Question

the accuracy of the measurement refers to how close the measured value is to the true or accepted value. for example, if you used a balance to find the mass of a known standard 100.00 g mass, and you got a reading of 75.95 g, your measurement would not be very accurate.
precision refers to how close together a group of measurement actually is to each other. precision has nothing to do with the true or accepted value of a measurement, so it is quite possible to be very precise and totally inaccurate. in many cases, when precision is high and accuracy is low, the fault can lie with the instrument. if a balance or a thermometer is not working correctly, they might consistently give inaccurate answers, resulting in high precision and low accuracy.
a dart - board analogy is often used to understand the difference between accuracy and precision. imagine a person throwing darts, trying to hit the bullseye. the closer the dart hits to the bullseye, the more accurate the tosses are. if the person misses the dartboard with every throw, but all of their shots land close together, they can still be very precise but not accurate.
heres another example: the tv weather forecaster says that it will be between 40 and 50 degrees today. the actual reading turns out to be 43. thus, the forecast was accurate, but not very precise. for tomorrow, the forecast is 52.47 degrees at 4 pm. it turns out to be 39.14 degrees. this forecast was very precise, but completely inaccurate.
problems:

1. below is a data table produced by three groups of students who were measuring the mass of a paper clip which had a known mass of 1.0003 g. the last row is the average of their measurements.

group 1	group 2	group 3	group 4
1.03 g	2.754158	10.13258 g	0.23 g
0.99 g	2.186357 g	10.13255 g	0.75 g
1.01 g	2.601267 g	10.13255 g	1.01 g

1a. which group(s) are the most accurate?
b. which group(s) are the most precise?
c. which group is the most accurate and precise?
2a. suppose a gps (global positioning system) which measures your position on earth, is not calibrated correctly. you take 3 readings at the same place and they are all close together but 14 miles from your actual position. explain your results in terms of precision and accuracy.
b. suppose you try out 3 gps units from different manufacturers. although they are supposed to read within 10 feet of your actual (known) position, unit 1 reads 110 feet north of your true position, unit 2 reads 10 feet west and unit 3 read 95 feet south. how could you take an average of the 3 measurements? (there could be more than one way to do this; pick what you think is the best one.) in terms of precision and accuracy explain your result.

Explanation:

Step1: Calculate accuracy for paper - clip mass

Accuracy is closeness to true value. For each group, find the average of measurements and compare to the true mass of 1.0003 g.
Group 1 average: $\frac{1.01 + 1.03+0.99 + 1.01}{4}=\frac{4.04}{4}=1.01$ g
Group 2 average: $\frac{2.863287+2.754158 + 2.186357+2.601267}{4}=\frac{10.405069}{4}=2.60126725$ g
Group 3 average: $\frac{10.13251+10.13258+10.13255+10.13255}{4}=\frac{40.53019}{4}=10.1325475$ g
Group 4 average: $\frac{2.05 + 0.23+0.75+1.01}{4}=\frac{4.04}{4}=1.01$ g
Group 1 and Group 4 have averages closest to 1.0003 g, so they are the most accurate.

Step2: Calculate precision for paper - clip mass

Precision is closeness of measurements to each other. Group 3 has measurements that are all very close to each other (10.13251 g, 10.13258 g, 10.13255 g, 10.13255 g), so Group 3 is the most precise.

Step3: Determine most accurate and precise for paper - clip mass

No group is both most accurate and most precise simultaneously.

Step4: Analyze GPS case 2a

The GPS readings are close together (precision is high) but 14 miles from the actual position (accuracy is low) because the instrument is not calibrated correctly.

Step5: Analyze GPS case 2b

Taking an average of the 3 GPS measurements is not straightforward as they are in different directions. In terms of accuracy, unit 2 is the best as it is only 10 feet away from the true position. In terms of precision, since the measurements are in different directions and magnitudes, it's hard to define precision in a simple way. But if we consider the spread of the values, unit 2 has the least deviation from the true position among the three.

Answer:

1a. Group 1 and Group 4
1b. Group 3
1c. None
2a. High precision, low accuracy because readings are close together but far from actual position due to incorrect calibration.
2b. Unit 2 is the best in terms of accuracy as it is closest to the true position. Precision is hard to define simply as measurements are in different directions, but unit 2 has least deviation.