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: $$\begin{array}{l}{\text { (1) } 22.52,22.48,22.54} \\ {\text { (2) } 22.64,22.58,22.62}\end{array}$$ (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?

Understanding the concepts of precision and accuracy is crucial when analyzing laboratory data. In a scientific context, accuracy refers to how close a measured value is to the true value, while precision indicates the repeatability of measurements, or how close a series of values are to each other, regardless of their accuracy.

Consider a dartboard analogy: If a series of darts are closely clustered on a target, this indicates high precision. If that cluster is at the bullseye, it also indicates high accuracy. However, if the cluster is away from the center, you have precision without accuracy. Conversely, if darts are scattered but one hits the bullseye, it reflects accuracy without precision.

In the textbook exercise, comparing the students' results to the true percentage of lead in the sample assesses accuracy, while looking at the spread of their measurements assesses precision. In essence, a set's accuracy is gauged by the average of its values against the known value, and its precision is judged by the consistency of the measurements within the set. To improve their laboratory analysis, students should aim for both high accuracy (getting close to the true value) and high precision (making consistent measurements).

Calculating averages is fundamental in chemistry as it allows for an estimation of central tendencies in sets of data. To calculate an average, also known as the mean, add up all the values in a set and divide by the number of values present.

In the given exercise, students calculated the average percentage of lead in a sample. The formulas used:

• For Set 1: \(Avg_1 = \frac{22.52 + 22.48 + 22.54}{3}\)
• For Set 2: \(Avg_2 = \frac{22.64 + 22.58 + 22.62}{3}\)

These averages help to summarize the data and provide a single value that represents the dataset. Chemists use averages to simplify the complexity inherent in data sets, thus deriving more approachable figures to work with. Such averages can then be used for further comparison, as done in the exercise to identify which set of results is more accurate.

Deviation in chemistry refers to the variation between individual measurements and the average of a dataset. To evaluate the precision of a given set of data, the deviation of each value is calculated first by determining the absolute difference between each value and the average.

The deviations for each data set in the exercise were calculated as follows:

• For Set 1: \(Deviation_1 = |22.52 - Avg_1|, |22.48 - Avg_1|, |22.54 - Avg_1|\)
• For Set 2: \(Deviation_2 = |22.64 - Avg_2|, |22.58 - Avg_2|, |22.62 - Avg_2|\)

After finding these individual deviations, the next step was to calculate the average deviation of each data set, indicating the typical amount of variation within the set. The formulas used to calculate average deviation were as follows:

• For Set 1: \(AvgDeviation_1 = \frac{Deviation_{1,1} + Deviation_{1,2} + Deviation_{1,3}}{3}\)
• For Set 2: \(AvgDeviation_2 = \frac{Deviation_{2,1} + Deviation_{2,2} + Deviation_{2,3}}{3}\)

The set with the smallest average deviation reflects greater precision, as the values are more clustered around the mean. By mastering deviation calculations, students enhance their ability to judge the reliability and consistency of their experimental results.