Question

Three students \((\mathrm{A}, \mathrm{B},\) and \(\mathrm{C})\) are asked to determine the volume of a sample of ethanol. Each student measures the volume three times with a graduated cylinder. The results in milliliters are: \(\mathrm{A}(87.1,88.2,87.6) ; \mathrm{B}(86.9,\) 87.1,87.2 ; \(\mathrm{C}(87.6,87.8,87.9)\). The true volume is \(87.0 \mathrm{~mL}\). Comment on the precision and the accuracy of each student's results.

Short answer

B has high precision and accuracy; A has low precision and accuracy; C has high precision but low accuracy.

Step-by-step solution

Understanding Precision and Accuracy

Precision refers to how close a set of measurements are to each other, indicating the consistency of the measurements. Accuracy refers to how close a measurement is to the true or accepted value.

Determine Precision for Student A

Student A's measurements are 87.1, 88.2, and 87.6 mL. To evaluate precision, calculate the range: the highest value (88.2) minus the lowest value (87.1), which equals 1.1. This range shows limited precision due to wider dispersion.

Determine Accuracy for Student A

Calculate the average of Student A's measurements: \(\frac{87.1 + 88.2 + 87.6}{3} = 87.63\). Compare this to the true volume of 87.0 mL. The average deviates by 0.63 mL from the true value, indicating low accuracy.

Determine Precision for Student B

Student B's measurements are 86.9, 87.1, and 87.2 mL. Calculate the range: 87.2 - 86.9 = 0.3. The small range suggests high precision.

Determine Accuracy for Student B

Calculate the average: \(\frac{86.9 + 87.1 + 87.2}{3} = 87.07\). The average is very close to the true volume, deviating only by 0.07 mL, indicating high accuracy.

Determine Precision for Student C

Student C's measurements are 87.6, 87.8, and 87.9 mL. Calculate the range: 87.9 - 87.6 = 0.3. A small range like this indicates high precision.

Determine Accuracy for Student C

Calculate the average: \(\frac{87.6 + 87.8 + 87.9}{3} = 87.77\). The average is 0.77 mL away from the true volume, indicating low accuracy.

Summary Conclusion

Student B's results demonstrate high precision and high accuracy. Student A's measurements have low precision and low accuracy. Student C shows high precision but low accuracy.

Key concepts

Precision in Chemistry

In the context of chemistry, precision refers to how reproducible a set of measurements are, essentially showing the consistency.
For instance, if you measure the same substance multiple times and get results that are very close to each other, your measurements are precise.
Applying this to our example:

• Student A had measurements that ranged from 87.1 to 88.2 mL, resulting in a range of 1.1 mL, which suggests low precision as the results are not tightly packed together.
• Student B's measurements varied from 86.9 to 87.2 mL, with a minimal range of 0.3 mL, indicating higher precision.
• Similarly, Student C achieved a range of 0.3 mL across his attempts, thus demonstrating high precision.

Understanding precision is valuable in research and experiments where repeatability ensures the reliability of data outputs.

Accuracy in Chemistry

Accuracy in chemistry is about how close a measured value is to the true or accepted value.
It helps in validating the correctness of measurements and whether they are rooted in reality.
Considering the students' data:

• Student A had an average of 87.63 mL, differing by 0.63 from the true value of 87.0 mL, thus showing low accuracy despite attempts to hit the true value.
• Student B's average was very close to the true value at 87.07 mL, achieving a deviation of just 0.07 mL, indicating high accuracy.
• Meanwhile, Student C recorded an average of 87.77 mL, which was 0.77 mL off from the true value, resulting in low accuracy despite consistent readings.

In real-world applications, accuracy is critical when exact measurements are necessary for safety and efficacy, such as in pharmaceuticals.

Measurement Error Analysis

Understanding measurement error analysis is crucial in determining both precision and accuracy.
It involves recognizing the types of errors that may occur in experiments and how they impact results.
Common sources of errors include observational mistakes, instrument failures, and environmental conditions.

• Random errors, which affect precision, are unpredictable fluctuations in the measurements.
• Systematic errors, on the other hand, consistently skew results in a particular direction, affecting accuracy.

By examining the students' results, we see how random errors might lead to the varied precision observed in Student A's work, while systematic errors may contribute to the consistent deviation from the true value in Student C's results.
Analyzing these errors helps in improving methodologies and enhancing the reliability of experimental data.