Problem 58 - Introductory Chemistry Atoms First

Question

Three people measure the distance from Main Street to Market Avenue using their best estimates. Their data are \(2.5\) miles, \(2.5\) miles, and \(2.6\) miles. Survey charts show the actual distance to be \(1.8\) miles. Characterize the collected data in terms of accuracy and precision.

Step-by-Step Solution

Answer

The collected data is characterized by low accuracy, as the mean of the estimated distances is 0.733 miles away from the actual distance of 1.8 miles. However, the data has high precision, as the differences between the estimated distances are very small (0.1 miles or less).

Step 1: Calculate the mean of the estimated distances.

To calculate the mean, sum up the three given distances and divide the result by the number of measurements. Mean = \(\frac{2.5 + 2.5 + 2.6}{3} = \frac{7.6}{3} = 2.533\) miles

Step 2: Compare the mean to the actual distance to determine the accuracy.

The actual distance is given as 1.8 miles. We can find the absolute difference between the mean and the actual distance: Difference = \(|2.533 - 1.8| = 0.733\) miles This difference indicates that the estimated distances are not very accurate as they are quite far from the actual distance of 1.8 miles.

Step 3: Analyze the variation of estimated distances to determine the precision.

Observe the three evaluated distances: 1. 2.5 miles 2. 2.5 miles 3. 2.6 miles All the evaluated distances are very close to each other. The differences among these measured values are only 0.1 miles or less, which makes them precise.

Step 4: Characterize the collected data.

Based on our analysis, we can characterize the collected data as follows: - The data have low accuracy since the mean of the estimated distances is far from the actual distance (0.733 miles away). - The data have high precision since the differences among the estimated distances are very small (0.1 miles or less).

Key Concepts

Mean Calculation

The concept of mean calculation is central to understanding various datasets. The mean is essentially the average of a series of numbers. It's perhaps the most commonly used metric in data analysis because it gives us an idea of the central value. To calculate the mean, you add up all the numbers in the dataset and then divide by the number of data points.
For example, if you have three measurements: 2.5, 2.5, and 2.6, you would calculate the mean as follows:

• Sum them up: 2.5 + 2.5 + 2.6 = 7.6
• Divide by the number of measurements (3): ewline \[\text{Mean} = \frac{7.6}{3} = 2.533 \text{ miles} \]

This mean value of 2.533 miles represents an average guess for the measurement taken by the group. Remember, while the mean provides a central value, it doesn't account for deviations from this average.

Error Analysis

Error analysis is critical for assessing the accuracy of measurements in datasets. Accuracy refers to how close a measured value is to the actual or true value. In our scenario, the mean of the estimated distances is 2.533 miles, but the actual distance is 1.8 miles. To determine accuracy, we calculate the absolute difference between the measured mean and the actual distance.

• Here, \[\text{Difference} = |2.533 - 1.8| = 0.733 \text{ miles}\]

This value of 0.733 miles indicates a comparatively large error, showing low accuracy. Accuracy can be improved by refining measuring tools or techniques. When analyzing any data, be aware that larger differences imply lesser accuracy. Error analysis helps in recognizing and quantifying these discrepancies, assisting in making informed decisions on how to correct or improve them.

Data Characterization

Characterizing the data involves understanding aspects such as precision and overall consistency of measurements. Precision refers to how close multiple measurements are to each other, regardless of how close they are to the actual value. In our analysis of three measures: 2.5, 2.5, and 2.6, we see a very small difference among them—specifically, a difference of only 0.1 miles.

• This means the measurements are precise.

It's important to note that data can be precise without being accurate. Precision shows consistency, but not correctness. For data characterization, both accuracy and precision are needed to fully understand the quality of dataset. In our exercise, while precision is high (consistent measurements), accuracy is low (measurements are far from the actual value). By performing data characterization, one can inform better strategies for data collection and methodology adjustments.