Tables Zero-dimensional data with position R.W. Oldford University - PowerPoint PPT Presentation

Tables Zero-dimensional data with position R.W. Oldford University of Waterloo

Magnitude Visual representations - labels Recall from last day, the ordering of the elementary tasks from most accurate to least accurate: 1. Position along a common scale 2. Position on identical but nonaligned scales 3. Lengths (N.B. line segments were all oriented horizontally or vertically, though nonaligned) 4. Angle Slope (not close to 0 , π/ 2 , or π radians) 5. Area 6. Volume 7. Colour density, colour saturation 8. Colour hue Missing from the list is the possibility of using “labels” which, provided there were not too many, were observed to work well with categorical data. E.g. “forty-two” or 42 Because it is simply read (by the trained person), using the number itself as “label” would probably have the most accurate (and fastest) decoding!

Magnitude Visual representations - numbers as labels Which is faster to decode? To compare values? Word numbers? Or symbolic numbers?

Magnitude Visual representations - numbers as labels Which is easier to compare values? left centre right decimal aligned rounded

Magnitude Visual representations - choosing positions Which is easier to compare magnitudes? unordered ascending descending

Magnitude Visual representations - choosing positions Which is easier to compare magnitudes? unordered ascending descending

Magnitude Visual representations - choosing positions Horizontal versus vertical for comparing magnitudes?

Modern numerals A surprisingly recent innovation ◮ Early European forms ◮ Roman numerals indicate century. ◮ from G.F . Hill’s The Development of Arabic Numerals in Europe (1915), p. 28. as recorded in Cajori (1928), p. 49.

Modern numerals Important Characteristics Characteristics permit ease of visual reasoning within the written system – hand calculation. E.g. consider Multiplication Long division At least for ‘natural’ everyday numbers.

Modern numerals Important Characteristics Calculating-Table by Gregor Reisch (c. 1467 - 1525) A woodcut illustration from Reisch’s Margarita Philosophica (1503) fea- turing Arithmetica instructing an “al- gorist” (Boethius) on the left and an “abacist” (Pythagoras) on right. These are two different types of arith- metic (Typus Arithmeticae) Algorism: “technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits.” (Wikipedia)

Modern numerals Search for an ostensive definition ◮ from Cajori (1928), p. 65. ◮ Earnest but fanciful hypotheses, post-hoc rationalizing. ◮ “They serve merely as entertaining illustrations of the operation of a pseudo-scientific imagination, uncontrolled by all the known facts.” Cajori (1928), p. 68. ◮ they are not themselves an ostensive definition, or pictorial form ◮ rather they are a learned symbolic abstraction ◮ must be distinguishable, easily learned, and easily constructed by pen and paper

Modern numerals Even positional representation is surprisingly recent innovation ◮ The nine numerals are enhanced with powers of 10 by surrounding them with that number of zeros. ◮ From Christoff Rudolff (Augsburg, 1574?) Künstliche Rechnung mit der Ziffer as taken from Cajori (1928), p. 56.

Modern numerals Cuneiform and positional representation One of the earliest written number systems was that of cuneiform, used by the Babylonians (circa 2500 BCE). A wedge-shape ( cuneus in Latin) tipped reed is pressed into wet clay and a single stroke indicates a single unit, the first number 1. (No zero.) A simple visual representation with a standardized layout of the symbols. 1 2 3 4 5 6 7 8 9 Needs a compression for larger numbers. 10 20 30 40 50 Ten is like two hands pressed together. Note standardized layout. Ends at 50. · · · 10 11 12 19 · · · Which is nice up until 59. After that positional representation is used! E.g. 70 =

Modern numerals Important Characteristics ◮ fixed and small base ◮ few characters to learn. ◮ Other bases, e.g. 12, might be even better. Babylonians (cuneiform) uses base 60. ◮ positional ◮ character encodes value ◮ position encodes magnitude (increasing right to left) ◮ standardized size and layout ◮ align in columns ◮ sequences separate in groups of 3 Allows them to be used dynamically as visual aids to reasoning. ◮ Visually executed algorithms rely on position (e.g. multiplication, long division, . . . ) ◮ Natural grouping by position ◮ Several pass sorting (by first digit, then second, . . . )

Tables of numbers A Babylonian innovation (a) Balance sheet (b) Pythagorean triplets ◮ Rows (sometimes unaligned) and columns, gridlines, indexing, headings. ◮ For reference. Static.

Modern Tables Another layout of digits. Tables are: ◮ for the record ◮ visual aids to reasoning. The first is more prevalent, the second more important. Like other layouts of numbers, tables should take advantage of the visual characteristics of the digits they display.

Modern Tables Example: Acidity of Ontario Lakes Background: One of the most pressing environmental problems facing large areas of North America is acidic precipitation. All of southern Ontario receives a steady bombardment of acids, acid forming gases and associated pollutants. The acids come down with rain, snow, fog, and small particles in the air. Man’s activities are responsible for the large majority of these acids. Smelters and coal-fired electric generating stations both in Canada and the United States spew millions of tonnes of sulphur dioxide into the atmosphere annually. Cars, trucks, and trains contribute more millions of tonnes of nitrogen dioxides. These gases react with sunlight, oxygen, ozone, water and other gases to form sulphuric and nitric acid – strong, corrosive acids. In unpolluted areas, rain and snow are naturally slightly acidic since carbon dioxide, which is a natural component of the atmosphere, dissolves in water to form weak carbonic acid. Water quality of lakes has developed in response to weathering processes induced by this weak acid. Rocks and minerals react with carbonic acid to form bicarbonate, which is found in natural waters everywhere. The complex biological communities in lakes, streams and forests have adapted and evolved in equilibrium with these natural conditions and processes. However, acid rain has seriously disturbed this equilibrium. There are over 250,000 lakes in Ontario. Thousands have been affected by acid rain, many of them in the Muskoka-Haliburton region, where there is a substantial cottage and tourist industry.

Acidity of Ontario Lakes Ontario Government Publication: “Acid Sensitivity Survey of Lakes in Ontario – 1989”.

Acidity of Ontario Lakes ◮ Rows ordered alphabetically by “County or District” ◮ Lots of redundant information. ◮ Difficult to see patterns, if any. Ontario Government Publication: “Acid Sensitivity Survey of Lakes in Ontario – 1989”.

Acidity of Ontario Lakes Arrange by region, remove uninteresting redundancy.

Acidity of Ontario Lakes Arrange by acidity.

Acidity of Ontario Lakes Arrange by acidity, remove further redundancy, annotate.

Modern Tables Conveying information visually Modern tables: ◮ need no longer be for the record (databases are) ◮ should be displayed as visual aids to reasoning. Like other layouts of numbers, tables should take advantage of the visual characteristics of the digits they display. ◮ Take advantage of rows. ◮ Align digits in columns. ◮ Show important individual numbers. ◮ Use white space to separate groups of numbers. ◮ Think hard about the information to be communicated.

Tables Analysis Consider the following table of “Sales data”: TABLE 1.1 Data in Four Areas and Eight Three-Month Periods in 1969-1970. 13-15 16-18 19-21 22-24 25-27 28-30 31-33 34-36 A 97.62 92.24 100.90 90.39 95.69 94.44 91.13 97.81 B 48.29 42.31 49.98 39.09 46.38 49.74 41.74 37.39 C 75.23 75.16 100.11 74.23 74.23 76.97 71.66 76.47 D 49.69 57.21 80.19 51.09 52.88 49.41 59.32 52.56 Analysis: ◮ Goal is to see some patterns in the data. ◮ Develop a summary description (“model”) for the pattern. ◮ Assess the agreement of the pattern with the data. Source: A.S.C. Ehrenberg (1975) Data Reduction: Analysing and Interpreting Statistical Data .

Analysis of table data Step 1 Separate row and column headings 13-15 16-18 19-21 22-24 25-27 28-30 31-33 34-36 A 97.62 92.24 100.90 90.39 95.69 94.44 91.13 97.81 B 48.29 42.31 49.98 39.09 46.38 49.74 41.74 37.39 C 75.23 75.16 100.11 74.23 74.23 76.97 71.66 76.47 D 49.69 57.21 80.19 51.09 52.88 49.41 59.32 52.56 TABLE 1.1 Data in Four Areas and Eight Three-Month Periods in 1969-1970. Lines and space. Next: Assign meaningful labels. Columns are 3 month periods numbered from 1968.

Analysis of table data Step 2 Meaningful labels. Separate years. Quarters (1969) Quarters (1970) Area 1 2 3 4 1 2 3 4 North 97.62 92.24 100.90 90.39 95.69 94.44 91.13 97.81 South 48.29 42.31 49.98 39.09 46.38 49.74 41.74 37.39 East 75.23 75.16 100.11 74.23 74.23 76.97 71.66 76.47 West 49.69 57.21 80.19 51.09 52.88 49.41 59.32 52.56 Gridlines added to separate years and define table. Table title unnecessary (a different one with different information might be added).