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A. Map Scale

B. Measurement Scales

C. Accuracy, Precision, and Huge Digits

D. Classification, Simplification and also Symbolization of Data

A. Map Scale

Map range is the partnership in between a unit of length on a map and also the matching length on the ground. We will use ideas of map range throughout the course, so it will pay you to study this section closely.

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1. Types of Map Scales

We have the right to relate map and ground with three various kinds of range. Verbal scale expresses in words a partnership between a map distance and also a ground distance. Typically it is alengthy the lines of:


One inch represents 16 miles.


Here it is implied that the one inch is on the map, and that one inch represents 16 miles on the ground. Verbal scales are typically discovered on popular atlases and maps.

The second type of range is a graphic scale, or bar scale. This reflects directly on the map the matching ground distance. For example:




The 3rd form of range is a representative fraction, or ratio range. Contrasted to the first 2, it is the a lot of abstract, yet also the many flexible. A representative fractivity, or RF, shows the partnership between among any type of unit on the map and among the very same unit on the ground. RFs may be displayed as an actual fraction, for instance 1/24,000, yet are usually created through a colon, as in 1:24,000. In this instance, one unit of any type of length (one mm, one cm, one inch, one foot, and so on.) on the map represents 24,000 of those exact same devices on the ground (24,000 mm, 24,000 cm, 24,000", 24,000", and so on.). The RF is flexible bereason you are not tied to any type of particular devices. You may work-related in any kind of unit you pick, either metric, English, or various other.

The RF is a referred to as a portion bereason it is just that--a fraction that mirrors how a lot the real civilization is decreased to fit on the map. A great compariboy is frequently made via range models of automobiles or aircraft. A 1/32-model of an auto is 1/32nd as big as the actual auto. In the exact same means, a 1:100,000-scale map is 1/100,000th as big as the ground location displayed on the map.

A associated idea is that of little range versus large scale. Geographers usage these terms in different ways than many type of human being. A large scale map is wright here the RF is reasonably huge. A 1:1200 map is therefore bigger scale than a 1:1,000,000 map. The 1:1,000,000 map would typically be referred to as a tiny scale map. This is true even though the 1:1,000,000 map would display a much larger location than the 1:1200 map.

Here is a dominion of thumb for size of scale by RF:



Size of ScaleRepresentative Franction (RF)
Large Scale1:25,000 or larger
Medium Scale1:1,000,000 to 1:25,000
Small Scale1:1,000,000 or smaller


Of course, what is little or large range is relative. I noticed a surveying text (Brinker & Wolf, 1984) that classed anypoint smaller than 1:12,000 as tiny range -- surveyors seldom job-related through anything smaller sized than this.

The large/tiny scale terminology have the right to end up being confutilizing when talking about large versus little areas. If you are talking about a phenomenon that occurs throughout a huge area, it is tempting to say it"s a massive phenomenon (e.g., "the forest blight is a large disease"). But considering that the map that would show this would certainly be small-range, it is much better to usage a various term to stop confusion. My favorite is "broad-range."

Many kind of maps encompass two or even all three forms of scales. USGS topographic maps have actually both bar scales and RFs.

2. Converting Between Scale Types

If you are offered one type of range, you should have the ability to derive or construct any kind of of the various other 2. This takes some exercise, and some troubles are consisted of in your lab exercises. Some examples are offered below.

A important action in doing any type of convariation that involves differing devices is to incorporate the devices in the trouble itself. You have the right to then cancel the devices by multiplying or splitting. This method you stop becoming confused about which convariation factors to use and how to usage them.

Verbal Scale to RF

The essential here is to write the verbal range as a portion, then transform so that both numerator and denominator have the same systems, and also the numerator has a 1.


(a) Convert verbal range of "1" to 18 miles" to RF





or 1:1,140,000.

Notice that the resulting fractivity is rounded so that the RF does not indicate more accuracy than the original precision warranted.


(b) Convert verbal scale of "15 cm to 1 km" to RF




or 1:6700.

In many kind of conversions you deserve to save actions if you remember added equivalencies.

For example, in (a) over, we could have provided the truth that 1 mile = 63,360 inches to skip a step.

Verbal Scale to Graphic Scale

Generally this is a reasonably straightforward job if the map provides us reasonable systems in the verbal scale. We have the right to usage the verbal scale favor a fraction to transform the ground distance to map distance.


(c) Convert verbal scale of "1 cm to 14 km" to a graphic scale.


One centimeter is a relatively tiny distance, so we most likely do not desire our bar range to have significant departments much smaller than this. A centimeter represents 14 km, so a division of 10 kilometres is most likely fine. Thus we want to find just how many type of centimeters reexisting 10 km.




In other words, we can recurrent our 10 kilometres increment on the bar scale by measuring off 0.71 cm on the map. We"d draw the initially tick at 0.71 cm, the second at 1.42 cm, and also so on:




RF to Graphic Scale

This adds an extra action to the example above. We have the right to discover the map-distance tantamount of a ground distance, but we also have to be mindful about selecting which ground distance we want to portray on the map. Perhaps it"s easiest to pick a smaller ground distance that you can then multiply to gain a reasonable bar range.


(d) Convert an RF of 1:250,000 to a graphic scale


If we aren"t certain what increments a bar range would have actually for this scale, we can start out, say, through finding the map equivalent of 1 mile:



This can work fine, through one mile marked off on the map every 0.25 inch; or, we might desire finer or wider increments, which we have the right to find by splitting or multiplying the .25" as required.

RF to Verbal Scale

Aacquire we need to select proper devices to convert right into. Many verbal scales are either "one inch represents ____ miles," or "one centimeter represents ___ kilometers." These are fairly simple to execute, since it indicates just that we transform the denominator of our RF to the bigger units.


(e) Convert from RF of 1:25,000 to a verbal range, in metric



1 centimeter on this map represents 1/4 of a kilometer on the ground.

Graphic Scale to RF

Here we must take a measurement from the bar scale to recognize the map distance that coincides to a ground distance.


(f) Find the RF range for the adhering to graphic scale



By measuring with a ruler, we find that 10 kilometers procedures 2.4 cm. We have the right to usage this relationship to uncover the RF for the bar scale:




3. Determining Scale from a Map or Photo

Some maps might come via no scale at all. Aerial photographs nearly never perform (unmuch less one was painted on the ground before the photo was taken!). How deserve to you derive a scale for use via the map or photo?

Actually the procedure is exceptionally comparable to the last example over. But instead of measuring alengthy a bar range, you should meacertain the size of a things on the map or photo whose actual size you recognize. This could be a footround area, a city block, or the Equator (if it"s a civilization map). Often you deserve to identify 1-mile-square sections in the US (see the account listed below, under Survey Systems). You might even must go out to the area mapped or pictured and meacertain the distance in between 2 identifiable objects.

Once you have actually the two ranges, you deserve to find the range as above. For one more instance, intend you have a map where the distance between 2 section-line roads is 3.5 inches on the map. We have the right to normally assume this is one mile on the ground (there are exceptions). The RF range is then:




One caveat (exception) for air photos is that this approach assumes the 2 locations are at the same elevation--or that the terrain is flat. If you are utilizing air photos, the terrain might not be level. If there are hills, also modeprice ones, the calculations can be thrvery own off. Keep this in mind for later on in the course.

Another means to calculate range on an unknown map or photo is to compare it to a map through a recognized range. For example, intend you have an air photo wright here the distance between two hills is 7.2 centimeters.You have actually a map of the very same area at 1:24,000, and also on the map the distance in between the hills is 2.4 centimeters.

The answer involves a small algebra. Since the ground distance is the very same on both photo and map, we deserve to develop an expression for this ground distance for both, and then put them on either side of an equation. The ground distance can be discovered by multiplying the map/photo distance by the scale (in this instance, by the inverse of the scale--notice exactly how this provides the units cancel correctly). We have to discover, for the photo, exactly how many ground units are stood for by one unit on the photo, so we use an x for this unwell-known amount and settle for it:








we deserve to cancel the devices on each side and divide by 7.2:






In various other words, the RF scale for the photo is 1:8,000.


4. Determining Distance and Area from Map & Scale

Map range isn"t much use in and of itself. We have the right to use a map"s range to determine ranges and areas on the map. Contrasted to converting in between scale types, calculating distance is simple. Area calculations are trickier, considering that we have to square the numbers.

Finding distance from map and also scale

As an instance, intend we have actually a map with a scale of 1:50,000. We meacertain the distance along a residential or commercial property boundary as 1.7 cm. What is the size in the actual world?




To discover ground distance, we must usage the map range to transform map distance to ground distance. Notice that aobtain we inverted the RF scale, so the systems will certainly cancel effectively. Once we multiply by the scale, we have to convert the ground distance to units suitable for ground measurement--in this case, from centimeters to kilometers.

We can additionally calculate distance from verbal and graphic scales. With verbal scales, we use the same procedure as above with the RF. The just distinction is that we need to usage the devices provided in the verbal scale (e.g., 1 inch to 17 miles). We"d most likely desire to meacertain our map distance in the exact same systems (in this case, inches) to make our convariation simple.

Graphic scales are most likely the scales most typically offered by laypersons. You can mark off a distance on the map and also compare it directly to the bar scale. You need not know exactly how many inches or centimeters the map distance is. The main drawback of bar scales is that they are usually short compared to the map itself, and also hence measuring longer ranges is difficult.

Finding location measurement from map and scale

Area need to be expressed in agenuine devices, which are generally distance units squared -- cm2 , mi2, and also so on. We should therefore used squared conversion determinants as soon as finding area from map measurements.

For example, intend we meacertain a rectangular piece of residential property that is 3 cm by 4 cm on a map. The map is at a scale of 1:24,000. What is the location of the parcel?

The location of the parcel on the map is










on the ground.

Because this is a large number, we might want to analyze to various other devices. Tbelow are 10,000 square meters per hectare, so the location is 69 hectares (ha) (a hectare is around 2.5 acres). Or, tright here are (1,000)2 = 1,000,000 square meters per square kilometer, so the area is also 0.69 km2.

Notice that by creating the units as component of the difficulty, and squaring them along with the numbers, our devices cancel effectively and also we finish up via a sensible answer.

Tright here is one more way to tackle location difficulties if you have actually distance dimensions favor 3 x 4 cm to begin out the difficulty. You can transform the distance dimensions to real-human being distances initially, and also then multiply them to discover the location. This makes the problem longer however maybe much easier.

B. Measurement Scales

1. Definition

Any kind of indevelopment on a map (or in a table, a list, a survey, etc.) deserve to be defined in regards to exactly how pieces of the indevelopment can be related to each other. Examples of mapped indevelopment incorporate land-use classes, road classes, city populations, and county areas.

We deserve to think of these forms of indevelopment as either categorical (or qualitative) or numerical (or quantitative). Categorical indevelopment have the right to just be defined qualitatively, whereas items of numerical information have the right to be compared quantitatively. That is, numerical indevelopment is on a real scale, such as distance (e.g., kilometers), location (e.g., hectares), or temperature (e.g., Celsius).

Example of categorical information:Land-Use categories of :1= forest, 2 = grassland also, 3 = urban;We cannot say that based upon the classes, 1 + 2 = 3 !We can just use the numbering to explain the items qualitatively.Example of Numerical indevelopment, city populations:Springfield = 100, Harmony = 200, Centerville = 300;We can say that Centerville has as many civilization as Springfield and Harmony.

2. Four Measurement Levels

The categorical and numerical forms have actually traditionally been additionally damaged dvery own, each right into 2 levels of measurement:

Categorical information deserve to either be nominal level or ordinal level . At the nominal level, categories sindicate describe various types of points, and cannot be compared to each other. Ordinal-level categories deserve to, on the various other hand also, be ranked against each various other.

An example of ordinal level would be road classes of interstate, major, second, and primitive; although 2 primary roadways don"t make an interstate, you can say in a feeling that interstates are at a higher level than the others, at leastern in terms of cost to build!

The land-usage classes would certainly be an example of nominal-level measurement. You probably wouldn"t rank the classes versus each other (unless you prefer one over the various other, in which situation they can be ordinal!).

Numerical indevelopment deserve to either be interval level or ratio level . This difference is subtle, and borders on fabricated. The only difference is that ratio-level information is on a scale that consists of a true zero, that is, a zero that truly represents a absence of whatever it is you"re measuring.

One of the few widespread examples of an interval range is temperature. You can compare 2 temperature readings quantitatively. For instance, 20 C is 15 degrees warmer than 5 C. But the Celsius scale has an arbitrary zero suggest (the freezing temperature of water, or 32 F). 0 C doesn"t intend a lack of temperature. For this reason, 20 C is not 4 times as heat as 5 C! (One temperature range, Kelvin, does function a true zero -- 0 K is -273 C, the temperature at which all molecular movement ceases). Fortunately, tbelow are few examples of this on maps. You could desire to incorporate an exemption for elevation, which have the right to be listed below sea level, or longitude, wbelow the 0° line (Prime Meridian) is arbitrarily attracted on the world.



Table 2: Measurement Levels




Measurement Level





Simple categories; cannot rank categories

Colors; land also use types; computer components



Distinct categories; canrank categores in order

Roadway classes; flavor preferences



Continuous scale; meacertain with numbers; realzero

Temperature; longitude, compass direction



Continuous scale; measurewith numbers; actual zero

Distance; weight; bank balance

Most numerical scales are ratio-level. We deserve to compare 20 kilometres through 10 km in any number of methods, consisting of saying the initially is twice as far as the second.

3. So What?

We will use this terminology typically to explain the kinds of information on maps and also in various other sources. You will certainly also encounter it in other places, I guarantee it!

Indevelopment regularly is interpreted from one level to an additional. The general ascendancy is that it"s much easier to interpret to a lower level (e.g., from proportion to ordinal) than to a higher level. In most instances, it"s impossible to go legitimately to a greater level. Why bother utilizing a lower-level scale? Often it conserves room and energy to usage a reduced range.

For example, if you have actually a table of city populaces in Sonoma County (i.e., ratio level) and also want to show population on a map, you could print a number alongside each city name. This would preserve your ratio-level information on the map itself. A more prevalent method is to have a set of significantly big signs for the city place, each symbol indicating a variety of population. These symbols would provide ordinal-level information around populace. This conveniently reduces the clutter on the map, but you cannot recover the original ratio-level numbers from the map.

C. Accuracy, Precision, and also Significant Digits

1. Accuracy versus Precision

Some people make a difference between accuracy and precision.

Accuracy would certainly be just how cshed a measured value is to the actual value.

Precision would certainly be the variety of digits reported with the worth. For instance, I included a thermometer to my automobile that shows the outside temperature to tenths of a level (e.g., 65.2° F). The thermometer"s precision is to tenths of a level Fahrenheit. But it frequently seems to give readings that are too high, probably as a result of the engine"s heat nearby. Its accuracy, then, might just be to within 5° F.

2. Monumental Digits

Huge digits are the number of digits in a value that have actual interpretation and also reflect the accuracy of the value or measurement. For instance, a distance measurement of 45.8 kilometres would have actually 3 substantial digits. The worth means that the measurement was made to an accuracy of tenths of a kilometer. If the instrument, such as an odometer, is not precise, the worth need to not include as many type of significant digits. Of course, if the instrument actions even more accurately than this, the worth must be reported with the proper precision.

The primary allude is to beware dimensions through very specific values once the accuracy can be dubbed right into question. This is particularly true when converting worths from one kind of range to an additional. For instance, if I meacertain a map distance as 11 inches with an plain leader, then desire to convert to centimeters, I uncover on my calculator:


(11 in.) * (2.54 cm/in.) = 27.94 cm


But 27.94 cm means high accuracy, which I didn"t have via the original measurement. The solution right here is to round the answer to around the exact same number of significant digits as the original measurement. In this situation, I have to more than likely say the distance is 28 cm.

More example of considerable digits:

Two considerable digits: 3700, 37, 3.7, 0.37, 0.0037Four considerable digits: 19120, 19.12, 0.001912, 19.00, 0.1900

Notice that zeroes added after the decimal point imply precision equal to any other digits. Normally, zeroes to the appropriate of various other digits (e.g., 3700) are not thought about significant, unmuch less a decimal allude is included and also more zeroes happen to the right of it (e.g., 3700.0 suggests 5 significant digits). What if we have a measurement of 1900 wright here the zeroes are significant? Traditionally, a bar is inserted over the zeroes to indicate definition.

Another solution is to usage scientific notation.

For instance, 3700 with the zeroes actually being substantial have the right to be written as 3.700 x 103. This breaks the number dvery own into the mantissa (the considerable digits, below 3.700) and also the exponent on the ten (below 3). In clinical notation, just one digit is inserted prior to the decimal allude in the mantissa. The exponent on the 10 suggests just how many areas the decimal allude should be relocated to reclaim the number to simple notation (in this situation, 3 places to the right; this is indistinguishable to multiplying the mantissa by 1000). Negative exponents suppose smaller numbers, and we transition the decimal suggest to the left. For example, 3.700 x 10-4 is the exact same as 0.0003700. Scientific notation is handy with incredibly large or small numbers. Numbers in clinical notation deserve to be offered simply like other numbers, as long as you follow the rules of exponents -- for multiplying, splitting, and so on.

D. Category, Simplification and Symbolization of Data

This is not a course in cartography, so we will certainly not deal in detail with exactly how data is classified and also simplified for mapping, and also just how symbols are created to suit those information. But you have to be aware that any map, consisting of topographic maps, is the result of many processing of the information behind the map. Here is a brief conversation of some aspects that go into information handling for maps.

1. Classification

Maps need to take the relevant data and assign it to a given number of classes, each with its own number of members. Instances are maps of land also offers and also of temperature. For categorical information favor land provides, it is basic to overfill the map reader through too many classes. Cartographers recommend a maximum of five or 6 different classes (for land also use, these might be woodland, rangeland, cropland, water, and urban). Numerical data shown on isarithmic maps (which display zones of equal values) have the right to be rather even more complicated. For instance, temperature areas deserve to trfinish from 80"s F to -20"s F. But the data should be presented with logical symbolization (check out below).

File deserve to be damaged down right into categories in 3 ways:

(a) Each category deserve to have an equal variety of members. For example, in a map of US population by state, we could have five categories of populace, each through ten says.

(b) Each category might expectations an equal interval of worths. In the population map, we would take the variety of populaces (highest state minus lowest state), and divide the range by 5. Each category would be one-fifth of the as a whole range. In this map, categories might have actually different numbers of states. The highest possible category might just have actually a few states (e.g., California, New York and also Texas), and other categories might have a big variety of claims.

(c) Finally, we can subjectively assign claims to classes, possibly by searching for "herbal breaks" in populace ranmajesties. Often map-makers start out through among the initially two approaches, then refine it by looking at the outcomes and also altering classes slightly. All of this need to reflect the purposes of the map. Different approaches deserve to develop radically different-looking maps (see any cartography message for examples).

2. Simplification

A second major job in map-making is simplifying the information behind the map. Category is one way of simplifying the data we use going into the map. Often this requires lowering the measurement level of the data (such as transcreating ratio-level information to ordinal-level information, as in our example of city populaces indicated by assorted circle sizes).

Another simplification task relates to the spatial aspects of the data. For example, mountainous roadways might take many type of twists and turns. A map that shows all these curves would certainly be cluttered and messy. We might desire to generalize the road spatially to clarify things. Beware, of course, those map individuals who curse you for those curves they didn"t expect! The cartographer"s job is to simplify without distorting and misleading the map user.

Sometimes the task might necessitate eliminating features altogether. For example, a small-range US map deserve to present just schosen cities; how carry out you choose which cities to include without offfinishing as well many people? I experienced a US map freshly that consisted of moderate-sized cities yet omitted my hometown, San Diego (now the 6th biggest city in the US).

3. Symbolization

Symbolization of indevelopment is one more essential component of communicating through maps. What signs are liked will certainly influence whether will certainly people will certainly understand also, or also usage, a map. Symbols should be favored so that the intfinished audience will understand also them. Different symbols must be provided for general-function maps than those intended for specialists in a area.

Symbols have the right to be put on a continuum in between intuitive and abstract. Intuitive signs suggest to the reader what they represent. Pictographs are recognizable representations of the actual objects. For instance, marshes are frequently displayed with swampy grass icons, or airports as planes. On the other extreme are abstract signs through no evident connection to what they recurrent. Examples are circles as cities, or streams as dotted lines. Some abstract icons have actually been provided so extensively to reexisting a specific function that, although abstract, they perform connect via the object in most people"s minds. A star within a circle, for circumstances, usually indicates a capital city.

Symbols deserve to also be classed according to their dimension: point (0-dimension), line (1-dimension) or area (2-dimension) symbols. Like the other two, point signs have the right to convey categorical or numerical information. Instances of categorical point icons are airports, campgrounds, springs, and also quarries. Variable-sized circles for city population would certainly be examples of numerical point icons.




Line signs have the right to convey indevelopment about direct attributes, such as roads, railroadways, and streams. Another necessary usage of lines on maps are isolines. Isolines (additionally called isarithms) connect points of equal value. You"ll often view these on a map of temperatures on the weather page of the newspaper. Eincredibly place along an isoline on this map has actually the same temperature, at least theoretically. This suggests likewise that locations in between 2 isolines have values somewright here between the values of the surrounding lines. Without extra information, we cannot say through confidence much more about intervening values. We deserve to estimate, or interpolate, the intervening values, for example by measuring the distance of a allude from each line. Tbelow is no guarantee, of course, that our estimate is really correct, so beware such approximates from isoline maps.




An important form of isoline for this course is the elevation contour . These isolines display equal elevations, and are had on all USGS topographic maps. The contour interval is the height difference between each contour. Contour intervals on USGS maps differ relying on the terrain. Typical intervals on 1:24,000-range maps are 20, 40 and also 80 feet. A 20-foot interval would certainly present, for example, contours at 0 (sea level), 20, 40, 60, 80, and also so on.

It takes some practice to feel comfortable working with elevation contours and various other isoline maps. We will have lab exercises to give you some exercise with them, including visualizing terrain and also constructing profiles, or two-dimensional views of just how elevation transforms on a line throughout the topographic map.

The areas in between isolines are often shaded or colored to portray the information more successfully. Temperature maps in a lot of newsdocuments are currently in shade, through warmer areas acquiring "warmer" colors, cooler areas "cooler" colors. These shaded-isoline areas are a blend of line and also location symbols.

Speaking of color and also shading, let us cover one last point about map symbolization. The rules about making use of color and shading are different for nominal versus higher-level data (ordinal, interval, and also ratio). If you are mapping nominal-level data, the categories have no certain order, and also you more than likely desire to maximize the comparison between categories. Land provides, for instance, need to be portrayed so that you have the right to sepaprice metropolitan, submetropolitan, forest, and also pasture, or whatever before categories you include on the map.

Ordinal, interval or proportion information must be mapped in a different way. Here tbelow is a definite ordering of categories. To use the temperature map aget, we have actually a trend from heat to cold in the zones we portray. We want the map reader to instantly appreciate this gradual trfinish in the information. Thus we pick an ordering of colors or signs that conveys this continuous trend in the data, for example: red-orange-yellow-green-blue-violet.

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Many kind of civilization make the mistake of reasoning they must choose colors that maximize contrast between categories, even through ordinal or higher-level data. Beware this mistake! You might be able to pick out categories easily through this strategy, however you destroy the impression of a trfinish in the data.