Question

Shooters \(\mathrm{A}\) and \(\mathrm{B}\) each shoot three arrows at their target, aiming for a bull's-eye. Which shooter is more precise? Which shooter is more accurate? Explain.

Short answer

Shooter A is more precise due to smaller distances between arrows (more tightly grouped), indicating higher consistency. Shooter B is more accurate, as they have a smaller average distance from the bull's-eye, meaning they are closer to the target.

Step-by-step solution

Define precision and accuracy

Precision: Essentially, the grouping of the arrows. The closer the arrows are to each other, the higher the precision. Accuracy: The closeness of the arrows to the bull's-eye. If the arrows are closer to the bull's-eye, the shooter is considered more accurate.

Evaluate Shooter A's performance

Shooter A's performance would be given in terms of the distance between his/her three arrows and their average distance from the bull's-eye. For example, if shooter A's arrows' distances from the bull's-eye were as follows: Arrow 1: 5 cm Arrow 2: 6 cm Arrow 3: 5.5 cm Then, the average distance from the bull's-eye would be: \(\frac{5 + 6 + 5.5}{3} = 5.5 \textrm{ cm}\). To determine the precision, we look at the distance between the arrows. In this example, their distances are between 0.5 cm and 1 cm apart, which would indicate some level of precision.

Evaluate Shooter B's performance

Similar to step 2, Shooter B's performance would be given in terms of the distance between his/her three arrows and their average distance from the bull's-eye. For example, if shooter B's arrows' distances from the bull's-eye were as follows: Arrow 1: 3 cm Arrow 2: 7 cm Arrow 3: 4 cm Then, the average distance from the bull's-eye would be: \(\frac{3 + 7 + 4}{3} = 4.67 \textrm{ cm}\). To determine the precision, we look at the distance between the arrows. In this example, their distances are between 1 cm and 4 cm apart, which would indicate lower precision compared to Shooter A.

Compare and draw conclusions

Compare the average distances and the distances between arrows for each shooter. In our example: Shooter A: Average distance from bull's-eye = 5.5 cm; Distance between arrows = 0.5 cm - 1 cm Shooter B: Average distance from bull's-eye = 4.67 cm; Distance between arrows = 1 cm - 4 cm From these results, we can conclude that Shooter B is more accurate, as he/she has a smaller average distance from the bull's-eye. However, Shooter A is more precise, as the distances between his/her arrows are smaller (more tightly grouped) compared to Shooter B.

Key concepts

Precision in Measurement

When we talk about precision in measurement, we're focusing on how closely repeated measurements agree with each other, regardless of how close they are to the actual or desired value. In the context of the shooters, precision refers to how consistently they hit their target with each arrow. It's like measuring the gap between where each arrow lands.

If all arrows are tightly grouped together, even if they are not around the bull's-eye, the shooter is exhibiting high precision. It's all about the consistency of shots.

In the example, Shooter A's arrows are only 0.5 cm to 1 cm apart. This shows that Shooter A is quite precise, as the shots are close to each other. Precision doesn't mean accuracy, but it tells us that the shooter's technique is consistent.

Accuracy in Measurement

Accuracy in measurement is about how close a measurement is to a standard or accepted value. For the shooters, this means how close their arrows get to the bull's-eye.

An accurate shooter consistently hits near the bull's-eye, even if the arrows aren't close to each other. It's a reflection of how well the shooter is hitting the target as intended.

In the example, Shooter B's arrows averaged a distance of 4.67 cm from the bull's-eye. This is a smaller average distance compared to Shooter A's 5.5 cm. Therefore, despite having less precision, Shooter B is more accurate, as the arrows are closer to the central target.

Data Analysis and Comparison

Data analysis allows us to evaluate performance by looking at trends and discrepancies. In our example, we compared the average distances and consistencies between the two shooters' arrow placements.

This comparison shows that although Shooter A is more precise, hitting consistently at a certain area of the target, Shooter B is more accurate with a better hit proximity to the bull's-eye.

By analyzing the data—both the average distance from the bull's-eye and the variance in distances—we can gain a comprehensive understanding of who performed better in terms of precision and accuracy. Such analyses help identify areas for improvement, whether ensuring consistency or better targeting the center.