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A NOIR walk across Manhattan

• Howard A. Spielman
• BI Review Magazine, March 2007

As a teacher, I find that questions from my students often challenge my thinking and inspire new insights. As a columnist, I find that questions from my editor do exactly the same thing.

Jim Ericson, Editor-in-Chief at BI Review, sent me an email with an incisive observation after reading my December, 2006 column. Jim noted: "Given your take on graphicacy, as an editor I look at literacy as it applies to the non-graphical usage of numbers ... there are interesting parallels." This is exactly the kind of thinking I hope to elicit in my readers, and there are some interesting insights that Jim inspired. In his email Jim was grappling with his experience in the use of the term 'ordinal' in a textual context, and the way that I had introduced the term 'ordinal' in a graphical context.

'Ordinal' in a Textual Context: The AP Style Manual

For textual communication, there are arbiters of style. The AP Stylebook, for example, is published by the Associated Press, a not-for-profit cooperative owned by its 1,500 U.S. daily newspaper members. More people write for the Associated Press than for any other news service in the world, and writers have bought more copies of The AP Stylebook than of any other journalism reference. The AP Stylebook has been called "The Bible of the newspaper industry." Jim sent me a scan of the section on ordinal numbers from his AP Stylebook.

The AP Stylebook has two rules regarding ordinal numbers:

• "-Spell out first through ninth when they indicate sequence in time or location: first base, the First Amendment, he was first in line. Starting with 10^th use figures."
• "-Use 1^st, 2^nd, 3^rd, etc. when the sequence has been assigned in forming names. The principle examples are geographic, military and political designations such as 1^st Ward, 7^th Fleet and 1^st Sgt."

In a textual context, the AP Stylebook thus distinguishes between when to use words (for first through ninth), and when to use numerals (e.g. 1^st, 2^nd, 10^th, 11^th). When we use an 'ordinal' number in a textual context, we are referring to a "sequence in time or location" or to a "sequence ... assigned in forming names." It would seem that a hierarchical sequence-such as the 'First Estate' or the 'Third World'-would also be a sequence of location, on the hierarchy.

It is important to think about the interval between, or the size of, the ordinal elements in our textual sequence. There are two possibilities:

• The distance between, or size of, textual ordinals could have no meaning. When we use a textual ordinal label we rarely have any concern about the interval of distance between, or size of Amendments, Wards, Fleets, Sergeants, Estates, or Worlds.
• The distance between, or size of, textual ordinals could be assumed to be equal. We have every reason to expect the interval from the first to third in line is the same as from the fourth to sixth in line, or that the interval of distance from first base to second base is the same as from second base to third base. Furthermore, first base and second base are always the same size.

The above is in distinct contrast to how an ordinal scale is understood in a graphical context.

• By definition, the distance between, or size of, graphical ordinals is assumed to be un-equal. What uniquely defines an ordinal scale, in a graphical context, is not the label for the points on the sequence, but the unequal intervals, or unequal size, of the elements in the sequence. The labels on an ordinal scale could easily be A, B, C, or 1, 2, 3, although they may be 1^st, 2^nd, 3^rd.

Missing this distinction between how ordinal numbers are used in a textual context and a graphical context, such as the data visualizations in our business and scientific meetings, is a common source of confusion. One major example of this was shown in my December 2006, column.

'Ordinal' in a Graphical Context: NOIR Revisited

In that December column I discussed NOIR, a mnemonic to help remember and understand the four scales of measurement found in most every database. The distinctions among these scales are critically important to help us understand our data and any visualization we prepare from our data.

• Nominal: A scale where each number represents a name (e.g., divisions in our company)
• Ordinal: A scale where each number represents an 'ordering' (e.g. Silver, Gold, or Platinum Customers, indicating a customer doing a small, medium, or large, amount of business with us).
• Interval: A scale such as degrees Celsius or degrees Fahrenheit with equal intervals but no discrete zero point, as zero on those scales does not mean the absence of heat. We create many interval scales in business.
• Ratio: A scale with equal intervals and a discrete zero, such as: counts of widgets, percent, dollars, pounds, yen, degrees Kelvin. We can divide one number on these scales by another to form a ratio.

By definition, an ordinal scale has elements in its sequence representing un-equal intervals, as equal intervals would indicate an interval scale. For example, to identify Silver, Gold, and Platinum Customers, we specify thresholds of purchasing dollars. Values such as $5M, $10M, and $25M might be typical in some company. Silver Customers thus represent a $4M wide band, Gold Customers a $15M wide band, and Platinum Customers could represent a very wide band indeed, especially if the largest customer is $100M. We have thus created an ordinal sequence of elements, representing three unequal bandwidths.

A Map of Manhattan

This is exactly where a walk across Manhattan may be exceedingly instructive, as it will illuminate the meaning and implications of NOIR and the use of 'ordinals.' Figure 1 shows a section of mid-town New York City in a Google map. The Queensboro Bridge, also known as the 59th Street Bridge, is at the East River towards the right edge of the map. Columbus Circle, at the southwest corner of Central Park is at the other end of 59th Street, at the upper left of the map. It is widely assumed that the avenues in Manhattan go north and south, and the numbered streets go east and west. Seeing the avenues in this map tilted about 29 degrees to the right of the vertical, makes it clear that the avenues are in fact tilted about 29 degrees east of true north. Still, we can speak of walking westward along the streets of New York.

Figure 1

Figure 2, a simplified version of Figure 1, highlights our issues in this column. The Queensboro Bridge, over the East River is at the right, and Columbus Circle, at the southwest corner of Central Park is at the left. On this simplified map, we will focus on the distance between the avenues across the horizontal axis, which is exactly to scale as the streets are laid out on the ground in New York City.

Figure 2

1. A Ratio Scale across some Avenues in New York City

We can walk westward out of Rockefeller University (Step 1 on Figure 2) and walk along 65^th Street. Walking the block between 1^st Avenue and 2^nd Avenue is perceived to be the same as walking the distance between 2^nd Avenue and 3r^d Avenue. There is an equal interval of distance between those avenues. In fact, in the area just north of the 59^th Street Bridge, York Avenue could actually be considered to be 'Zero Avenue.' These two observations are very important:

• A distinct 'zero' point for beginning our westward walk across Manhattan ... and
• Equal intervals of block-length from the zero point to the avenues numbered 1^st, 2^nd, and 3^rd.

For this range of our walk, the ordinal textual Avenue names (1^st, 2^nd, 3^rd) actually represent equal-interval sequence points that may be considered to be on a ratio scale. To walk from York Avenue to 3^rd Avenue, we would walk three times as far as walking only to 1^st Avenue. We can easily compute the ratio.

But, what happens if we wish to walk to 4^th Avenue or 5^th Avenue? Is it four or five times as long a walk as it is to 1^st Avenue?

2. A Nominal Scale across some Avenues in New York City

In New York City 4^th Avenue effectively does not exist. (Only the cognoscenti know that 4^th Avenue exists solely between 8^th Street and 14^th Street, for barely a quarter of a mile in lower Manhattan.) In our walk westward along 65^th Street (Step 2 in Figure 2) there are actually four blocks to be traversed between 3^rd Avenue and 5^th Avenue. We must cross the named avenues of Lexington Avenue, Park Avenue (a double-width avenue), and Madison Avenue. Since these avenues are "named," we have now shifted to a nominal scale.

Even if these avenues had been numbered, it is clear from Figure 2 that the length of these blocks is noticeably less than the length of the blocks between 1^st and 3^rd Avenues. We do not have equal intervals.

3. An Interval Scale across the Streets of New York City

Upon reaching Central Park in our walk, we can turn south along 5^th Avenue and walk from 65^th Street down to 59^th Street (Step 3 in Figure 2). Every one of these blocks may be considered to be one-twentieth of a mile. A walk along 5^th Avenue from 65^th Street down to 45^th Street would be a one mile walk, but to 59^th Street would be just over one-quarter of a mile. For over 9.5 miles, with rare exception, the numbered streets of Manhattan are laid out at equal intervals of 20 blocks to the mile. With this knowledge, of equal interval length blocks along the 'streets,' it is easy to estimate how far one may plan to walk north to south in New York. With unequal length blocks across the 'avenues' such estimates may only be done with the experience of having walked this distance before, or having studied it on a map.

4. An Interval Scale across some Avenues in New York City

For the last leg of our walk we turn west along 59^th Street and go along the southern edge of Central Park from 5^th Avenue to 6^th, 7^th, and 8^th Avenues, ending at Columbus Circle (Step 4 in Figure 2). We have returned to numbered avenues and, walking from one avenue to the next, we can make two observations:

• The distance from 5^th-to-6^th, 6^th-to-7^th, and 7^th-to-8^th Avenues are all equal intervals ... yet
• This interval is not the same as the interval of distance from 1^st-to-2^nd, and 2^nd-to-3^rd Avenues.

For this range of our walk, the ordinal textual names of our Avenues (5^th, 6^th, 7^th & 8^th) represent sequence points on an interval scale.

NOIR and the Avenues of Manhattan

In New York City, the textual ordinal label of an Avenue makes no implication about the spacing between the Avenues but only the ordering of the sequence. As pointed out above, this is NOT the meaning of an ordinal label in many textual applications. Based upon our textual experience, we see an ordinal label (e.g. 3^rd Avenue or 5^th Avenue) and wish to assume that it is an equal interval distance from the avenues before and after it. As we have observed, this assumption can be very wrong. This understanding is important to us in this graphical context since we are treating the avenue names as place-markers on a (geo)graphical map of the city and attempting to understand the distance between them to plan a walk.

On our walk, we found the avenues in different sections to be on different types of scales: step 1 a Ratio Scale, step 2 a Nominal Scale, and step 4 an Interval Scale. Taken together, the unequal intervals on our entire walk-across the avenues from York Avenue to 8^th Avenue-define this as an Ordinal Scale.

NOIR and Data Visualization

In the graphical context of data visualizations, we sometimes place an ordinal scale of measurement on an axis (but it could also be used in other ways, such as on a color key or the size of floating bubbles). The ability to spot an ordinal scale of measurement is very important, because our technical tools, in our BI software, do not distinguish an ordinal scale from interval or ratio scales. Furthermore, we do not have the experience of getting tired, as we walk across town in New York City, to sensitize us to the unequal intervals found on many data visualizations.

When ordinal scales are used in data visualizations they invariably cause great difficulties. An example of this in the context of a CEO presentation on employee hiring may be seen in my December, 2006 column. I discussed another example of this problem, taken from a medical journal, in my August, 2006 column.

Why every executive should walk west across Manhattan (at least once)

Sure, the exercise is great. More important to a career spent studying data visualizations in the boardroom though, is the visceral sense of the meaning of an ordinal scale in a (geo)graphical context. We are so easily seduced by the labels '1^st' through '8^th' on the Avenues of New York to make the unfortunate assumption of an equal distance interval between each successive avenue. We must develop a clear understanding of the distinction between the use of 'ordinals' in both the textual and graphical contexts.

Howard A. Spielman, M.B.A., Ph.D., President of Management Semiotics International Inc., can be reached at HASpielman@ManagementSemiotics.com.

For more information on related topics, visit the following channels:

• Data Visualization
• Leadership Zone