Hellbent on Measurement

tape-measure-on-a-catAny variable that you record in a dataset will have some scale of measurement. Scales of measurement are, simply put, the ways that associated numbers relate to each other. Scales are properties of numbers, not the objects being measured. You could measure the same attribute of an object using more than one scale. For example, say you were doing a study involving cats and wanted to have a measure of each cat’s age. If you knew their actual birth dates, you could calculate their real ages in years, months, and days. If you didn’t know their birth dates, you could have a veterinarian or other knowledgeable individual estimate their ages in years. If you didn’t need even that level of precision, you could simply classify the cats as kittens, adult cats, or mature cats.

Understanding scales of measurement is important for a couple of reasons. Use a scale that has too many divisions and you’ll be fooled by the illusion of precision. Use a scale that has too few divisions and you’ll be dumbing down the data. Most importantly, though, scales of measurement determine, in part, what statistical methods might be applied to a set of measurements. If you want to do a certain type of statistical analysis on a variable, you have to use an appropriate scale for the variable. There are a few intricacies involved with measurement scales, so for now, just know that you have to understand a variable’s scale of measurement in order to analyze those data and interpret what it all means.

sound-27302bac00000578-3021300-image-a-35_1427885938930In Statistics 101, you’ll learn that there are four types of measurement scales – nominal, ordinal, interval, and ratio. This isn’t entirely true. The four-scale classification, described by Stevens (1946)[1], is just one way that scales are categorized, though it’s mentioned in almost every college-level introduction to statistics. There are actually a variety of other measurement scales, some differing in only obscure details.

The most basic classification of measurement scales involves whether or not the scale defines (1) groups having no mathematical relationship to each other, called grouping scales, or (2) a progression of measurement levels within a group, called progression scales.

Grouping (Nominal) Scales

Grouping scales define groups, which are finite, usually independent, and non-overlapping (discrete). Nominal scales are grouping scales. They represent categories, names, and other sets of associated attributes. None of the levels within a grouping scale have any sequential relationship to any of the other levels. One level isn’t greater than or less than another level.

Examples of properties that would be measured on a qualitative scale include:

  • Names—Kyle, Stan, Eric, Kennycup-beaker-vv9areh
  • Sex—female, male
  • Identification—PINs, product serial numbers
  • Locations—Wolf Creek, Area 51, undisclosed secure location
  • Car styles—sedan, pickup, SUV, limo, station wagon
  • Organization—company, office, department, team

Grouping scales are sometimes subdivided by the number of measurement levels. Discrete scales have a finite number of levels. For example, sex has two levels, male and female. Discrete scales with two levels are also called binary or dichotomous scales. Discrete scales with more than two levels are called categorical scales.

Variables measured on grouping scales can be used for counts and statistics based on counts, like percentages. They are also used to subdivide variables measured on progression scales.

ordinal-progression

Progression Scales

Progression or continuous scales define some mathematical progression. The number of possible levels may be finite or infinite. They can be limited to integers or use an integer and any number of decimal points after the integer. Ordinal, interval, and ratio scales are all progression scales.

Ordinal Scales

Ordinal scales have levels that are ordered. The levels denote a ranking or some sequence. One measurement may be greater than or less than another. However, the intervals between the measurements might not be constant.

Examples of properties that would be measured on an ordinal scale include:

  • Time—business quarter, geologic period, football quarters
  • Rankings—first place, second place, third place …
  • Thickness—geologic strata, atmospheric layers
  • Survey responses—very good, somewhat good, average, somewhat bad, very bad

Sometimes the intervals between levels of an ordinal scale are so different they can be treated as if they were grouping scales. Consider geologic time. It’s divided into eon, eras, periods, epochs, and ages, but the divisions aren’t the same lengths. Some periods are four times longer than others and the lengths can change as more is learned about the history of Earth. The units of the scale are also different in different parts of the world. Then there’s Moh’s scale of mineral hardness. It consists of ten levels. However, the interval between levels 1 and 8 is about the same as the interval between levels 8 and 9. The interval between levels 9 and 10 is four times greater than the interval between levels 8 and 9. Geologists must be a bunch of really creative people who aren’t bound by convention.

More frequently, the intervals between levels of an ordinal scale are the same, in theory or reality. Rankings, game segments like innings and periods, business quarters and fiscal years, are all examples.

cup-70694-002132826jpg-vf2a

Counts and statistics based on medians and percentiles can be calculated for ordinal scales. This includes most types of nonparametric statistics. However, there are situations in which averages and standard deviations are used. Surveys present one of those situations because the responses can be considered to be either grouping or progression scales depending on how the levels are defined. Say you have a survey question that has five possible responses:

  • Very good
  • Good
  • No opinion
  • Poor
  • Very poor

This is a grouping scale because the No Opinion response is not part of a progression. But, if the responses were:scale-twitch_scale

  • Very good
  • Good
  • Fair
  • Poor
  • Very poor

The scale could be recoded as Very Good=5, Good=4, Fair=3, Poor=2, and Very Poor=1 allowing statistical analyses to be conducted. If it were believed that the intervals between levels were not constant, analyses should be limited to counts and statistics based on medians and percentiles. If the intervals between levels were believed to be fairly constant, calculating averages and standard deviations might be legitimate. This is one of the points of contention with Stevens’s categories of scales. A given measurement’s scale might be perceived differently by different users.

Ratio Scales

Ratio scales are the top end of progression scales. Their levels consist of integers followed by any number of decimal points. Ratios and arithmetic operations are meaningful. Zero is a constant and a reference to an absence of the attribute the scale measures.

Measurements made by most kinds of meters or other types of measuring device are probably ratio scales. Examples of variables measured on ratio scales include:five

  • Concentrations, densities, masses, and weights
  • Durations in seconds, minutes, hours, or days
  • Lengths, areas, and volumes

Any type of statistic can be calculated for variables measured on a ratio scale.

Other Scales of Measurement

Understanding different types of measurement scales can help you select appropriate techniques for an analysis, especially if you’re a statistical novice. Stevens’s classification of scales works for many applications but it should be viewed as guidance rather than gospel. Interval scales in particular are an exception to the progression of scales form ordinal to ratio scales, and there are other exception scales as well. The following sections describe interval scales and a few scales that don’t quite fit into Stevens’s taxonomy.

Interval Scales

Interval measurements are ordered like ordinal measurements and the intervals between the measurements are equal. However, there is no natural zero point and ratios have no physical meaning. The classical example of an interval scale is temperature in degrees Fahrenheit or Centigrade. The intervals between each Fahrenheit degree are equal, but the zero point (-32 degrees) is arbitrary. Elevation is sometimes considered to be an interval scale temperature-should-hospital_e2d565717fa09970because the choice of sea level as the zero elevation is arbitrary. Time can also be thought of as an interval scale.

Some statisticians consider log-interval scales of measurement, in which the intervals between levels are constant in terms of logarithms, to be a subset of interval scales. Earthquake intensity (Richter and Mercali scales) and pH are examples of log-interval scales.

Statistics for ordinal scales and statistics based on means, variances, and correlations can be calculated for interval scales.

Counts

count-the-cats

Counts are like ratio scales in that they have a zero point, constant intervals and ratios are meaningful, but there are no fractional units. Any statistic that produces a fractional count is meaningless. The classic example of a meaningless count statistic is that the average family includes 2.3 children. Counts are usually treated as ratio scales, but the result of any calculation is rounded off to the nearest whole unit.

Restricted-Range Scales

A constrained or restricted-range scale is a type of scale that is continuous only within a finite range. Probabilities are examples of constrained scales because any number is valid between the fixed endpoints of 0 and 1. Numbers outside this range are not possible. Percentages can be considered constrained or unconstrained depending on how the ratio is defined. For example, percentages for opinion polls are restricted to the range 0 to 100 percent. Percentages that describe corporate profits can be negative (i.e., losses) or virtually infinite (as in windfall profits). Restricted-range scales must be handled with special statistical techniques, such as logistical regression, that account for fixed scale endpoints.cat-bicycle

Cyclic Scales

Cyclic scales are scales in which sets of units repeat.

Repeating Units

Some cyclic scales consist of repeating levels for measuring open-ended quantities. Day of the week, month of the year, and season are examples. Time isn’t the only dimension with repeating scales, either. Musical scales, for instance, repeat yet have very different properties compared to time scales.

Repeating scales can be analyzed either by (1) treating them as an ordinal scale or (2) ignoring the repeating nature of the measure and transforming them into non-repeating linear units, such as day 1, day 2, and so on, or using a specialized statistical technique. The objective of the statistical analysis dictates which approach should be used. The first approach might be used to identify seasonality or determine if some measurement is different on one day or month rather than another. For example, this approach would be used to determine if work done on Fridays had higher numbers of defects than work done on other days. The second approach might be used to examine temcompass-20130531-182857poral trends. The third approach is used by statisticians who want to show off.

Orientation Scales

Orientation scales are a special type of cyclic scale. Degrees on a compass, for example, are a cyclic scale in which 0 degrees and 360 degrees are the same. Special formulas are required to calculate measures of central tendency and dispersion on circles and spheres.

Concatenated Numbers and Text

Concatenated numbers and text are not scales in the true sense of variable measurement, but they are part of every data analysis in one way or another. Concatenated numbers contain multiple pieces of information, which must be treated as a nominal scale unless the information can be extracted into separate variables. Examples of concatenated numbers include social security numbers, telephone numbers, sample IDs, date ranges, latitude/longitude, and depth or elevation intervals. Likewise, labels can sometimes be parsed into useful data elements. Names and addresses are good examples.

Time Scales

Time scales have some very quirky properties. You might think that time is measured on a ratio scale given its ever finer divisions (i.e., hours, minutes, seconds), yet it doesn’t make sense to refer to a ratio of two times any more than the ratio of two location coordinates. The starting point is also arbitrary. This sounds like an interval scale.

time-daylight-savings-time-cat-checks-for-accuracy

Time is like a one-dimensional location coordinate but it can also be linear or cyclic. Year is linear, so it’s at least an ordinal scale. For example, 1953 happened once and will never recur. Some time scales, though, repeat. Day 8 is the same as day 1. Month 13 is the same as month 1. So, time can also be treated as being measured on a nominal scale.

Time units are also used for durations, which are measured on a ratio scale. Durations can be used in ratios, they have a starting point of zero, and they don’t repeat (eight days aren’t the same as one day).

Time formats can be difficult to deal with. Most data analysis software offer a dozen or more different formats for what you see. Behind the spreadsheet format, though, the database has a number, which is the distance the time value is from an arbitrary starting point. Convert a date-time format to a number format, and you’ll see the number that corresponds with the date. The software formatting allows you to recognize values as times while the numbers allow the software to calculate statistics. This quirk of time formatting also presents a potential for disaster if you use more than one piece of software because different programs use different starting dates for their time calculations. Always check that the formatted dates are the same between applications.location-d71_2271

Location Scales

Just as there is time and duration, there is location and distance (or length), but there are a few twists. Time is one-dimensional; at least as we now know it. Distance can be one-, two-, or three-dimensional. Distance can be in a straight line (“as the crow flies”) or along a path (such as driving distance). Distances are usually measured in English inches, feet, yards, and miles or metric centimeters, meters, and kilometers. Locations, though, are another matter. Defining the location of a unique point on a two-dimensional surface (i.e., a plane) requires at least two variables. The variables can represent coordinates (northing/easting, latitude/longitude) or distance and direction from a fixed starting point. Of the coordinate systems, only the northing/easting scheme is a simple, non-concatenated scale that can be used for classical statistical analysis. However, this type of scale is usually not used for published maps, which can be a problem because virtually all environmental data are inherently location-dependent and multidimefly-c2bac65889b946dec4996a0a248e2ba0nsional. Thus, coordinate systems usually have to be converted for one to the other. Geostatistical applications, for example, are based on distance and direction measurements but these measurements are calculated from spatial coordinates.

At least three variables are needed to define a unique point location in a three-dimensional volume, so a variable for depth (or height) must be added to the location coordinates. Often, however, a property of an object occurs over a range of depths (or heights or elevations) rather than a finite point. Unfortunately, depth range is a concatenated number (e.g., 2-4 feet). It’s always better to use two variables to represent starting depth and ending depth. Thus, it may take four variables to define an environmental space, such as the sampled interval of a well or soil boring.

Selecting Scales

In the simplest taxonomy, almost all scales act either to group data othe_cat_stairsr represent the progression in a variable’s attribute, whether simple, ordinal-scale levels or more expansive ratio-scale levels. One way to view these differences is this: nominal (grouping) scales are like stone outcrops, randomly scattered around a garden area. Ordinal scales are like garden steps. You can only be on a step not between steps, and the steps lead progressively upward or downward. There may be many steps or just a few. Ratio scales are like a garden path or ramp. You can be anywhere along the path, at high levels or low. You can move forward or back, in small or large intervals.

Somewhere between those simple, discrete ordinal scales and the finely-divided ratio scales, however, are quite a few types of scales that don’t meet either definition. Just ask yourself these questions to understand the scale you will be dealing with:

  • Does the scale represent a progression of values? If not, the scale is a grouping scale.
  • Are the scale intervals approximately equal? If not, the scale is may be treated as a grouping scale.
  • Is there a constant zero (or other reference point) representing the absence of the attribute being measured? If not, the scale is may be treated like an interval scale.
  • Are the limits of the scale limited in any way? Is there a scale minimum or maximum? Are negative numbers prohibited? If so, you may have to use special statistical approaches to analyze data measured on the scale.
  • Are the scale values cyclic or repeating? If so, you may have to use special statistical approaches to analyze data measured on the scale.
  • Are ratios and other mathematical operations that produce fractional scale levels permissible? If so, you have a ratio scale.

Some people think that an attribute can be measured in only one way. This is untrue more often than it is not. Consider the example of color. To an auto manufacturer, color is measured ontape-itskhnz a nominal scale. You can buy one of their cars painted red or blue or silver or black. To a gemologist, the color of a diamond is graded on an ordinal scale from D (colorless) to Z (light yellow). To an artist, color is measured on an interval scale because their color wheel contains the sequence: red, red-orange, orange, orange-yellow, yellow, yellow-green, green, green-blue, blue, blue-violet, violet, and violet-red. To a physicist, colors are measured by a continuous spectrum of light frequencies, which employ a ratio scale.

Using a different scale than what might be the convention can provide advantages. Consider this example. Soil texture is usually measured on a nominal scale that defines groups such as loam, sandy loam, clay loam, and silty clay. The information can be made quantitative by recording the percentages of sand, silt, and clay (which define the texture) instead of just the classification. The nominal-scale measure is much easier to collect in the field and is one variable to manage rather than three. On the other hand, the progression-scale measures can be analyzed in more ways. Correlating the clay content of a soil to crop growth, soil moisture, or a pollutant concentration can be done only if soil texture is measured on a progression scale.

scale-fat-cat-on-a-scale-30630

If a choice can be made on which type of scale to use, use a ratio scale. Ratio scales are usually best because they provide the most information and can be rescaled easily as ordinal scales. For example, many sports organize contestants using weight class, measured on an ordinal scale, instead of weight, which is measured on a ratio scale. Weight is still measured at weigh-in using a ratio scale but is converted to the ordinal-scale weight classes for simplicity. In contrast, it’s usually not possible to upgrade an ordinal scale to a ratio scale unless the ordinal scale has equal intervals and calculation of percentages or z-scores makes sense. You couldn’t just estimate a contestant’s weight of 178.2 pounds from a weight class of 170-185 pounds.

If you can’t measure an attribute on a ratio or interval scale, think about hobook-92e6f20565c2a9472dda6410939a44a6w an ordinal scale could be applied. You can almost always devise an ordinal scale to characterize an attribute; you just have to be creative. Think of opinion surveys. If you can measure opinions, you can measure anything.

[1] Stevens, S. S. 1946. On the theory of scales of measurement. Science v. 103, No. 2684, p. 677–680.

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About statswithcats

Charlie Kufs has been crunching numbers for over thirty years. He currently works as a statistician.
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2 Responses to Hellbent on Measurement

  1. Very nice.

    If you’re looking for examples, temperature is one where “If a choice can be made on which type of scale to use, use a ratio scale” doesn’t always hold. You *can* measure temperature on a ratio scale in Kelvin, and it’s important in some areas of science, but in most applications it makes a lot more sense to use degrees Fahrenheit or degrees Celsius on an interval scale.

    Also, probably at a higher level than Stats 101, there are non-obvious complications with ordinal data when it comes to comparing distributions rather than comparing values (http://notstatschat.tumblr.com/post/63237480043/rock-paper-scissors-wilcoxon-test). B

  2. Pingback: How to Describe Numbers | Stats With Cats Blog

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