# 1.5 Uncertainty in Measurement

Even the most carefully taken measurements are always inexact. This can be a consequence of inaccurately calibrated instruments, human error, or any number of other factors.

Two terms are used to describe the quality of measurements: precision and accuracy. Precision is a measure of how closely individual measurements agree with one another. Accuracy refers to how closely individually measured numbers agree with the correct or "true" value.

Whatever the source, all measurements contain error. Thus, all measured numbers contain uncertainty. It is important that these numbers be reported in such a way as to convey the magnitude of this uncertainty.

Consider a fourth-grade student who, when asked by his teacher how old the Earth is, replies "Four billion and three years old." (The student had been told by a first-grade teacher three years earlier that the Earth was four billion years old.) Obviously, we don't know the age of the Earth to the year, so it is not appropriate to report a number that suggests we do.

In order to convey the appropriate uncertainty in a reported number, we must report it to the correct number of significant figures. The number 83.4 has three digits. All three digits are significant. The 8 and the 3 are "certain digits" while the 4 is the "uncertain digit." As written, this number implies an uncertainty of plus or minus 0.1, or an error of 1 part in 834. Thus, measured quantities are generally reported in such a way that only the last digit is uncertain. All digits, including the uncertain one, are called significant figures.

Guidelines

1. Nonzero digits are always significant–457 cm (3 significant figures); 2.5 g (2 significant figures).
2. Zeros between nonzero digits are always significant–1005 kg (4 significant figures); 1.03 cm (3 significant figures).
3. Zeros at the beginning of a number are never significant; they merely indicate the position of the decimal point–0.02 g (one significant figure); 0.0026 cm (2 significant figures).
4. Zeros that fall at the end of a number and after the decimal point are always significant–0.0200 g (3 significant figures); 3.0 cm (2 significant figures).
5. When a number ends in zeros but contains no decimal point, the zeros may or may not be significant–130 cm (2 or 3 significant figures); 10,300 g (3, 4, or 5 significant figures).

To avoid ambiguity with regard to the number of significant figures in a number with tailing zeros but no decimal point, such as 700, we use scientific (or exponential) notation to express the number. If we are reporting the number 700 to three significant figures, we can leave it written as is, or we can express it as 7.00 102. There is no ambiguity in the latter regarding the number of significant figures, because zeros after a decimal point are always significant. However, if there really should be only two significant figures, we can express this number as 7.0 102. Likewise, if there should be only one significant figure, we can write 7 102. Scientific notation is convenient for expressing a measurement in the appropriate number of significant figures. It is also useful when reporting extremely large and extremely small numbers. It would be most inconvenient for us to have to write all of the zeros in the number 1.91 10-24 (0.00000000000000000000000191).

## Questions

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When measured numbers are used in a calculation, the final answer cannot have any greater certainty than the measured numbers that went into the calculation. In other words, the precision of the result is limited by the precision of the measurements used to obtain that result. For example: If we measure the length of one side of a cube and find it to be 1.35 cm; and we then calculate the volume of the cube using this measured length, we get an answer of 2.460375 cm3. Our original measurement had three significant figures. The implied uncertainty in 1.35 is 1 part in 135. If we report the volume of the cube to seven significant figures, we are implying an uncertainty of 1 part in over two million! We can't do that. In order to report results of calculations so as to imply a realistic degree of uncertainty, we must follow the following rules.

1. When multiplying or dividing measured numbers, the answer must have the same number of significant figures as the measured number with the fewest significant figures.
2. When adding or subtracting, the answer can have only as many places to the right of the decimal point as the measured number with the smallest number of places to the right of the decimal point.

Using these rules, we would report the volume of the cube in the example above as 2.46 cm3. Use the Significant Figures activity to practice reporting calculated numbers to the appropriate number of significant figures.

## Significant Figures

Instructions
1. Select addition, subtraction, multiplication, or division.
2. Enter two numbers, and predict the result to the proper number of significant figures.

Determine the answer for each calculation to the correct number of significant figures. Click on the box to see the correct answer for each.

## Questions

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How many significant figures should there be in the answer to the following problem?

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Not all numbers are measured numbers. There are some numbers that arise as a result of counting or as a result of a definition. If there are three birds in a cage, there is no uncertainty in the number of birds. Likewise, there is no uncertainty in the number of items in a dozen. These are exact numbers, and they are taken to have an infinite number of significant figures (exact numbers should never limit the number of significant figures you report in a calculated answer).