Question

Analyze Data Students collected mass data for a group of coins. The mass of a single coin is 5.00 g. Determine the accuracy and precision of the measurements $\begin{array}{|ccccc|}\hline \text{Number of coins}5& 10& 20& 30& 50\\ \text{Mass (g)}23.2& 54.5& 105.9& 154.5& 246.2\\ \hline\end{array}$

Short answer

The accuracy of the mass measurements can be determined by calculating the percent errors, which are 7.2%, 9.0%, 5.9%, 3.0%, and 1.52% for 5, 10, 20, 30, and 50 coins, respectively. The precision of the measurements can be found by calculating the standard deviation, which is approximately 37.71 g.

Step-by-step solution

Calculate True Values

Calculate the true mass value for each group of coins by multiplying the number of coins by the mass of a single coin (5.00 g): - true_mass(5) = 5 * 5.00 g = 25.00 g - true_mass(10) = 10 * 5.00 g = 50.00 g - true_mass(20) = 20 * 5.00 g = 100.00 g - true_mass(30) = 30 * 5.00 g = 150.00 g - true_mass(50) = 50 * 5.00 g = 250.00 g

Determine the Accuracy

Calculate the percent error for each measured mass value by dividing the absolute difference between the true value and the measured value by the true value, and then multiply by 100 to get the percentage: - Percent Error(5) = |(23.2 - 25.0)| / 25.0 * 100 = 7.2 % - Percent Error(10) = |(54.5 - 50.0)| / 50.0 * 100 = 9.0 % - Percent Error(20) = |(105.9 - 100.0)| / 100.0 * 100 = 5.9 % - Percent Error(30) = |(154.5 - 150.0)| / 150.0 * 100 = 3.0 % - Percent Error(50) = |(246.2 - 250.0)| / 250.0 * 100 = 1.52 %

Determine the Precision

Calculate the standard deviation for the measured mass values: 1. Calculate the mean of the measured mass values: mean(Measured_mass) = (23.2 + 54.5 + 105.9 + 154.5 + 246.2) / 5 = 116.86 g 2. Calculate the variance of the measured mass values: - variance = (Sum of square differences between each value and the mean) / (Number of values - 1) - variance = ((23.2 - 116.86)^2 + (54.5 - 116.86)^2 + (105.9 - 116.86)^2 + (154.5 - 116.86)^2 + (246.2 - 116.86)^2) / (5 - 1) = 5688.635 / 4 - variance = 1422.15875 3. Calculate the standard deviation: - standard deviation = sqrt(variance) = sqrt(1422.15875) ≈ 37.71 g

Report the Results

Provide the accuracy and precision values for the mass measurements: - Accuracy: - Percent Error(5) = 7.2 % - Percent Error(10) = 9.0 % - Percent Error(20) = 5.9 % - Percent Error(30) = 3.0 % - Percent Error(50) = 1.52 % - Precision: - Standard Deviation = 37.71 g

Key concepts

Percent Error Calculation

Percent error is a measure of how accurate a measured or experimental value is compared to a true or accepted value. It's particularly useful in lab settings, where knowing the accuracy of your results is critical. To calculate percent error, we take the absolute difference between the true value and the measured value, divide it by the true value, and then multiply by 100 to express the error as a percentage.

Using the coin mass measurement example, the percent error gives us insight into how close each mass measurement is to the expected true mass. Lower percent error values indicate that the measurements are more accurate. This is crucial for students to understand since it reveals how much trust we can put into the individual measurements of an experiment or study.

Standard Deviation

The standard deviation is a significant statistical tool that measures the amount of variation or dispersion within a set of data. In the context of our coin example, it helps to assess the precision of the mass measurements. To find the standard deviation, we first work out the mean of the measured values, then we calculate each value's variance from the mean, square these variances, and find their average. The square root of that average gives us the standard deviation.

What does that tell us? A small standard deviation indicates that the measurements are clustered closely around the mean, which means they are precise. Conversely, a larger standard deviation would suggest a wider spread of measurements, which implies less precision. Therefore, teaching students about standard deviation is imperative for them to understand the reliability and consistency of their data.

True Mass Value

True mass value is considered the actual or theoretical exact mass of an object, which experimental results aim to measure as closely as possible. It's the reference point for accuracy assessments in measurements. In a classroom setting, understanding the concept of true mass value is vital.

For instance, knowing that a single coin has a true mass of 5.00 g allows students to calculate expected values for larger groups of coins. It's an excellent way to explain the importance of standard units and the need for precise measurement instruments. Recognizing the true mass value serves as the basis for identifying and correcting systematic errors in experimental measurements.

Variance in Data

Variance measures how much the numbers in a set of data are spread out from their mean. Mathematically, it's the average of the squared differences from the mean. Variance is directly linked to standard deviation. In fact, the standard deviation is the square root of the variance.

Grasping the concept of variance is vital, as it lays the groundwork for understanding the standard deviation and thus the consistency of data. With the measurement of coin masses, a high variance would suggest some measurements were very far off from others, hinting at potential errors or inconsistencies in the measurement process. Simplifying these statistical concepts helps in creating a solid foundation for students to interpret and analyze experimental data effectively.