Problem 11 - Chemistry for Allied Health

Question

The density of a copper sample was determined by three different students (shown below). Each performed the measurement three times and is reported below (all values in units of \(\mathrm{g} / \mathrm{cm}^{3}\) ). The accepted value for the density of copper is \(8.92 \mathrm{~g} / \mathrm{cm}^{3}\). a. Determine if each student's data is accurate, precise, neither or both. b. What is the average density based on Justin's data? c. What is the average density based on Jane's data? o Jane: \(8.94,8.89,8.91\) o Justin: 8.32, 8.31, \(8.34\) o Julia: \(8.64,9.71\), and \(9.10\)

Step-by-Step Solution

Answer

Jane's data is accurate and precise; Justin's is precise but not accurate; Julia's is neither accurate nor precise. Justin's average density is 8.32 \(\mathrm{~g/cm^3}\) and Jane's average density is 8.91 \(\mathrm{~g/cm^3}\).

Step 1: Calculate Average Density for Each Student

To determine the average density for each student, add up their three density measurements and divide by 3 (the number of measurements). For Jane, the average is \( \frac{8.94 + 8.89 + 8.91}{3} = 8.91 \mathrm{~g/cm^3}\). For Justin, \( \frac{8.32 + 8.31 + 8.34}{3} \approx 8.32 \mathrm{~g/cm^3}\), and for Julia, \( \frac{8.64 + 9.71 + 9.10}{3} = 9.15 \mathrm{~g/cm^3}\).

Step 2: Determine Accuracy of Each Student's Data

Compare each student's average density to the accepted value of \(8.92 \mathrm{~g/cm^3}\). If it's close (considering some minor experimental error) to the accepted value, their data is considered accurate. Close typically means within 0.05 to 0.1 units, depending on acceptable experimental error in your context.* Jane's data is accurate (average is 8.91).* Justin's data is not accurate (average is 8.32, quite different).* Julia's data is not accurate (average is 9.15, also significantly different).

Step 3: Assess Precision of Each Student's Data

Precision refers to how close the measurements are to each other, which can be observed by examining the range or variance of their measurements. * Jane's measurements: 8.94, 8.89, 8.91 (very close to each other, very precise). * Justin's measurements: 8.32, 8.31, 8.34 (also very close to each other, very precise). * Julia's measurements: 8.64, 9.71, 9.10 (quite spread out, not precise).

Step 4: Conclusion on Accuracy and Precision

Summarize the findings for each student: * Jane's data: Accurate and precise. * Justin's data: Not accurate, but precise. * Julia's data: Not accurate, and not precise.

Step 5: Calculate Justin's Average Density

Justin's average density has already been calculated in Step 1. It is 8.32 \(\mathrm{~g/cm^3}\).

Step 6: Calculate Jane's Average Density

Jane's average density has already been calculated in Step 1. It is 8.91 \(\mathrm{~g/cm^3}\).

Key Concepts

Understanding Accuracy vs. Precision

In science, accuracy and precision are crucial concepts that describe two distinct aspects of measurement quality. Many students often get confused between these two terms, but it is essential to differentiate them for accurate data interpretation and analysis. Accuracy refers to how close a measured value is to the true or accepted value. In the context of this exercise, the accepted density for copper is 8.92 g/cm³, and a student's measurements would be considered accurate if their average is close to this value.
Precision, on the other hand, describes how close repeated measurements are to each other. It is possible for measurements to be precise, but not accurate if they are consistent yet far from the true value. Similarly, measurements can be accurate on average but not precise if individual values are scattered.
In the given exercise, Jane's measurements are both accurate and precise as they are close to each other and the accepted value. Justin’s measurements are precise; they are close to each other but not to the copper's true density. Julia's results are neither accurate nor precise, with significant spread and deviation from the accepted value.

Insight into Error Analysis in Measurements

Error analysis is a critical part of experimental science. It helps to identify and quantify uncertainties in experimental data. In practical terms, making precise and accurate measurements involves understanding potential errors and minimizing them whenever possible. Every measurement has some degree of uncertainty, which could stem from instrumental limitations, environmental conditions, or human error.
Errors can be systematic, which affects the accuracy, or random, which affects the precision. Systematic errors are consistent and reproducible inaccuracies that occur in the same direction. Random errors cause measurement variations and influence precision. In this exercise, the calculation of average densities reflects both accuracy and precision considerations. Adjusting for errors helps in minimizing uncertainty and improving the reliability of results. Therefore, understanding these concepts helps in interpreting the results accurately, as seen with Jane's data being accurately close to the accepted value, while Justin and Julia faced deviations influencing their accuracy.

Mastering Average Density Calculation

Calculating the average density is fundamental in measurements involving multiple attempts to determine a singular value. To find the average density, one needs to sum the individual measurements and divide by the number of observations. This average reflects the most representative value of the data set.
The calculation is simple: for Jane, adding 8.94, 8.89, and 8.91 and dividing by 3 gives an average density of 8.91 g/cm³. Similarly, Justin’s average density is calculated by summing 8.32, 8.31, and 8.34, resulting in an average of approximately 8.32 g/cm³. Julia’s density calculations follow the same formula but result in a significantly different average due to her varied measurements.
Understanding how to compute the average density correctly helps in comparing how experimental data aligns with accepted standards. It is a key element in determining both the accuracy and precision of the measurements, ultimately guiding decision-making in scientific research.