Question

Is the use of significant figures in each of the following statements appropriate? Why or why not? (a) The 2005 circulation of National Geographic was \(7,812,564\). (b) \(\mathrm{On}\) July 1, 2005, the population of Cook County, Illinois, was \(5,303,683 .\) (c) In the United States, \(0.621 \%\) of the population has the surname Brown.

Short answer

In summary, the appropriateness of the significant figures used in each statement depends on the context and the level of precision required. For statements (a) and (b), the use of seven significant figures can be considered appropriate as they provide precise numbers for circulation and population, while for statement (c), the use of three significant figures for the percentage is likely to be an unnecessary level of precision and could be simplified to one or two significant figures (like 0.62% or 0.6%).

Step-by-step solution

Statement (a) Analysis

For statement (a), the circulation of National Geographic is given as 7,812,564 in 2005. The number has seven significant figures.

Statement (a) Evaluation

All of the digits provided in this number could be important to know – for example, if someone wanted to know the exact number of copies distributed that year. It could also be argued that an approximation would be sufficient for most purposes (for example, 7.81 million or 7,810,000). So, it really depends on the context in which this number is being used. In general, it can be considered appropriate as it provides a precise number of the circulation.

Statement (b) Analysis

For statement (b), the population of Cook County, Illinois, on July 1, 2005, is given as 5,303,683. The number has seven significant figures.

Statement (b) Evaluation

Similar to statement (a), it depends on the context in which this number is being used. If it is important to know the exact population, then the use of seven significant figures is appropriate. However, for most purposes, an approximation could be sufficient (for example, 5.30 million or 5,300,000). So, it can be considered appropriate as it provides a precise number for the population.

Statement (c) Analysis

For statement (c), we are given that 0.621% of the population in the United States has the surname Brown. This value has three significant figures.

Statement (c) Evaluation

In this case, the percentage is provided with three significant figures, which is likely to be an unnecessary level of precision. For most purposes, rounding to one or two significant figures would be acceptable (for example, 0.62% or 0.6%). In this case, the use of significant figures is not entirely appropriate and can be simplified without losing much precision.

Key concepts

Accuracy in Scientific Notation

Understanding the accuracy in scientific notation is crucial when dealing with very large or very small numbers. Scientific notation expresses numbers as a product of two factors: a coefficient that typically ranges from 1 to 10, and 10 raised to the power of an exponent. The coefficient itself will typically include all the significant figures of the original number, representing the accuracy of the number.

For instance, suppose we have the population of a city recorded as 8,540,000. In scientific notation, this might be written as 8.54 x 10^6, where '8.54' carries three significant figures. The accuracy of this representation lies in how closely these figures reflect the true or measured value. It's vital to know the precision of the original measurement to determine if all significant figures are necessary or if they are overestimating the accuracy of the number.

Precision in Data Representation

Precision in data representation refers to how detailed a number is. It is related to the concept of significant figures, which includes all the digits in a number that contribute to its precision starting with the first non-zero digit.

For example, in the case of National Geographic's circulation being 7,812,564, this level of precision might be excessive for general knowledge but could be essential for accounting purposes. Precision is dictated by the need for detail. If the purpose is to understand trends or for large-scale comparisons, rounded numbers such as 7.81 million may be more appropriate and still sufficiently precise. Precision should not be confused with accuracy; a number can be very precise but not accurate to the true value if there is an error in measurement or recording.

Contextual Use of Significant Figures

The contextual use of significant figures is about understanding when and why precision is needed. As seen in the example of Cook County's population of 5,303,683, attention to significant figures ensures we present data that matches the required information specificity. In scientific and technical fields, having too many or too few significant figures can lead to errors or misunderstandings.

When dealing with large populations, the precision to the exact person is seldom necessary, and an estimation with fewer significant figures is more practical. However, in legal or administrative contexts, exact figures are often required. The reported percentage of 0.621% having the surname Brown is another case where the amount of significant figures should reflect the precision of the research method. If the data collection method only justifies a precision to the nearest tenth of a percent, the reported figure should be 0.6%, aligning with the accuracy of the underlying data.