between bridge supports, and the sum of the span lengths
is the total length, both of which are quantitative and continuous variables. The sufficiency
rating is a mathematical construct on a 0-to-100 percentage scale that accounts for several
factors describing the soundness of the bridge to stay in service. This scale would also
represent a continuous quantitative -
One last note about qualitative and quantitative aspects to data... The questions that we want
to answer, those questions that spur data collection in the first place, tend to be
qualitative questions. And answering them often requires us to collect quantitative
data that in some way serves as a proxy to addressing that qualitative question. For
example, perhaps we want to know whether one evacuation route is better than another. That’s
a qualitative question: which one is better? To answer, we have to determine what quantitative
variable or variables might help us decide which is better. Perhaps the quantitative
variable we can use as a proxy could be “average evacuation time per resident.” We could
simulate several evacuation scenarios for a large city, collecting the evacuation time
for each of the simulated residents, then make a comparison. In another example, we
might be interested in determining what material we should use to build a particular component
part out of. We could build the component out of several materials, put them in place
in our system, then observe different variables including deformation of the component under
use, the weight the component adds to the system, and its time until failure, among
others. With that collection of quantitative variables, we can draw a conclusion about
the qualitative question broadly asked as “what material should we use?” As engineers,
we should always start with the root question we want to ask, then determine what variables
we need to collect to best answer that question.