# Measurement systems—values and what they represent

The type of measurement system used may have a dramatic effect on the interpretation of the resulting values. A distance of 20 kilometers is twice as far as 10 kilometers, and something that weighs 100 pounds is one-third as much as something that weighs 300 pounds. But someone who came in first place in a race may not have done three times as well as someone in third place, and soil with a pH of 3 is not half as acidic as soil with a pH of 6. To carry this even further, someone that is 60 years old is twice as old as someone that is 30 years old. But the older of the two individuals can only be twice as old as the younger individual just once in a lifetime.

The significance of this discussion on numbers is that all numbers cannot be treated the same. It is important for you to know the type of measurement system being used in the raster dataset so that the appropriate operations and functions can be implemented and the results will be predictable. Measurement values can be broken into four types: ratio, interval, ordinal, and nominal.

ArcGIS Spatial Analyst does not distinguish between the four different types of measurements when asked to process or manipulate the values. Most mathematical operations work well on ratio values, but when interval, ordinal, or nominal values are multiplied, divided, or evaluated for the square root, the results are typically meaningless. On the other hand, subtraction, addition, and Boolean determinations can be meaningful when used on interval and ordinal values. Attribute handling within and between raster datasets is most effective and efficient when using nominal measurements.

## Ratio

The values from the ratio measurement system are derived relative to a fixed zero point on a linear scale. Mathematical operations can be used on these values with predictable and meaningful results. Examples of ratio measurements are age, distance, weight, and volume.

## Interval

Time of day, calendar years, the Fahrenheit temperature scale, and pH values are all examples of interval measurements. These are values on a linear calibrated scale, but they are not relative to a true zero point in time or space. Because there is no true zero point, relative comparisons can be made between the measurements, but ratio and proportion determinations are not as useful.

## Ordinal

Ordinal values determine position. These measurements show place, such as first, second, and third, but they do not establish magnitude or relative proportions. How much better, worse, prettier, healthier, or stronger something is cannot be demonstrated from ordinal numbers. For example, a runner who was first place in a race probably did not run twice as fast as the second-place runner. Knowing the winners only by place, you do not know how much faster the first-place runner was compared with the second-place runner.

## Nominal

Values associated with this measurement system are used to identify one instance from another. They may also establish the group, class, member, or category with which the object is associated. These values are qualities, not quantities, with no relation to a fixed point or a linear scale. Coding schemes for land use, soil types, or any other attribute qualify as nominal measurements. Other nominal values are social security numbers, ZIP Codes, and telephone numbers.

## Discrete versus continuous data

A second subdivision of the values assigned to each cell are the values representing discrete or continuous data.

### Discrete data

Discrete data, sometimes called categorical data, most often represents objects. These objects usually belong to a class (for example, soil type), a category (for example, land-use type), or a group (for example, political party). A categorical object has known and definable boundaries.

An integer value is normally associated with each cell in a discrete raster dataset. Most integer raster datasets can have a table that carries additional attribute information. Floating-point values can be used to represent discrete data, but this is not common.

Discrete data is best represented by ordinal or nominal numbers.

### Continuous data

A continuous raster dataset or surface can be represented by a raster with floating-point values (referred to as a floating-point raster dataset) or occasionally by integer values. The value for each cell in the dataset is based on a fixed point (such as sea level), a compass direction, or the distance of each location from a phenomenon in a specified measurement system (such as the noise in decibels monitored at various sites near an airport). Examples of continuous surfaces are elevation, aspect, slope, the radiation levels from a nuclear plant, and the salt concentration from a salt marsh as it moves inland.

Floating-point raster datasets do not have a table associated with them because most, if not all, cell values are unique, and the nature of continuous data excludes other associated attributes.

Continuous data is best represented by ratio and interval values.

Many times, meaningless results will occur when combining discrete and continuous data, for instance, adding land use (discrete data) to elevation (continuous data). A value of 104 on the resulting raster dataset could have been derived from adding single-family housing land-use type, with a value of 4, to an elevation of 100.