4 Order of magnitude
Fig 4. 1 potentially addresses a critical worldwide issue: high child mortality. Mortality1 ranges widely, from around 30 deaths per 1000 children to …? It is hard to tell from the graph, but it looks like the bottom end of the range is very low: about one death per 1000. Everyone would like to reduce mortality to the lowest possible level. Experience shows that vast reductions from 30 deaths per 1000 are possible. The question is, how to do it? Understanding which factors contribute to low mortality, and which do not, can help us figure out what to do.
Fig 4. 1 is interactive; hovering the cursor over a dot causes the country name to appear, along with the precise numerical value for mortality and GDP.
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The strategy that underlies such graphics, indeed much of the power of data graphics, comes from ☞ sorting ☜ the data values according to each quantity. Sorting helps the eye identify points that are close together in one quantity, then check whether they are close or spread out along the other quantity. Similarly, sorting facilitates checking whether points that are widely separated in one quantity are also widely separated in the other quantity. Sorting helps us identify points that fall in the same region of the coordinate plane: they are close together in the two-dimensional graphics frame.
For instance, Fig 4. 1 shows that countries with small GDP tend to have large mortality rates. It is also true that countries with low mortality rates tend to have high GDP. The three countries with high mortality and high GDP in 1960 are, when we look into things, similar in other ways as well: all three are oil-rich.
Fig 4. 2 shows the same data but has a strikingly different appearance. The data points are much more spread out across the graphics frame. Subtle patterns are visible that involve many more countries. For instance, comparing countries with relatively low GDPs in 1960 and 2020 suggests that changes in mortality are not associated with changes in GDP.
This chapter is about the technique used in Fig 4. 2 that makes patterns much easier to see. It is all about focusing on the number of digits, especially zeros.
4.1 Counting digits
When quantities are about the same size, it is easy to compare them to see which is bigger or smaller. For most people, the cognitive load of such comparisons is very low for values in the range 1-100. In an instant, a person can pick the larger of two quantities. Try it, which is larger: 72 or 68? No work at all.
The cognitive load rises for quantities that span a much larger range, say 1 to 10,000 or larger.
When quantities are a mixture of the very large and the very small, comparisons require attention. For instance, consider Table 4. 1, which shows the gross domestic product (GDP) of a handful of countries in 2024. The table standardizes values from different countries to a standard unit: United States dollars (USD). [Source: World Bank]
| Country | GDP in USD |
|---|---|
| Albania | 27177740000.00 |
| Australia | 1752193000000 |
| Bhutan | 3019250000.00 |
| China | 18743803000000 |
| Estonia | 42764930000.00 |
| Germany | 4659929000000 |
In Table 4. 1, the numbers are “left justified”, that is, their leading digits line pup in a column. Unfortunately, the leading digits—27 for Albania and 18 for China—are little help in comparing the values.
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Now consider the GDP numbers, written in a different format. Table 4. 2 makes it easy to see which countries have very large values and which comparatively smaller values. First, we have sorted the values. Sorting helps the eye compare every value to every other value in an instant.
| Country | GDP in USD |
|---|---|
| China | 18,743,803,000,000 |
| Germany | 4,659,929,000,000 |
| Australia | 1,752,193,000,000 |
| Estonia | 42,764,930,000 |
| Albania | 27,177,740,000 |
| Bhutan | 3,019,250,000 |
Table 4. 2 also makes it clear why looking at the leading digits is not always helpful in comparing values. Instead, the first thing to look at is the number of digits. Because the values are all aligned to the right, we do not even need to count the number of digits; we can treat them as if we are looking at a bar graph. It is necessary to look to the leading digits only when the number of digits is the same.
4.2 Unit prefixes and scientific notation
As noted in the previous section, people are quick to compare or sort numbers with 1, 2, or even three digits. Bigger numbers, however, require shifting to another mode: counting digits. Similarly, comparing tiny numbers—such as 0.000032 and 0.00018—involves counting the zeros that immediately follow the decimal point.
There are two widely used techniques for formatting numbers that include information about the number of digits directly in the number. The logic of these two techniques will bring us very close to explaining how to construct graphics like Fig 4. 2.
The most familiar of the two techniques involves unit prefixes. Every day, we deal with both very large and very small distances. For instance, using base SI units, the width of a typical room is about 3 m, while a bike ride to a hardware store is about 2500 m, depending on the neighborhood. The screw bought there to fix a bike could have a diameter of 0.004 m. If these numbers sound strange, it is because everyday practice is to change the units in order to avoid counting zeros. Instead of 0.004 meters, say 4 millimeters. Rather than 2500 meters, say 2.5 kilometers.
The standard prefixes “milli” and “kilo” mean, respectively, one-thousandth and one thousand. The milli prefix—often abbreviated simply as m—moves the decimal point 3 places to the right: 0.004 m is the same as 4 mm. Likewise, kilo (abbreviated as k) moves the decimal three places to the left: 2500 m becomes 2.5 km.
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The system was introduced on a limited basis by the revolutionary government of France. (See Table 4. 3.) It has been extended to handle larger and smaller quantities on several occasions.
| Abbreviation | Prefix | Moves decimal point | Year of introduction |
|---|---|---|---|
| p | pico | 12 places to right | 1960 |
| n | nano | 9 places to right | 1960 |
| \(\mu\) | micro | 6 places to right | 1883 |
| m | milli | 3 places to right | 1794 |
| c | centi | 2 places to right | 1795 |
| d | deci | 1 place to right | 1795 |
| …. | … | … | … |
| da | deca | 1 place to left | 1795 |
| h | hecto | 2 places to left | 1795 |
| k | kilo | 3 places to left | 1795 |
| M | mega | 6 places to left | 1873 |
| G | giga | 9 places to left | 1960 |
| T | terra | 12 places to left | 1960 |
Only a handful of prefixes are in everyday use: micro, milli, kilo, mega, giga. These cover a broad range, from a tiny dose of medicine to the wealth of ultra-rich individuals. Notice that the most used prefixes each shift the decimal point by three places, enabling the range of quantities to eliminate leading or trailing zeros and be represented as numbers with one, two, or three digits plus the prefix. The 1795 prefixes that move the decimal point by 1 or 2 places have mostly fallen out of use. The most common exception is centi, which is widely used for length: centimeter (cm). The prefix hecto is in regular use to refer to area; a “hectare” is an area of 100 m squared, that is, 10,000 m2.
For temperatures, on the other hand, the prefix system is not used. Even without prefixes, we can write everyday temperatures with two digits.
As regards time, we have an idiomatic system for avoiding zeros: seconds, minutes, hours, days, weeks, months, years, decades, centuries, millennia. The right-moving prefixes, however, are used for time: milliseconds, microseconds, and such.
When talking about money, it is common to replace the Latin prefixes with words like “thousand,” “million,” “billion,” and “trillion.” Unfortunately, such words differ from language to language. The French use “milliard” and “billion” to mean what in English is a billion and a trillion, respectively. The Japanese use 4-digit groups for monetary quantities. “Man” is 1,0000 (that is, 10,000), “oku” is 1,0000,0000 (that is, one-hundred million), “cho” is 1,0000,0000,0000, which aligns with English “billion.”
People can easily confuse prefixes they do not use often. Such mistakes can lead to serious problems. Mistake milligrams for micrograms, and the dose of medicine will be 1,000 times too small or too large. News reports and politicians frequently confuse “million” with “billion.”2
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The prefix system works best when applied to units with short names, such as meters, grams, watts, joules, and so on. Things can get out of hand with derived units. For instance, a fundamental quantity used in quantum physics is Planck’s constant, which has the unusual dimension [energy] \(\times\) [time].
Planck’s constant: 0.000000000000000000000000000000000662607015 m2 kg / s
Doing calculations with such numbers is error-prone, since it is easy to omit or add zeros and hard to see at a glance that this has occurred. Writing it more compactly with prefixes is not much help:
0.663 femtometers2 gm / s
☞ Scientific notation ☜ is helpful here. In scientific notation, there are two separate numerals: A decimal numeral called the ☞ mantissa ☜ and an integer numeral called the ☞ order of magnitude ☜. (The order of magnitude is also called the ☞ exponent ☜. We prefer “order of magnitude” as giving a better indication of the use of the integer numeral.) Proper form requires writing the mantissa with a single digit 1 to 9 to the left of the decimal point. (There can be any number of digits, including 0, to the right of the decimal point.) The mantissa in Planck’s constant is 6.62607015. The order of magnitude, in contrast, is always an integer: -34 for Planck’s constant. The printed format for scientific notation is verbose, always involving the punctuation \(\color{blue}{\fbox{ }}\color{black}{\times 10}^{\,\color{magenta}{\fbox{ }}}\). The boxes (\(\color{blue}{\fbox{ }}\) and \(\color{magenta}{^{\fbox{ }}}\) show the position for the mantissa and order of magnitude.)
- Planck’s constant: \(\color{blue}{6.62607015} \color{black}{\times 10}^{\color{magenta}{-34}}\) m2 kg / s
- China GDP for 2023: \(\color{blue}{1.87} \color{black}{\times 10}^{\color{magenta}{13}}\) USD
- Value of German Mark in USD during an episode of hyperinflation: January 1923 \(\color{blue}{1.4}\color{black}{\times 10}^{\color{magenta}{-4}}\) USD; October 1923 \(\color{blue}{2.0} \color{black}{\times 10}^{\color{magenta}{-10}}\) USD;
A flaw of this notation is that the most important number, the order of magnitude, is written in the smallest font. In computer notation, however, the punctuation is dramatically streamlined: Planck \(\mathtt{\color{blue}{6.62607015}\color{black}{e}\!\color{magenta}{-\!34}}\); China \(\mathtt{\color{blue}{1.87}\color{black}{e}\color{magenta}{13}}\); German Mark \(\mathtt{\color{blue}{1.4}\color{black}{e}\color{magenta}{-4}}\).
Public health workers and sociologists use a variant of scientific notation. A mortality rate of 0.0063 (deaths per person) will instead be presented as 63 deaths per 10,000 people, avoiding the need to count digits. For a rare disease, we present its prevalence across different units, e.g., cases per 100,000 people or 1,000,000 people.
4.3 Magnitude as a single number
A ☞ numeral ☜ is the printed form of a number that uses the digits 0-9 and often some punctuation. Scientific notation presents two numerals for a single value: the mantissa and the exponent. It is convenient to condense this into a single numeral, which we call the ☞ magnitude ☜. In contrast, the numeral placed in the exponent position tells the order of magnitude and is, in scientific notation, always an integer: \(\ldots, -2, -1, 0, 1, 2, 3, \ldots\).
Our proposed single-numeral, magnitude form of a value will combine the exponent and the mantissa. To illustrate, consider the scientific-notation form of the 2023 GDP of China: \(\mathtt{\color{blue}{1.87}e\color{magenta}{13}}\) USD. This value is somewhat greater than \(\mathtt{\color{blue}{1}\color{black}{e}\color{magenta}{13}}\) USD but less than \(\mathtt{\color{blue}{10}\color{black}{e}\color{magenta}{13}}\) USD. Of course, \(\mathtt{\color{blue}{10}\color{black}{e}\color{magenta}{13}}\) ought to be written \(\mathtt{\color{blue}{1}\color{black}{e}\color{magenta}{14}}\). Let us define the magnitude of a number like \(\mathtt{\color{blue}{1}\color{black}{e}\color{magenta}{13}}\) to be 13, that is, the value of the exponent. Similarly, the magnitude of \(\mathtt{\color{blue}{1}\color{black}{e}\color{magenta}{14}}\) is 14.
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So what is the magnitude of \(\mathtt{\color{blue}{1.87}\color{black}{e}\color{magenta}{13}}\)? Since it is between \(\mathtt{\color{blue}{1}\color{black}{e}\color{magenta}{13}}\) and \(\mathtt{\color{blue}{1}\color{black}{e}\color{magenta}{14}}\), its magnitude should be between 13 and 14: something like 13._ _. The digits to fill in the blanks are, to be honest, by no means obvious. Indeed, the answer was invented as recently as 1600 by John Napier (1550-1617) as an arithmetical aid. The magnitude of \(\mathtt{\color{blue}{1.87}\color{black}{e}\color{magenta}{13}}\) turns out to be 13.272, between 13 and 14, but closer to 13.
We have set things up here so that the R function that calculates the magnitude3 of a number is magnitude().
4.4 Magnitude graphics
The method used to produce Fig 4. 2, the intelligible version of the mortality-versus-GDP graph, is to set the axes’ scales to reflect the magnitude of the quantity. The magnitude axes spread the data points, but the tick mark labels lose their intuitive meaning. For instance, the 1 on the vertical scale in Fig 4. 2 really means ten deaths per thousand children.
To avoid this problem, it is common to label the axes in a magnitude plot with numerical values rather than magnitudes. Fig 4. 3 shows an example based on the mortality versus GDP data. Note that the data points are spread out in the same way as when magnitude is used for the labels; only the labels and tick-mark positions differ.
4.5 Order of magnitude phenomena
Many phenomena and their associated quantities are best thought of in terms of magnitude rather than number. An example: Animals, including humans, go about the world in varying states of illumination, from the bright sunlight of high noon to the dim shadows of a half-moon. To see in such diverse conditions, the eye needs to respond to light intensity over many orders of magnitude.
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The lux is the unit of illuminance in the Système international. Table 4. 4 shows the illumination in a range of familiar outdoor settings.
| Illuminance | Condition |
|---|---|
| 110,000 lux | Bright sunlight |
| 20,000 lux | Shade illuminated by clear blue sky, midday |
| 1,000 lux | Typical overcast day, midday |
| 400 lux | Sunrise or sunset on a clear day (ambient illumination) |
| 0.25 lux | A full Moon, clear night sky |
| 0.01 lux | A quarter Moon, clear night sky |
For a creature active both night and day, the eye needs to be sensitive over a range of illumination spanning more than seven orders of magnitude. To accomplish this, the eyes use several mechanisms: pupil contraction or dilation accounts for about one order of magnitude; photopic (color, cones) versus scotopic (black-and-white, rods, nighttime) accounts for about three orders of magnitude; adaptation over minutes (1 order); and squinting (1 order).
More impressively, human perception of sound spans more than 16 orders of magnitude in the energy impinging on the eardrum. The energy density of perceptible sound ranges from the threshold of hearing at 0.000000000001 W/m2 to a conversational level of 0.000001 W/m2 to 0.1 W/m2 in the front rows of a rock concert. However, in terms of our subjective perception of loudness, each order-of-magnitude change is perceived the same way, whether it is from street traffic to a vacuum cleaner or from a whisper to everyday conversation. (The unit of sound measurement is the decibel (dB), with 10 decibels corresponding to an order of magnitude in the energy density of sound.)
| Situation | Energy level (dB) |
|---|---|
| Rustling leaves | 10 dB |
| Whisper | 20 dB |
| Mosquito buzz | 40 dB |
| Normal conversation | 60 dB |
| Busy street traffic | 70 dB |
| Vacuum cleaner | 80 dB |
| Large orchestra | 98 dB |
| Earphones (high level) | 100 dB |
| Rock concert | 110 dB |
| Jackhammer | 130 dB |
| Military jet takeoff | 140 dB |
6, 60, 600, and 6000 miles per hour are quantities that differ in size by orders of magnitude. Such differences often point to a substantial change in context. A jog is 6 mph, a car on a highway goes 60 mph, a cruising commercial jet goes 600 mph, and a rocket passes through 6000 mph on its way to orbital velocity. The range from an infant’s crawl to highway cruising covers four orders of magnitude in speed.
Of course, many phenomena are not usefully represented by orders of magnitude. For example, the difference between normal body temperature and high fever is 0.01 orders of magnitude in temperature.4 An increase of 1 order of magnitude in blood pressure from the normal level would cause instant death! The difference between a very tall adult and a very short adult is about 1/4 of an order of magnitude.
Comparing with orders of magnitude makes sense when the relevant comparison is a ratio. “A car is 10 times faster than a person” refers to the ratio of speeds. In contrast, we generally use differences to compare quantities such as body temperature, blood pressure, and adult height. Fever is only 2\(^\circ\)C higher in temperature than normal. A 30 mmHg increase in blood pressure is worth noting as an indication of hypertension.5 A very tall and a very short adult differ by about 3 feet.
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Thinking in terms of orders of magnitude is appropriate when working with a set of objects whose sizes span one or many factors of 2. Comparing baseball and basketball players? No need for orders of magnitude. However, in considering the weights of mammals, from bats to whales, orders of magnitude will reveal patterns that the whales would obscure on a number-line scale.
Magnitudes are inappropriate whenever zero or negative values have meaning for the quantity under consideration. Consider this short list of numbers and their magnitudes:
| Quantity | 1000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 |
|---|---|---|---|---|---|---|---|
| Magnitude | 3 | 2 | 1 | 0 | -1 | -2 | -3 |
Continuing the list to cover 0.0001, 0.00001, and so on will never reach a column where a zero or negative quantity appears.
New terms {
Footnotes
Defined as the rate of death of children under five years in age.↩︎
I once overheard a transaction at a London cheese shop. The American customer, perhaps familiar with kilograms and kilometers, had likely been told that 100 grams of cheese is a sensible amount to buy, but instead ordered “100 kilograms of Stilton.” The shopkeeper was surprised and confused.↩︎
For the R chunks in this book and related materials, you can use
magnitude(). However, in general computing, use another name for that function:log10(). We are using “magnitude” to avoid the triggering of anxiety that many people feel when they hear “logarithm”.↩︎We are using the Kelvin scale, which is the only meaningful scale for a ratio of temperatures.↩︎
In medicine, blood pressure is measured using old-fashioned, non-SI units based on the height of a column of mercury that will balance blood pressure. The “mmHg” is short for “millimeters of mercury,”↩︎