Problem 69 - Chemistry: The Central Science in SI Units, Global Edition

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

The more accurate data set is (1), with an average percentage of 22.5133, which is closer to the true percentage of 22.52%. Both data sets have an average absolute deviation of 0.0222, so they are equally precise.

Step 1: Calculate the average percentage for each data set

We will find the average percentage of each data set by summing the values and dividing by the number of values (3 in both cases). For data set (1): \(\frac{22.52+22.48+22.54}{3}=22.5133\) For data set (2): \(\frac{22.64+22.58+22.62}{3}=22.6133\)

Step 2: Determine which set is more accurate

To determine which set is more accurate, we need to compare the average percentages to the true percentage (22.52%). The closer the average value to the true value, the more accurate the data set. Data set (1) average percentage: 22.5133 Data set (2) average percentage: 22.6133 Data set (1) is closer to the true percentage of 22.52%, so it is more accurate.

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

We will now calculate the average absolute deviation to determine which set is more precise. First, find the absolute deviation of each value from its data set's average value. Then, find the average of these absolute deviations. For data set (1): - |22.52 - 22.5133| = 0.0067 - |22.48 - 22.5133| = 0.0333 - |22.54 - 22.5133| = 0.0267 The average absolute deviation for data set (1): \(\frac{0.0067+0.0333+0.0267}{3} = 0.0222\) For data set (2): - |22.64 - 22.6133| = 0.0267 - |22.58 - 22.6133| = 0.0333 - |22.62 - 22.6133| = 0.0067 The average absolute deviation for data set (2): \(\frac{0.0267+0.0333+0.0067}{3} = 0.0222\)

Step 4: Determine which set is more precise

To determine which set is more precise, we need to compare the average absolute deviations. The lower the average absolute deviation, the more precise the data set. Data set (1) average absolute deviation: 0.0222 Data set (2) average absolute deviation: 0.0222 Both data sets have the same average absolute deviation, so they are equally precise.

Key Concepts

Understanding Accuracy

In the realm of analytical chemistry, accuracy is a crucial factor. It indicates how close a measured value is to the true or accepted value. Think of it like playing darts. If your dart lands close to the center of the bullseye, you're accurate. When measuring chemical properties, this means your results are close to the actual value you're trying to determine.

In the exercise, the true percentage is given as 22.52%. We found that data set (1) had an average measurement of 22.5133%, which was closer to the true value than data set (2)'s average of 22.6133%. Therefore, data set (1) is more accurate because its average value is nearer to the true percentage.

For better accuracy, scientists ensure their instruments are well-calibrated and they follow proper procedures. Mistakes or errors can throw off the results, so accuracy also often involves improving measuring techniques and reducing human error.

The Importance of Precision

Precision, unlike accuracy, focuses on the consistency or repeatability of measurements. Even if your results aren't close to the true value, they can still be precise if they are tightly grouped together.

Going back to our dart analogy, if all your darts hit the same spot even if they're far from the bullseye, you're very precise! In calculations, precision is examined by looking at the spread of measurements. Narrower spreads symbolize higher precision.

In this exercise, precision was analyzed through the calculation of absolute deviations, which measures how far each individual value in a set is from the average of the set. Interestingly, both data sets (1) and (2) had the same average absolute deviation of 0.0222, indicating equal precision. This means both students reliably measured close to each other, even if the averages differed.

To enhance precision in lab results, repeated trials and averaging multiple measurements can help, as well as ensuring consistent lab environments and reagent purity.

Decoding Absolute Deviation

Absolute deviation helps to understand precision better by indicating variation among a set of measurements. This is computed by first finding the absolute difference each measurement has from the average of its data set. By averaging these differences, we get the average absolute deviation.

In our example, for data set (1), the deviations were calculated as follows: the differences of each individual value from 22.5133% (which is the average of the set) were noted and then averaged to find an absolute deviation of 0.0222. Data set (2) underwent a similar process, resulting in the same deviation.

Interpreting this data, we see that both data sets display the same level of precision. The consistency between their measurements is parallel, even if the actual average values differ slightly.

This measurement is invaluable in assessing measurement quality in labs. It reinforces that while two experimental setups can be equally precise, their accuracy may very well differ based on external factors or initial estimates.