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 tell 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?

Short answer

The average percentage for Set 1 is \(22.5133\%\) and for Set 2 is \(22.6133\%\). Set 1 is more accurate since its average is closer to the true value of \(22.52\%\). The average absolute deviations for both sets are \(0.0222\), indicating that both sets are equally precise.

Step-by-step solution

Calculate the average percentage for each set of data

To calculate the average percentage of lead in each set of data, we have to sum up the numbers and divide by the number of measurements (3). Set 1: \( \frac{22.52 + 22.48 + 22.54}{3} \) Set 2: \( \frac{22.64 + 22.58 + 22.62}{3} \)

Determine which set is more accurate based on the average

Calculate the averages from Step 1: Set 1: \( \frac{22.52 + 22.48 + 22.54}{3} = 22.5133 \) Set 2: \( \frac{22.64 + 22.58 + 22.62}{3} = 22.6133 \) The true percentage is 22.52%. Therefore, Set 1 with an average of 22.5133% is more accurate.

Calculate the average absolute deviations for each set

To calculate the average of the absolute deviations of each measurement from the average, we first need to calculate the absolute deviation for each measurement, and then calculate the average of these deviations. Set 1: |22.52 - 22.5133| = 0.0067 |22.48 - 22.5133| = 0.0333 |22.54 - 22.5133| = 0.0267 \( \frac{0.0067 + 0.0333 + 0.0267}{3} \) Set 2: |22.64 - 22.6133| = 0.0267 |22.58 - 22.6133| = 0.0333 |22.62 - 22.6133| = 0.0067 \( \frac{0.0267 + 0.0333 + 0.0067}{3} \)

Determine which set is more precise

Calculate the average absolute deviations: Set 1: \( \frac{0.0067 + 0.0333 + 0.0267}{3} = 0.0222 \) Set 2: \( \frac{0.0267 + 0.0333 + 0.0067}{3} = 0.0222 \) The average absolute deviations for both sets are 0.0222, which implies that both sets of data are equally precise.

Key concepts

Percentage Calculation

In scientific experiments and measurements, understanding how closely your results match the expected or known value is crucial. This is where percentage calculation becomes important. When you calculate the average percentage of a set of data, you might sum up all the measurements and divide by the number of readings. For example:

• In Set 1, the calculations would be: \[ \frac{22.52 + 22.48 + 22.54}{3} = 22.5133\% \]
• In Set 2, the calculation is: \[ \frac{22.64 + 22.58 + 22.62}{3} = 22.6133\% \]

The calculated average provides an estimate of the sample's composition, and you can compare it against a known standard (like the true percentage of 22.52%). The closer your calculated percentage is to this true value, the more accurate your measurements are. Here, Set 1 is more accurate with an average percentage of 22.5133%, as it is closer to the true value.

Average Deviation

While accuracy focuses on how close measurements are to a true value, precision concerns the consistency of the results. Average deviation is a metric that indicates precision by showing how close the measurements are to each other within the same dataset.To calculate it, you consider the absolute deviations of each measurement from their set's average, then compute the average of these deviations. For instance, in Set 1, you get: - Deviations: - \(|22.52 - 22.5133| = 0.0067\) - \(|22.48 - 22.5133| = 0.0333\) - \(|22.54 - 22.5133| = 0.0267\)- Average deviation: \[ \frac{0.0067 + 0.0333 + 0.0267}{3} = 0.0222 \]Set 2 follows a similar process:- Deviations: - \(|22.64 - 22.6133| = 0.0267\) - \(|22.58 - 22.6133| = 0.0333\) - \(|22.62 - 22.6133| = 0.0067\)- Average deviation: \[ \frac{0.0267 + 0.0333 + 0.0067}{3} = 0.0222 \]Surprisingly, both sets have the same average deviation, indicating equal precision in the measurements.

Data Analysis

Analyzing data accurately involves interpreting both accuracy and precision. With measurements, these concepts help understand and compare datasets, leading to better experimental insights. In our example:

• **Set 1** is more accurate because its average is closer to the true value of 22.52%.
• Both **Set 1** and **Set 2** have equal precision as their average deviations are the same.

Data analysis helps in determining factors that might be affecting results, like identifying measurement flaws or inconsistencies in procedure.
To enhance analysis: - Always compare your data averages against known true values for accuracy. - Calculate deviations to understand precision better. - Review experimental steps to find sources of errors if data isn't meeting expectations.
Through effective data analysis, you can achieve reliable results and better understand the trends and variations within your measurements.