Significant Figures, Propagation of Error, Graphs and Graphing


 Loren Miller
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1 Chapter Two Significant Figures, Propagation of Error, Graphs and Graphing Every measurement has an error associated with it. If you were to put an object on a balance and weight it several times you will not get the exact same answer. Generally, there are three kinds of errors. Each of these will be explained below. Personal Error Sometimes you make a mistake. Students want to report these mistakes as sources of error in an experiment. Refrain from doing so. If you make a mistake, fix it. Your mistakes are not considered a source of error and neither are the concentrations or composition of the reagents you use. While you are not responsible for the quality of the reagents you use, your instructor is, and if an error is found, it can be fixed. Therefore, errors in composition or mistakes you make during an experiment like spilling a reagent or missing an endpoint are never reported as sources of error. Method Error Sometimes when you do an experiment the method you used produces a consistent error in the data. For example, if you mix two solutions together you can produce a solid. One way of retrieving the solid would be to filter the entire solution and collect the solid on the filter paper, but to weigh the solid, you must scrape it off of the filter paper onto a weigh boat. It is obvious that you will never be able to scrape all of the solid off of the filter paper so what you weigh will always be a little bit light. The method used to do this experiment produces a consistently low value for the mass of the solid produced. This is called a method error. Method errors are always unidirectional, that is, they are always too large or too small. As such, it is possible to figure out how much too large or too small the error is and
2 account for them. As a consequence, method errors are not usually considered as sources of error when an experiment is being done. Random Error Random errors arise as random fluctuations in the measurement of your data. As explained earlier, if you weigh an object several times you will get several different answers. The values will fluctuate around some average value. Sometimes the value will be too high and other times it will be low. There is no consistent direction for these kinds of errors. They will be random. If many measurements are made (many = 20 or more measurements) then statistically the average measurement is considered to be the actual value. Usually, we do not run an experiment 20 times or take 20 measurements. Usually, we take three or maybe four measurements. The question then is, how well do we know the value that we have just measured? We begin by taking an average and we expect the average to be very close to the actual value, but random errors produce fluctuations in the data and to be completely accurate we need to tell the reader the size of these fluctuations. But as we will soon find, even the average is not necessarily a good measure of the actual value. Just how accurate is this data and how precise is it? Some of you might be saying, What s the difference? Accuracy and Precision Accuracy and precision are very different from one another. Precision measure how close one measured value is to another. Accuracy measures how close the average value is to the accepted value. You should note that scientists do not use the term true value. True values do not exist in science. Scientists do experiments that confirm values that then become accepted but a true value never exists. Consider the following dartboards. In the first case the darts are very close to one another but they do not hit the bulls eye. If the bull s eye represents the accepted value then the darts are not very accurate, but since the thrower is able to cluster the darts close to one another the thrower is being precise if not accurate. In the second case the darts are both accurate and precise since they are clustered close to one another in the bull s eye. It is possible to be accurate and not precise as shown in the third case. Here we see that the darts are scattered around the dartboard but the average of all these throws is in the bull s eye. So while the thrower was not very precise, the average was very accurate. In the last case we see darts that are neither precise nor accurate. The average of the throws are not in the bull s eye and the darts are not very close to one another. This is the worst possible case for a scientist. Science strives for both accuracy and precision but this can sometimes be difficult to determine. When determining a
3 value that has never been measured we hope that we are being both accurate and precise. Often though, we are only being precise and only time will tell if our data is Precise but not Accurate Accurate and Precise Accurate but not Precise Not Accurate or Precise also accurate. When enough people have done our experiment and confirmed that they have measured the same value then the value that we measured may eventually become the accepted value. Only by comparing our value to an accepted value can we determine if we are being accurate. Significant Figures One way of indicating how well a number is known is by using significant figures. In general, significant figures mean counting the number of digits in a number. For example, look at the number of significant digits in each of the following numbers Number Sig. Figs , Counting the number of digits in any number is relatively easy but zeros are always a problem. There are two cases when zeros present a challenge to science students and both share something in common; the zeros are placeholders. Lets take each case one at a time.
4 Zeros at the End of Numbers without Decimal Points Number Sig. Figs To figure out the number of significant figures in each of the following numbers given above, start on the left hand side and begin counting numbers that are not zeros. The zeros in each of these problems are placeholders. How else do you write 100 than with two zeros? The zeros are placeholders. Zeros at the End of Numbers with Decimal Points Number Sig. Figs If a number has a decimal point after a bunch of numbers then all the numbers, including the zeros, are significant. The decimal point makes all the difference. It tells you that you know that these are zeros and they are not just placeholders. It is the difference between saying that you have about $100 and that you have exactly $ If you say that you have about $100 then nobody would be surprised if you actually had $ But if you said that you had exactly $ but actually had $109.57, then you would be lying. You can see that having about $100 and having $ is really two very different statements. In the first number, $100 the zeros are simply placeholders and you are only approximating $100, but with $ the extra zeros have meaning and you are trying to say that you know, to the penny, how much money you have. Therefore, if you know that you have $ and you wrote down that you had just $100, you would be wrong. You would be conveying the wrong information about the amount of money you had. In science it is important to write down all the numbers that you know, even if those numbers are zero. The zeros give the reader extra information. If you weigh something and find that it weighs grams, you are saying that you know the last digit is zero so it is important to write it down. If you decided to write grams when the
5 number is really grams then you would lose some information about the mass you just weighed. Truth is, grams grams. In the first number, you know the last digit is a zero. In the second number, you only know that the last digit is a 5 and you have no idea what the next number might be. You just lost some information about the mass you just weighed. ALWAYS write down all the number a balance gives you, even if they are zeros. No Numbers Before the Decimal Point Number Sig. Figs In the case where there are no numbers before the decimal point (except a zero) then all the zeros that appear before numbers begin to appear are not significant and can be ignored. So starting on the left, you continue moving right until you hit the first nonzero number and begin counting (in bold in the example above). This gives you the significant figures in each number. Addition and Subtraction Now that we have established how to count significant figures we can now turn our attention on how to use them in our work. In addition and subtraction, the result is rounded off to the last common digit occurring furthest to the right in all components. Another way to state this rule is as follows: in addition and subtraction, the result is rounded off so that it has the same number of decimal places as the measurement having the fewest decimal places (or digits to the right). For example, 100 (assume 3 significant figures) (5 significant figures) = which should be rounded to 124 (3 significant figures). Note, however, that it is possible two numbers have no common digits (significant figures in the same digit column). When combining measurements with different degrees of accuracy and precision, the accuracy of the final answer can be no greater than the least accurate measurement. This principle can be translated into a simple rule for addition and subtraction: When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.
6 (a) g H2O g salt g solution You will notice in (a) that the first number has 4 significant figures and the second has 3 significant figures. The answer has 4 significant figures because the least accurate number is actually the because it is only known to the tenths place while the second number is known to the thousandth. We therefore report our answer to the hundredth place which results in a number with 4 significant figures. (b) 56.0 g H2O g salt g solution In case (b) we have two numbers with 3 significant numbers each. They are both known to the tenths place so our answer must be reported to the tenths place. Doing so produces a number with 4 significant figures. Therefore, it is possible to increase the total number of significant figures when adding numbers g salt + weigh boat g weigh boat g salt = 87.5 g salt In similar fashion to (b) above, when subtracting two numbers it is possible to lose a significant figure. In this case we have two numbers that have 4 significant figures each but when one is subtracted from the other and reported to the tenths place (the least accurate of the two numbers) the result is a number with only 3 significant figures. Multiplication and Division In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures. For example, 3.0 (2 significant figures ) (4 significant figures) = which should be rounded to 38 (2 significant figures). This rule applies to more complicated examples,
7 23.6 x = => x 3.6 Since the number 3.6 has the least number of significant figures in this problem (2 significant figures) the answer must be to 2 significant figures also. So the answer is rounded to 12. Rounding Numbers There are two methods used to round numbers. The simpler, and more common, of the two uses the following rule, a) Round down if the last digit is 0, 1, 2, 3, or 4. b) Round up if the last digit is 5, 6, 7, 8, or 9. Therefore, from the previous example, Significant Figures unrounded original number Significant Figures 4 rounds down Significant Figures 3 rounds down Significant Figures 7 rounds up 12 2 Significant Figures 6 rounds up 10 1 Significant Figure 2 rounds down Banker s Method There is another method for rounding that is called the Banker s Method that is more accurate than the simple method of rounding shown above but it is used less often. This method addresses how we round the number 5. Since the number 5 sits in the middle of our number line, it is argued that rounding it up puts a greater emphasis on the numbers that are 5 or greater and leads to answers that are slightly too large. To compensate, a rule dealing just with then number 5 has been formulated that deals with the error caused by always rounding up. The rule is a simple one, Banker s Rule If the digit to be rounded is 5, make the preceding digit an even number. It is best to give an example of how this is applied. Suppose you have the following two numbers, and Using the Banker s Rule, both numbers would round to In the first case, 3.215, the digit previous to the 5 is odd (it is 1) so the 1 gets rounded up
8 to 2, making it an even number. In the next number, 3.225, the digit previous to the 5 is even (it is 2), so the number is not changed and is just left as a 2. The result is that actually gets rounded down to So we see that using the Banker s method, both numbers, and 3.225, get rounded to Statistically, using this method, half of the numbers would be rounded up and the other half would be rounded down so the error that arises from always rounding up at 5 has been fixed. While this method generally leads to better answers, it is not commonly used. Most people simply round up at the number 5. Your instructor will tell you which method he or she prefers. Exact Numbers In the lab and elsewhere you may be asked to find the average of a set of numbers. The process is simple, and up the numbers and then divide by how many numbers have been added. For example, = = Average But the question comes up, how many significant figures should we report? The tendency of a student is to look for the least number of significant figures in the problem and report the answer to that number of significant figures. In this problem, the number with the least number of significant figures is the division by 3 when taking the average. But rounding your answer to just 1 significant figure would be a great mistake. The number 3 does not represent just one significant figure. It is an exact number. That is, it is actually for as many digits as you need. Numbers like these are exact. In this case you have exactly 3 numbers that you are averaging. There is no such thing as 3.1 numbers that you are averaging. You have either 3 numbers or 4 numbers but not 3.1. So the number 3 in this example is exactly 3 to as many significant figures as you need. So your answer is limited by the (4 significant figures) and not by the exact number, 3, so the answer should be rounded to 4 significant figures (25.07). We use exact numbers all the time. A dozen eggs, 5 people, and a sixpack of beer all represent exact numbers. Exact numbers never limit the number of significant figures you report.
9 A Practical Example Using Significant Figures We began this section by talking about the errors that arise when measurements are made and how to report numerical values based on these measurements. In general, we report answers to problems based on the least number of significant digits in our problem. As a simple example of how this might work in a problem involving some kind of chemistry, why don t we see how we might report the density of water using some common glassware found in a lab. Buret Graduated Cylinder Erlenmeyer Flask ml 50.0 ml 50 ml Suppose that we want to measure the density of water using each of these pieces of glassware so we fill each of them to their 50 ml mark and then pour that water out into another container and weigh it. These data are shown in the following table, Density of Water using Various Glassware Glassware Sig. Figs Volume Mass Mass/Volume Reported Density Buret ml g g/ml g/ml Grad Cylinder ml g g/ml g/ml Erlenmeyer Flask 1 50 ml g g/ml 1 g/ml Each of these pieces of glassware can be read to varying degrees of accuracy. The buret is the most accurate measure of volume so we write ml to indicate that it can be
10 read to 4 significant figures and 2 digits past the decimal point. The marks found on an Erlenmeyer flask are only approximate volume measurements. They can be off by as much 20 ml or more so we must indicate this by writing our volume as 50 ml which is only 1 significant figure. As a consequence, using an Erlenmeyer flask to measure an accurate volume is not advised. Burets and pipets are the best way to measure volume and so are graduated cylinders. Other glassware is far less accurate and should not be used unless the exact volume is not important. You will notice that our mass readings are all 5 significant figures because a balance is capable of giving you this many digits in its reading. Always write down all the digits given to you by a balance. Don t ever round and never leave off zeros if they are the last digit (53.3 grams is NOT the same as grams). The zeros are significant and they convey information about how well you know your number. Always write down every digit given to you by a balance. When we calculate the density of water we divide the mass of the water by its volume. Your calculator will give a bunch of numbers but not all of these numbers are significant so they must be rounded to the proper number of significant figures. In each of the three examples, the least number of significant digits is found in the measurement of the volume of water being weighed. For the buret, this means that we must write our density to 4 significant figures so g/ml gets rounded to g/ml, and this is the density we would report. Similarly a graduated cylinder can be read to just 3 significant figures so, even though the mass has been measured to 5 significant figures, we can only report 3 digits in our final answer. In this case, for the graduated cylinder, we would round to 3 significant figures since that is the best we can read the volume. In this case, using a graduated cylinder, our reported density would be rounded to g/ml. Finally, since an Erlenmeyer flask can only be read to 1 significant figure, our reported density would be just 1 g/ml to reflect this fact. Significant digits give us a quick way of determining the number of digits we should write down when doing a calculation, and while the answers we get are always close to the right answer, sometimes they are not. More importantly, significant figures have an important theoretical basis that cannot be seen using the simple methods employed here. Therefore we must reexamine significant figures in light of the actual error found in each measurement and using statistical methods, determine the number of significant figures to report in an answer. These statistical methods only apply to random errors. Let us investigate random errors further.
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