Exact and Inexact Numbers

In scientific work, numbers are groups in two categories: exact numbers and inexact numbers. An exact number is a number that has a value with no uncertainty in it; that is, it is known exactly.

Examples:There are exactly 12 objects in a dozen, not 12.01 or 12.02.

There can be 7 people in a room, but never 6.99 or 7.02.

An inexact number is a number that has a value with a degree of uncertainty in it. Inexact numbers result anytime a measurement is made.

It is impossible to make an exact measurement; some uncertainty will always be present.

Flaws in measuring-device construction, improper calibration of an instrument, and the skills (or lack of skills) possessed by a person using a measuring device all contribute to (uncertainty).

Errors in measurement can be classified as either random errors or systematic errors.

A random error is an error originating from uncontrollable variables in an experiment. Such errors result in experimental values that fluctuate about the true value. A variation in the angle from which a measurement scale is viewed will cause random error.

Momentary changes in air currents, atmospheric pressure, or temperature near a sensitive balance for weighting would cause random errors. The net result of random errors, which can never be completely eliminated, is a decrease in the precision of a series of measurements.

A systematic error is an error originating from controllable variables in an experiment. They are “constant” errors that occur again and again. A flaw in a piece of equipment, such as a chipped weight in a balance, would cause systematic error. All readings would be off by a specific amount because of the flaw.

Accuracy, Precision, and Error

Precision is an indicator of how close a series of measurements on the same object are to each other.

Accuracy is an indicator of how close a measurement (or the average of multiple measurements) comes to a true or accepted value.

The activity of throwing darts at a target illustrates nicely the difference between these two terms.

Accuracy refers to how close the darts are to the center (bull's-eye) of the target. Precision refers to how close the darts are to each other.

The difference between precision and accuracy.

Practice Problem

Solution

Student A: poor accuracy, poor precision Student B: good accuracy, good precision Student C: poor accuracy, good precision

Poor accuracyPoor precision