Problem 60 - Chemistry The Central Science

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. \((\mathbf{b})\) 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

The average percentage for Set 1 is \(22.513\%\) and for Set 2 is \(22.613\%\). Based on these averages, Set 1 is more accurate as it is closer to the true percentage of 22.52%. The average deviation for Set 1 is \(0.020\) and for Set 2 is \(0.036\). Comparing these values, Set 1 is more precise as it has a smaller average deviation. In conclusion, Set 1 is both more accurate and precise compared to Set 2.

Step 1: Calculate the average percentage for each set of data

To calculate the average percentage for each set of data, we will add the three measurements and divide by the number of measurements (3) for both sets of data. Set 1: \[ \textbf{Average Set 1} = \frac{22.52+22.48+22.54}{3} \] Set 2: \[ \textbf{Average Set 2} = \frac{22.64+22.58+22.62}{3} \]

Step 2: Determine which set is more accurate based on average

To determine which set is more accurate, we will compare the calculated average for each set with the true percentage (22.52%).

Step 3: Calculate the average deviation for each set of data

To calculate the average deviation for each set of data, we first need to find the absolute deviation of each measurement from the average for that data set. Then, we will average those absolute deviations. Set 1: \[ \textbf{Average Deviation Set 1} = \frac{|22.52 - \textbf{Average Set 1}| + |22.48 - \textbf{Average Set 1}| + |22.54 - \textbf{Average Set 1}|}{3} \] Set 2: \[ \textbf{Average Deviation Set 2} = \frac{|22.64 - \textbf{Average Set 2}| + |22.58 - \textbf{Average Set 2}| + |22.62 - \textbf{Average Set 2}|}{3} \]

Step 4: Determine which set is more precise based on average deviation

To determine which set is more precise, we will compare the calculated average deviation for each set of data. The set with the smaller average deviation will be more precise. Comparing the results from Step 3 will allow us to determine which set of data is more precise.

Key Concepts

Average Percentage Calculation

Understanding how to calculate the average percentage is essential for various scientific data analysis tasks. In the context of the laboratory exercise provided, calculating the average involves summing the individual percentages obtained from measurements and then dividing by the quantity of those measurements.

Let's apply this to the given data. For the first set, the calculations would be as follows:
\[\begin{equation}Average\ Set\ 1 = \frac{22.52 + 22.48 + 22.54}{3}\end{equation}\]
Similarly, for the second set:
\[\begin{equation}Average\ Set\ 2 = \frac{22.64 + 22.58 + 22.62}{3}\end{equation}\]
It's crucial that the steps are followed meticulously, as an accurate average forms the basis for further calculations of accuracy and precision, as will be discussed in the following sections.

Accuracy Assessment

Accuracy assessment is about how close a measured value comes to the actual or true value. In scientific measurements, achieving high accuracy is often essential. To assess accuracy, one compares the average calculated from a data set with a known standard or true value.

In the context of the laboratory data being analyzed, the true percentage of lead in a sample is known to be 22.52%. To determine which set of data is more accurate, compare the averages calculated for both sets to this true value.

A smaller difference between the average and the true value indicates a higher level of accuracy. Improvement advice for accuracy would be to use more precise instruments, follow protocols strictly, or conduct more trials to minimize discrepancies.

Precision Evaluation

Precision evaluation, unlike accuracy, does not involve the true value, but rather how close the set of measurements are to each other. Precision is a measure of the reproducibility of data. To evaluate precision in the given laboratory exercise, calculate the average deviation of each measurement from the average calculated for that data set.

For Set 1, we have:
\[\begin{equation}Average\ Deviation\ Set\ 1 = \frac{|22.52 - Average\ Set\ 1| + |22.48 - Average\ Set\ 1| + |22.54 - Average\ Set\ 1|}{3}\end{equation}\]
And for Set 2:
\[\begin{equation}Average\ Deviation\ Set\ 2 = \frac{|22.64 - Average\ Set\ 2| + |22.58 - Average\ Set\ 2| + |22.62 - Average\ Set\ 2|}{3}\end{equation}\]
The data set with the lower average deviation is more precise. To improve precision, ensure consistent experimental conditions, proper calibration of instruments and eliminate or minimize external variables that could introduce variation into the measurements.