The system of significant figures plays an important role in the uncertainty of measurements. Significant figures in a measurement comprise of all known digits, plus a last digit that is estimated. Essentially, the more significant figures in a value, the less uncertainty there is. It is vital that all calculations be recorded with the correct number of significant figures, because the answers often depend on the number of significant figures in the values used in the calculations. For this reason, there are several rules everyone must follow when utilizing significant figures:
First, all nonzero integers are always considered as significant figures. Leading zeroes are never significant. However, zeroes appearing between nonzero digits or sandwich zeroes are always significant. Trailing zeroes do not count as significant figures if they are there to serve solely as place holders. In cases when definitions or counting numbers are used there will be an unlimited number of significant figures.
Rounding significant figures have their own rules as well. If the number to be rounded is less than half, the number stays the same. For example 13.22 rounded to three significant numbers would be 13.2. If the number to be rounded is more than half, the rounded number will increase by one. So, 13.46 rounded to three significant figures will become 13.5. There are special cases where the number you round to has exactly half to the right of it such has 13.2500, then at this point if the round to number is even, everything after the two is dropped. If it was an odd number, the number would be increased by one. Rounding is only done after all calculations are completed.
When adding and subtracting significant figures, the answer should be rounded to the same number of decimal places as the measurement with the least number of decimal places. In multiplication and division however, you would round to the number of significant figures as the measurement with the least number of significant figures in the operation.
In all three objects that were measured; the black box, the water in the graduated cylinder and the area of the table, an increase in significant figures were noticeable. For example, the black box was measured with Ruler A, which was calibrated by only one meter; the dimensions of the box were .3 x .3 x .3 meters. Using the rules of uncertainty, and employing the system of scientific notation it becomes 3 x 10 -3 meters cubed. Since the directions want the volume in liters; dimensional analysis can be used to convert the answer into 30 L. The answer has only one significant figure. Using Ruler D, after all calculations the answer comes out to be 22.78 liters. The numbers of significant figures increase to four, which also increases precision. It is accurate in that it is far closer to the correct volume of 22.4 liters. The rest of the calculations can be done in the same manner, using the basic rule of significant figures and conversions, and similar results will be produced. (“See attached data” for in depth calculations).
From this, an observation can be made. The more calibrated the instrument the more precise and accurate results can be found, assuming that no human error is involved. This is why when using the various rulers; precision and accuracy go up as the instruments used goes from the least calibrated (Ruler A) to the most calibrated (Ruler D).
After pooling all the results, it was concluded that the measurements from the table were precise because they were grouped close together. The correct volume of the box was 22.4 liters. Using the other groups to compare the volume of the black boxes, the whole class was in a good range of being accurate, only being ± .38 from the real value of 22.4 liters. Also, since the class was ± .38 from the true value, it meant that the numbers were fairly close together implying results were precise.
During the experiment, the same measuring devices were used to determine the calculations of the various objects. Yet, there were still some differences between results. This was mostly due to defects in some of the items that were being measured. For example, the table had curvatures on the corners of the tables, and those who measured the length and the width from the very edge of the table could have added or lost a few units in their calculations. To ensure that everyone gets the closest results possible, the instructions could have said to measure a few inches inside the area to avoid people from measuring right on the edge where the curve is present.
When performing measurements on the black box, the same problem arose. The boxes were not new, and because of that they had started to chip off on the corners. The rough corners gave off different edges, which resulted in a variety of different numbers amongst the groups. The calculations can still be considered precise and accurate, but since the instructions gave no specific location to measure the dimensions the answers were slightly off, because some people used the edge with the dull corners and some avoided it by measuring a few inches away from the edge. It seems if the instructions could have been specified to say where to measure; such as away from the edge, the answers would have been more precise and accurate.
Conclusion
Through this experiment by using various calibrated measuring devices, it can be concluded that the more highly calibrated the instrument is, the more certain, accurate and precise the results will be, because of the increase in significant figures. Although the class’s results were fairly precise and accurate in their calculations, there was room for error, because of the condition of the objects, such as denting and dull corners.
Post Laboratory Questions
1. From looking at the four calculated areas of the table, the closely related results proved that the set was precise. But, the accuracy of these results can not be determined, because the true value of the area of the table is unknown.
2. As the instrument got more calibrated, the answers started getting more and more precise. It went from 30 liters to 22 to 22.7 and finally 22.78 liters. The final three answers are very much precise, but the first one is off due to the fact that it was only allowed to round to one significant figure. Even so, the set of numbers can be named precise, because as a whole they are pretty much close together. Although the measurement was off by .38 liters, this number is not that significant and 22.78 liters can still be counted as an accurate record of the box’s volume.
3. Overall, everybody in the class had fairly precise and accurate results. All of the answers were 22 and then somewhat liters, suggesting that the results were precise. Without knowing the true value, accuracy would be unknown. When comparing the results to the true value of 22. 4 liters, the results ranged from 22.18 to 22.78 liters which was ± .38 from 22.4 liters. The fact that all the results had less than a .5 difference amongst each other showed the answers were impressive and proved to be accurate and precise against the claimed volume of the box.
http://wordnet.princeton.edu/perl/webwn?s=measure
www.sec.noaa.gov/forecast_verification/Glossary.html
