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 (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?

Verified step by step guidance

1. Calculate the average value for each data set. For set (1), add the values 22.52, 22.48, and 22.54, then divide by 3. For set (2), add the values 22.64, 22.58, and 22.62, then divide by 3.
2. Determine the absolute deviation of each measurement from the average value for each data set. For each measurement, subtract the average value calculated in step 1 and take the absolute value of the result.
3. Calculate the average of the absolute deviations for each data set. Add the absolute deviations calculated in step 2 for each set, then divide by 3.
4. Compare the average absolute deviations of the two data sets. The set with the smaller average absolute deviation is more precise.
5. Conclude which data set is more precise based on the comparison of average absolute deviations.

Key Concepts

Accuracy vs. Precision

Accuracy refers to how close a measured value is to the true value, while precision indicates the consistency of repeated measurements. In this context, the true percentage of lead is 22.52%, and the students' results need to be evaluated for how closely they cluster around this true value (accuracy) and how consistently they yield similar results (precision).

Mean and Average Deviation

The mean is the average of a set of values, calculated by summing all measurements and dividing by the number of measurements. The average deviation is a measure of precision, calculated by finding the absolute differences between each measurement and the mean, then averaging those differences. This helps quantify how spread out the measurements are from the mean.

Statistical Analysis of Data Sets

Statistical analysis involves evaluating data sets to draw conclusions about their characteristics. In this case, comparing the average deviations of the two sets of measurements will determine which set is more precise. A smaller average deviation indicates that the measurements are more closely grouped around the mean, reflecting higher precision.