Question

Is the use of significant figures in each of the following statements appropriate? Why or why not? (a) Apple sold 22,727,000 iPods during the last three months of 2008 . (b) New York City receives 49.7 inches of rain, on average, per year. (c) In the United States, \(0.621 \%\) of the population has the surname Brown. (d) You calculate your grade point average to be \(3.87562 .\)

Short answer

In summary, the use of significant figures is appropriate in statements (a), (b), and (c) because they represent exact counts or reasonable approximations for the given context. However, statement (d) has too many significant figures, as grade point averages are usually rounded to two decimal places for easier comparison and understanding.

Step-by-step solution

Statement (a) Analysis

For statement (a), Apple sold 22,727,000 iPods during the last three months of 2008. This number has a lot of significant figures, and it is appropriate because it represents an exact count of iPods sold. There is no estimation involved, and the number is precise and accurate.

Statement (b) Analysis

In statement (b), New York City receives 49.7 inches of rain, on average, per year. This value has three significant figures and is appropriate because the rainfall amount is an average, meaning there is some level of estimation involved. The use of only one decimal place provides a balance between precision and readability, which is suitable for this type of data.

Statement (c) Analysis

In statement (c), 0.621% of the population in the United States has the surname Brown. This value has three significant figures, with one decimal place. Since this result represents a percentage of the population, which is an approximation, it is appropriate to provide the information with the given precision.

Statement (d) Analysis

For statement (d), the calculated grade point average is 3.87562. This number has six significant figures, which is likely too many for this context. Typically, grade point averages are rounded or truncated to two decimal places (e.g., 3.88), which provides sufficient precision for most purposes while remaining easy to understand and compare. In conclusion, statements (a), (b), and (c) have appropriate uses of significant figures, while statement (d) has a higher precision than typically needed for grade point averages.

Key concepts

Scientific Notation

When dealing with very large or very small numbers in science and mathematics, scientific notation becomes an invaluable tool. It simplifies numbers by expressing them as a product of a number between 1 and 10 and a power of ten. For instance, the distance between the Earth and the Sun, approximately 149,600,000 kilometers, can be written in scientific notation as \(1.496 \times 10^8\) kilometers, making it more readable and manageable.

Understanding how to convert numbers into scientific notation is essential, especially in exercises like the one from the textbook, where incredibly precise data is presented. If one were to convert the number of iPods sold by Apple into scientific notation, it would be represented as \(2.2727 \times 10^7\), maintaining all significant figures but in a more concise form. Remember, scientific notation is not only about brevity but also about clarity in representation, allowing one to instantly grasp the scale of the number.

Measurement Precision

The precision of a measurement refers to the smallest interval that can be measured reliably, and it often depends on the instrument used for measuring. Precision is closely tied to the concept of significant figures, which are digits that carry meaning contributing to its measurement accuracy. For example, in our exercise, the amount of rainfall in New York City is given to one decimal place. This suggests the measurement instrument can reliably measure the rainfall to the nearest tenth of an inch.

It's vital to understand that while more significant figures can imply higher precision, they may not always reflect greater accuracy. Precision indicates the level of detail in the measurement, but not necessarily how close the measurement is to the true or accepted value. Therefore, in contexts like calculating a grade point average, where the number of significant figures used was six, it shows an artificially high level of precision that isn't typically applicable in real-world scenarios, such as academic records, where two decimal places suffice.

Statistical Data Representation

Representing statistical data accurately is critical, as it conveys information through numbers that must reflect reality as closely as possible. The concept of significant figures plays a role here as well. In our example, the percentage of the U.S. population with the surname Brown is given to three significant figures. This is appropriate for statistical representation where details are important, yet the data must remain digestible and clear.

Statistical data should always strive for a balance between precision and practicality. Too few significant figures can lead to loss of important detail, whereas too many can imply a level of accuracy that is not supported by the data collection method. In the exercise, we're advised that for most practical purposes, rounding statistics to a reasonable number of significant figures helps maintain readability and ensures the data remains statistically significant and relevant.