Problem 70 - Chemistry The Central Science In Si Units

Question

Is the use of significant figures in each of the following statements appropriate? (a) The 2005 circulation of National Geographic was 7,812,564. (b) 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. (d) You calculate your grade point average to be 3.87562 .

Step-by-Step Solution

Answer

Statements (a) and (c) use appropriate significant figures, while (b) and (d) do not.

Step 1: Evaluating Statement (a)

In statement (a), the circulation of 7,812,564 given to the exact digit implies that each of those figures is significant. This is likely accurate as circulation numbers can be tracked precisely. Thus, the use of significant figures is appropriate.

Step 2: Evaluating Statement (b)

Statement (b) provides a population number for Cook County recorded to the exact digit, 5,303,683. Population figures are subject to change and rounding; hence, expressing them without a decimal or any indication of approximation suggests unnecessary precision. The use of significant figures is somewhat inappropriate.

Step 3: Evaluating Statement (c)

In statement (c), the percentage 0.621% is provided with three significant figures. Given that percentage calculations generally involve measurement, a figure with three significant digits is practical, making the use of significant figures appropriate.

Step 4: Evaluating Statement (d)

Statement (d) provides a GPA calculated to five decimal places, 3.87562. Generally, GPAs are recorded typically up to two or three decimal places for practical and comparative purposes. The use of five decimal places thus implies better-than-necessary precision. The use of significant figures is inappropriate.

Key Concepts

Precision in Measurements

Precision in measurements is critical in various fields as it reflects the exactness of a given value. In the context of measurements, precision refers to how close multiple measurements of the same item are to each other.
For example, when discussing circulation numbers for magazines or newspapers, reporting figures down to every single digit can be appropriate. This is because we often track exact sales which can be precisely counted and thus, can justify highly accurate data.
If we say the circulation is exactly 7,812,564, it indicates it's a countable and accurate total where every digit is significant.

• Precision ensures reliable and consistent results.
• Precision in significant figures implies each recorded digit has meaningful accuracy.
• Overly precise data that doesn't reflect reality can lead to misunderstandings.

Population Data Accuracy

Population figures are inherently dynamic and often estimated for specific points in time. Presenting a population with exact precision, such as specifying Cook County’s population exactly as 5,303,683, might seem unnecessarily precise.
Population continuously changes due to births, deaths, and migration therefore recording to so many significant figures could be misleading.
Instead, providing a rounded number would better represent general population counts.

• Rounding population figures can prevent projecting false precision.
• Population estimates often serve as an approximation, not an exact count.
• Understanding this aids in setting realistic expectations for data accuracy.

GPA Calculation Precision

Calculating GPA (Grade Point Average) involves determining a student’s academic performance. GPA calculations typically round to two or three decimal places, which offer enough precision for most academic and comparative purposes.
In the example where the GPA is calculated as 3.87562, this level of precision is excessive and unnecessary.
Being overly precise with five decimal places doesn't add value and can complicate the comparison between students.

• Standard GPA display rounds to two decimal places for clarity.
• Over-precise GPAs can cause perceived inaccuracies or stress.
• Rounding offers clarity and aligns with most educational institutions’ standards.

Significance in Percentage Data

Presenting percentage data calls for careful consideration regarding precision. In the context where 0.621% of the population has a particular surname like 'Brown,' offering three significant figures is usually sufficient.
This level of precision is appropriate because percentage values often derive from smaller sample sizes and measurements that benefit from clear accuracy.
Being precise enough helps convey clear data without assuming undue accuracy that more significant figures might falsely imply.

• Percentages typically include up to three significant figures.
• Three figures mitigate effects of rounding errors in small percentages.
• Balanced precision avoids confusion and exaggeration of data reliability.