1.6: Levels of Measurement
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Define and distinguish among nominal, ordinal, interval, and ratio levels of measurement
- Give examples of errors that can be made by failing to understand the proper use of measurement levels
Types of Measurement
When we want to learn more about a topic, we need to collect the right kind of data through measurement or observation, but how we measure something depends on the variable we are studying. For example, to measure how long it takes someone to respond to a sound, we would probably use a stopwatch. But a stopwatch would not be helpful if we wanted to measure someone's opinion about a political candidate. In that case, a rating scale would be our best option, with choices like "very favorable" or "somewhat favorable." In the instance of measuring someone's favorite color, using a rating scale would seem odd. Simply recording the name of the color they prefer would be a better method.
As you can see, measurements can look very different. To help organize them, they can be grouped into categories called levels of measurement. These categories are called levels because levels build on each other, meaning each level includes the features of the previous one. Similar to how animal-->mammal-->dog-->golden retriever becomes more specific, the levels of measurement move from nominal to ordinal, then to interval, and finally to ratio. We'll start with the broadest level, nominal, and work our way down to the most specific category, ratio.
Nominal Scale
A nominal scale is used when we only name or label categories. Some examples include gender, seasons, favorite color, and religion. The key idea is that these categories have no order. For example, when classifying people according to their favorite color, there is no measurement sense in which green is placed "ahead of" blue implying that it is "more" or "better". These categories are simply different. Nominal data represents the most basic level of measurement.
Ordinal Scale
An ordinal scale includes categories that can be ordered. For example, a researcher might want to measure customer satisfaction after a shopping experience by asking people to choose one of the following options: "very dissatisfied", "somewhat dissatisfied", "somewhat satisfied", or "very satisfied." Because there is an order, ordinal data allows us to compare responses. We can say that one person is more satisfied than another. However, ordinal scales do not tell us how much more satisfied one person is compared to another.
The differences between levels on an ordinal scale are not guaranteed to be equal. For example, the difference between "very dissatisfied" and "somewhat dissatisfied" may not be the same as the difference between "somewhat dissatisfied" and "somewhat satisfied." The scale does not give us enough information to measure the size of these differences. Even if we change the responses to numbers, such as rating satisfaction from \(1\) to \(4\), the problem remains. The difference between \(1\) and \(2\) does not necessarily mean the same thing as the difference between \(3\) and \(4\). Using numbers does not change the meaning of the scale and the data is still ordinal.
Classify the following variables as either nominal or ordinal. Explain.
- Eye color
- Answer
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Nominal, since there is no meaningful order to eye color (as normally denoted green, blue, hazel, and brown).
- BMI weight type (underweight, healthy, overweight, obese, severely obese)
- Answer
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Ordinal, there is meaningful order to BMI weight types, each listed category indicates an increasing BMI.
- Shirt size (S, M, L, XL)
- Answer
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Ordinal, there is meaningful order to shirt sizes, each listed value indicates a larger or smaller shirt (depending on the ordering).
- Phone number
- Answer
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Nominal, phone numbers are granted based on availability so they have no meaningful order.
Interval Scale
Hopefully you have noticed that qualitative data is measured using nominal or ordinal scales. When we work with quantitative data (where differences between values matter), we need to make further distinctions.
An interval scale is a numerical scale where equal differences between numbers mean the same thing everywhere on a scale. A common example is the Farenheit temperature scale. The difference between \(30\) degrees and \(40\) degrees represents the same temperature difference as the difference between \(80\) degrees and \(90\) degrees. This is because each \(10\)-degree increase means the same amount of change (in terms of kinetic energy of molecules).
However, interval scales have an important limitation: zero does not mean "none". On an interval scale, the value \(0\) is chosen for convenience and does not represent the absence of the quantity being measured. For example \(0\) degrees Farenheit does not mean there is no temperature, it simply marks a point on the scale based on how the system was created.
To differentiate interval scale from our next level of measurement, it is important to note that on interval scales, a measurement of zero does not mean "nothing" (or the absence of something), so it does not make sense to multiply or divide values on an interval scale. For instance, saying that \(80\) degrees Farenheit is "twice as hot" as \(40\) degrees Farenheit is not meaningful as the ratios do not represent anything real about temperature.
Years on a calendar are another example of interval data. Citizens of the United States remember the year \(1776\) as the year the Declaration of Independence was signed. Jews remember the year \(70\) as the year the temple was destroyed. The first recorded Olympic games occurred in \(776\)BC. The differences between these years are meaningful; the Declaration of Independence occurred \(1706\) years after the Romans razed the temple and \(2552\) years after the first recorded Olympic games. However, the year \(0\) does not mean that time stopped or began. Because zero does not represent the absence of time, years are measured on an interval scale.
Ratio Scale
The ratio scale is the highest and most informative level of measurement. It includes all the features of the other scales but adds one important property: a true zero point. On a ratio scale, zero means the complete absence of the quantity being measured.
Like nominal scales, ratio scales label categories using numbers. Like ordinal scales, the numbers can be ordered. Like interval scales, equal differences mean the same thing. In addition, ratio scales allow meaningful comparisons using multiplication and division.
For example, the Kelvin temperature scale is a ratio scale. Zero Kelvin represents absolute zero, where there is no heat energy at all. Because of this, it makes sense to say that \(80\) K has twice the kinetic energy as \(40\) K, therefore it is twice as hot.
Another example of a ratio scale is money. If you have $\(0\), that means you have no money. Since money has a true zero, we can say that someone with \(50\) cents has twice as much money as someone with \(25\) cents.
Ratio scales are very common. Most quantities of scientific interest tend to be ratio: distance, speed, weight, mass, pressure, volume, area, energy, population; these are all variables measured on ratio scales.
Identify a variable on the ratio scale that can take on both negative and positive numbers.
- Answer
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Note: Answers may vary.
Because ratio scales have a zero that truly means "none," it may seem like values on a ratio scale cannot be negative. This idea makes sense at first, but is not always true.
Some ratio measurements include direction, and these are called vector measurements. Vectors can have both positive and negative values while still being measured on a ratio scale. Examples include displacement, velocity, and acceleration. The sign shows direction, not the absence of quantity.
There are also ratio-scale variables that are not vectors but can still be positive or negative. Electric charge and money in a bank account both being examples. A negative balance means debt in a bank account, but a value of \(0\) still means there is no money in an account.
Consequences of Levels of Measurement
Why do we care so much about the type of scale used to measure a variable? The main reason is that the level of measurement affects which statistics we can use and what those statistics actually mean. Some calculations make sense for certain types of data but not for others. For example, imagine a study where five children are asked to choose their favorite color from blue, red, yellow, green, and purple. To record teh answers, the researcher assigns a number to each color, like this:
Table \(\PageIndex{1}\): Guide for Encoding Colors as Numbers
| Color | Code |
|---|---|
| Blue | \(1\) |
| Red | \(2\) |
| Yellow | \(3\) |
| Green | \(4\) |
| Purple | \(5\) |
This means that if a child said her favorite color was "Red," then the response was coded as \(2,\) if the child said her favorite color was "Purple," then the response was coded as \(5,\) and so forth. Consider the following hypothetical data:
Table \(\PageIndex{2}\): Favorite Colors and Code from Sample of \(5\) Children
| Subject | Color | Code |
|---|---|---|
| \(1\) | Blue | \(1\) |
| \(2\) | Blue | \(1\) |
| \(3\) | Green | \(4\) |
| \(4\) | Green | \(4\) |
| \(5\) | Purple | \(5\) |
Each code is a number, so nothing stops us from computing the average code assigned to the children. The average happens to be \(3,\) but you can see that it would not make sense to conclude that the average favorite color is yellow (the color with a code of \(3).\) This kind of mistake happens because "favorite color" is measured on a nominal scale. Averaging the numbers used as labels does not make sense. It would be like counting the letters in the name of a snake and using that number to decide how long the snake is - it just doesn't measure what we want to know.
Similarly, we can ask whether it makes sense to find the average of numbers measured on an ordinal scale. The answer to this question differs based on your field of study. For example, online reviews often report an average rating from \(1\) to \(5\) stars, and surveys with scales from \(0\) (strongly disagree) to \(10\) (strongly agree) are often summarized using an average.
However, this can be misleading. Suppose one runner finishes four races in \(1^{st},\) \(2^{nd},\) \(3^{rd},\) and \(4^{th}\) place. The average finish would be \(2.5^{th}.\) Now suppose another runner finishes \(3^{rd},\) \(3^{rd},\) \(1^{st},\) and \(3^{rd}\) in the same races and also has an average finish of \(2.5^{th}.\) Does this mean the two runners are equally fast? Not necessarily.
The problem is that ordinal data only tells us the order, not how big the differences are. One runner may have lost badly in some races and barely lost in others, but we cannot tell from placement alone. Because of this, averaging ordinal data should be done with caution, and conclusions based on those averages should be carefully questioned.
In contrast, once we reach interval and ratio levels of measurement, calculating an average is clear and meaningful. As we continue, it is important to always consider the level of measurement when deciding which descriptive statistics makes sense to use.