Problem 69 - Chemistry The Central Science In Si Units

Question

Two students determine the percentage of lead in a sample as a laboratory exercise. The true percentage is \(22.52 \% .\) The students' results for three determinations are as follows: (1) 22.52,22.48,22.54 (2) 22.64,22.58,22.62 (a) Calculate the average percentage for each set of data and state which set is the more accurate based on the average. (b) Precision can be judged by examining the average of the deviations from the average value for that data set. (Calculate the average value for each data set; then calculate the average value of the absolute deviations of each measurement from the average.) Which set is more precise?

Step-by-Step Solution

Answer

Set 1 is more accurate; Set 2 is more precise.

Step 1: Calculate Average Percentage for Each Set

To find the average percentage of lead in each student's data, sum the values for each set and divide by the number of measurements (3). - For Set 1: \[\text{Average 1} = \frac{22.52 + 22.48 + 22.54}{3} = \frac{67.54}{3} = 22.51\%\]- For Set 2: \[\text{Average 2} = \frac{22.64 + 22.58 + 22.62}{3} = \frac{67.84}{3} = 22.613\%\]

Step 2: Determine More Accurate Set Based on Average

Compare the averages to the true value, which is 22.52%. The set with an average closer to the true value is more accurate.- Absolute deviation from true value for Set 1: \[|22.51 - 22.52| = 0.01\%\]- Absolute deviation for Set 2: \[|22.613 - 22.52| = 0.093\%\]Set 1 is more accurate because its average is closer to the true percentage.

Step 3: Calculate Average Absolutely Deviations for Precision

Calculate the average of the absolute deviations from their respective average. - For Set 1:\[\text{Deviations: } |22.52 - 22.51|, |22.48 - 22.51|, |22.54 - 22.51|\]Absolute deviations are: 0.01, 0.03, and 0.03.\[\text{Average Deviation 1} = \frac{0.01 + 0.03 + 0.03}{3} = 0.0233\]- For Set 2:\[\text{Deviations: } |22.64 - 22.613|, |22.58 - 22.613|, |22.62 - 22.613|\]Absolute deviations are: 0.027, 0.033, and 0.007.\[\text{Average Deviation 2} = \frac{0.027 + 0.033 + 0.007}{3} = 0.0223\]

Step 4: Determine More Precise Set Based on Deviations

Compare the average absolute deviations calculated for each set. The set with the smaller average deviation is more precise. - Average Deviation for Set 1: 0.0233 - Average Deviation for Set 2: 0.0223 Set 2 is more precise because it has a smaller average deviation from its mean value.

Key Concepts

Precision and Accuracy

In the world of analytical chemistry, understanding the concepts of precision and accuracy is crucial for interpreting experimental results effectively. Accuracy refers to how close a measurement is to the true value or the accepted standard. In our exercise, the true percentage of lead is 22.52%. Student sets are compared based on their calculated average values:

• Set 1 averaged at 22.51%, which is extremely close to the true value.
• Set 2 averaged at 22.613%, which deviates more significantly from 22.52%.

The closer the average is to the true value, the more accurate the results are. Here, Set 1 is more accurate because its average is nearer to the true percentage value. Directly connected to accuracy is precision, which deals with how close repeated measurements are to one another. Precision does not consider the true value, but consistency among results.

Statistical Analysis

Statistical analysis plays a powerful role in interpreting experimental data. In evaluating the precision of a data set, we often calculate the average absolute deviation. This statistical measure gives insights into how interspersed the data points are around their own mean:

• For Set 1, the average absolute deviation is calculated as 0.0233%.
• For Set 2, it is slightly lower at 0.0223%.

A lower average deviation indicates that the individual measurements are clustered more closely to the average value, which translates to better precision. In our exercise, Set 2 shows tighter clustering of data, hence it is more precise. The analysis involves summing absolute differences of each data point from the mean and dividing by the number of observations, providing a snapshot of data consistency.

Experimental Data Interpretation

Interpreting experimental data involves evaluating both accuracy and precision to form a comprehensive understanding of the results. This process is central to analytical chemistry and allows scientists to trust the results they gather.
In an experiment, data must be scrutinized for both its closeness to the true value and the reliability of outcomes. By comparing these values, chemists can identify potential sources of error and suggest improvements.
In our given exercise, despite Set 1 being more accurate, Set 2's greater precision indicates more consistent measuring techniques possibly due to better control of experimental conditions or methodology.
Experimental data interpretation also involves looking at external factors that might affect the results, such as instrument calibration, operator skill level, or environmental conditions. Understanding these factors aids in refining methods and gaining more reliable outcomes in future experiments. Through detailed analysis of both accuracy and precision, chemists can continuously improve the quality of their analytical processes.