Accuracy and Precision
Accuracy and Precision
Sometimes in science we use words that you think you know, but we have more specific meanings for them. For example, most people think that "velocity" and "speed" mean the same thing. If you had physics with us, however, you should know that they are not (velocity includes not only speed, but direction as well). Accuracy and precision is another case like that. Most people think of them as meaning the same thing, but they really are quite different. Accuracy is how close a measurement comes to the correct result. This assumes of course, that you know the correct result. Usually, you don't; so it is not always possible to determine accuracy. Here is a link to the Wikipedia entry on the age of the Earth. I would like you to read the opening 2 sections (through “early calculations”).
Precision is how close all of your measurements are to each other. You can always determine the precision of measurements by repeating them several times. Increased precision corresponds to an increased number of significant figures. How would you determine the ACCURACY of a balance? To do this, you would have to have something that is a known standard mass (these can be purchased). All this is an object that has been made to very high precision to be a mass we have agreed upon as “standard.” You would determine it's mass with your balance and see if the result you get is the same as the accepted mass. You might want to take ten measurements of the standard mass and record each. Then you could take the average measurement and compare it to the accepted amount. You might find that the balance was consistently off in one direction. Suppose a 20.00g standard mass always comes out between 19.20 and 19.45. You could say that the instrument is consistently off by between -.80 to -.55g. So it is consistently under mass on average by -.0675. However, there is an interesting significant figure question about how far I should take this. Should I report -0.675 or round it to -0.68? In practice, you would calibrate the balance to read the correct mass. Alternatively, you could simply add the .068 to every measurement. This is similar to what you would do if you knew your clock always ran 5 minutes fast...you would just take off 5 minutes to get the correct time (which, again can only be stated because we have agreed upon standards). How would you determine the PRECISION of a balance? Let's look at those ten measurements again. The range of the answers is between 19.20 and 19.45g. The range is .25g, or +/-0.125g. In practice, we would say that this balance is precise to the tenths of a gram. This would not be an especially precise balance. ours should give us variation of less than 0.1g. Usually, we expect a well maintained instrument to show variation only in the last digit to which it gives information. So, if the balance shows results to the hundredths place, we expect the precision to be in that range. This is analogous to a ruler marked to the nearest millimeter. We expect that we can consistently determine length to the nearest mm, and then estimate the next digit. The last digit of any measurement is soft. That is, it is always an estimate. You will notice this when you use the balance. As a result, any calculations based on this will also be limited by the precision of the actual measurement.
Discussion questions: Which would you rather have: a balance that is imprecise, or one that is precise, but consistently off by a known amount? Often, we see political polls reported +/- X%, let's say 2%. Is that precision or accuracy? Hint: does anyone know what the correct answer is? Significant figures and precision It may be obvious to you from the discussion above that the significant figures with which I report a measurement and its units essentially tell you the precision. If my balance reports a number in the 0.001g place (milligram), I can say that it is precise to number of milligrams.
Remember, significant figures are not some secret code that you have to learn in order to get the full credit on a question. Significant figures are the way that we make sure we convey accurately what we know, and don't imply that we know things we don't. It’s how we convey how precise our measurements are. When writing a number, you write every digit that you measured. When you perform mathematical operations, you make sure that your answer reflects what you actually measured in the first place. In the same way that a chain is only as strong as its weakest link, the final answer you get can only be as precise as the "worst" measurements you made. Whenever you write a nonzero integer, it is assumed that it represents a number that you actually measured. For example, suppose you use a ruler to measure how long your big toe is. Let's say that you measure 3.15 cm. You derive this number because clearly you can measure 3 cm. The .1 cm is the same as 1 mm, which you can read easily on the ruler. Furthermore you can see that the end of your toe is between the .1 and .2 cm mark. Take a stab at it, and arrive at 3.15 cm. All of those digits are significant, because you measure them (though, as in most cases, the last number is an estimate, or will result from rounding). Now suppose another person does the same thing and finds that their toe measures out at 3.05 cm. Their toe is shorter than your toe by 1 mm. You actually measured that difference. Therefore the zero trapped in between the number three and the number five is also significant. The way to think of it is like this: did I measure closely enough to know the difference between 3.05 and 3.15? If the answer is yes, as it is in this case, the new zero really has meaning and is therefore significant. Trapped or “captive” zeroes are always significant. What about leading zeros? Leading zeros do not tell you anything about the value that you measured; they just tell you the scale you are working in. Suppose you decided to report the length of your toe in meters. In order to do this you would need to divide by hundred because there are 100 cm in 1 m. Therefore you would report the number as 0.0315 cm. You certainly have more digits now, but are they significant? Once again, remember that a number is significant if you actually measured it. You measured the number to three significant figures. When you divide the number by 100, did you learn anything more about the length of your toe? Did your number of suddenly have more information in its simply by dividing it by 100? Obviously, it didn't. All those leading zeros do is tell you what scale you are in. They are place holders. Converting from centimeters to meters cannot give you more information. Leading zeros are never significant. The final example is the one that causes the most consternation among students: trailing zeros. Often, I have heard students say that trailing zeros are significant because there is a decimal point in the number. That's not correct; or at least, that’s not the reason that the trailing zeroes are significant. Trailing zeros are significant if you measured them. The problem is, I need a way to tell you whether or not I measured them. If I write the number 100.0, I don't need that last zero to preserve the value of the number. Since I am an honest person, you can safely assume that I would only write that zero in the tenths place if I had measured to the tenths place. If I write the number 100, you don't know if I measured that number to the hundreds place, to the tens place, or all the way to the units place. Let’s say I only measured to the tenths. I cannot simply remove the 0 in the units place; that would change the value. I have to leave that last 0 there even though I didn’t measure it. That really chafes at me, since I’m an honest man. So if I don't give you any more information, you have to assume that the number written as 100 was measured only to the 100s place. That is I'm only sure that it's not 200, but I'm not sure that it's not 110, or even 90. The question then is what to do when I need the trailing zeros to hold the place (as I do for the leading zeros in the case of the number 0.016), but didn't measure them, or measured only some of them? The simplest thing to do is to use scientific notation. If I measured 100 to the tens place (two significant figures), I write 1.0x102. If I measured to the units place (three sig figs), I write 1.00x102. So, every digit in the first part, called the “coefficient,” is always significant. The exponent just tells you how many other places you need to hold. For example, in the number 6.02x1023, all three digits in the coefficient are significant. The 23 has no bearing on the the significant figures, but tells you would need to write 21 trailing zeroes after the 602. In the number 6.626x10-34, there are 4 sig. figs. The -36 just tells you that you would need 35 leading zeroes. In addition to being much easier to write than 602000000000000000000000, 6.02x1023 makes easily apparent the number of significant figures.
Exact numbers: these are definitions, like 12 in a dozen, or 1000 mm in a meter. These numbers have an infinite number of sig. figs.
Mathematical operations and Sig Figs. When adding and subtracting, you line the numbers up so that you are adding the right columns (10s place to 10s place etc.). This is one of the few things that scientific notation makes more difficult. Suppose you add 1.3 cm to 0.24 cm. Line it up. What happened to the 4 in the bottom number? Well, what am I adding it to? The top number is not 1.30; I didn’t measure it that well. It could be 1.32, or 1.37...I don’t know. So, I don’t know what to put in the hundredths place. Since I don’t know what the value is, and I’m an honest man, I leave that place blank. The story for multiplication is harder to explain but easier to execute. You simply have to provide the answer to the number of significant digits of the least precise number. So, 1.38 (3 sig. fig) times 0.10 (2 sig. figs) would yield the result 0.14 (2 significant figures).
Order of operation: This is important: whenever you complete an addition or subtraction step, adjust the significant figures before you move on. You may do runs of multiplication or division and just report the answer to as many sig. figs you have in your “worst” number. So The answer in the numerator is 11.1 (3 sig. fig), so the final answer will be in 3 sig fig. If the denominator were 5.4, the answer would be 2.0, only two sig figs.
We will talk more about significant figures in person. See you soon.
Add Questions and Comments here:
Sometimes in science we use words that you think you know, but we have more specific meanings for them. For example, most people think that "velocity" and "speed" mean the same thing. If you had physics with us, however, you should know that they are not (velocity includes not only speed, but direction as well). Accuracy and precision is another case like that. Most people think of them as meaning the same thing, but they really are quite different. Accuracy is how close a measurement comes to the correct result. This assumes of course, that you know the correct result. Usually, you don't; so it is not always possible to determine accuracy. Here is a link to the Wikipedia entry on the age of the Earth. I would like you to read the opening 2 sections (through “early calculations”).
Precision is how close all of your measurements are to each other. You can always determine the precision of measurements by repeating them several times. Increased precision corresponds to an increased number of significant figures. How would you determine the ACCURACY of a balance? To do this, you would have to have something that is a known standard mass (these can be purchased). All this is an object that has been made to very high precision to be a mass we have agreed upon as “standard.” You would determine it's mass with your balance and see if the result you get is the same as the accepted mass. You might want to take ten measurements of the standard mass and record each. Then you could take the average measurement and compare it to the accepted amount. You might find that the balance was consistently off in one direction. Suppose a 20.00g standard mass always comes out between 19.20 and 19.45. You could say that the instrument is consistently off by between -.80 to -.55g. So it is consistently under mass on average by -.0675. However, there is an interesting significant figure question about how far I should take this. Should I report -0.675 or round it to -0.68? In practice, you would calibrate the balance to read the correct mass. Alternatively, you could simply add the .068 to every measurement. This is similar to what you would do if you knew your clock always ran 5 minutes fast...you would just take off 5 minutes to get the correct time (which, again can only be stated because we have agreed upon standards). How would you determine the PRECISION of a balance? Let's look at those ten measurements again. The range of the answers is between 19.20 and 19.45g. The range is .25g, or +/-0.125g. In practice, we would say that this balance is precise to the tenths of a gram. This would not be an especially precise balance. ours should give us variation of less than 0.1g. Usually, we expect a well maintained instrument to show variation only in the last digit to which it gives information. So, if the balance shows results to the hundredths place, we expect the precision to be in that range. This is analogous to a ruler marked to the nearest millimeter. We expect that we can consistently determine length to the nearest mm, and then estimate the next digit. The last digit of any measurement is soft. That is, it is always an estimate. You will notice this when you use the balance. As a result, any calculations based on this will also be limited by the precision of the actual measurement.
Discussion questions: Which would you rather have: a balance that is imprecise, or one that is precise, but consistently off by a known amount? Often, we see political polls reported +/- X%, let's say 2%. Is that precision or accuracy? Hint: does anyone know what the correct answer is? Significant figures and precision It may be obvious to you from the discussion above that the significant figures with which I report a measurement and its units essentially tell you the precision. If my balance reports a number in the 0.001g place (milligram), I can say that it is precise to number of milligrams.
Significant Figures
Remember, significant figures are not some secret code that you have to learn in order to get the full credit on a question. Significant figures are the way that we make sure we convey accurately what we know, and don't imply that we know things we don't. It’s how we convey how precise our measurements are. When writing a number, you write every digit that you measured. When you perform mathematical operations, you make sure that your answer reflects what you actually measured in the first place. In the same way that a chain is only as strong as its weakest link, the final answer you get can only be as precise as the "worst" measurements you made. Whenever you write a nonzero integer, it is assumed that it represents a number that you actually measured. For example, suppose you use a ruler to measure how long your big toe is. Let's say that you measure 3.15 cm. You derive this number because clearly you can measure 3 cm. The .1 cm is the same as 1 mm, which you can read easily on the ruler. Furthermore you can see that the end of your toe is between the .1 and .2 cm mark. Take a stab at it, and arrive at 3.15 cm. All of those digits are significant, because you measure them (though, as in most cases, the last number is an estimate, or will result from rounding). Now suppose another person does the same thing and finds that their toe measures out at 3.05 cm. Their toe is shorter than your toe by 1 mm. You actually measured that difference. Therefore the zero trapped in between the number three and the number five is also significant. The way to think of it is like this: did I measure closely enough to know the difference between 3.05 and 3.15? If the answer is yes, as it is in this case, the new zero really has meaning and is therefore significant. Trapped or “captive” zeroes are always significant. What about leading zeros? Leading zeros do not tell you anything about the value that you measured; they just tell you the scale you are working in. Suppose you decided to report the length of your toe in meters. In order to do this you would need to divide by hundred because there are 100 cm in 1 m. Therefore you would report the number as 0.0315 cm. You certainly have more digits now, but are they significant? Once again, remember that a number is significant if you actually measured it. You measured the number to three significant figures. When you divide the number by 100, did you learn anything more about the length of your toe? Did your number of suddenly have more information in its simply by dividing it by 100? Obviously, it didn't. All those leading zeros do is tell you what scale you are in. They are place holders. Converting from centimeters to meters cannot give you more information. Leading zeros are never significant. The final example is the one that causes the most consternation among students: trailing zeros. Often, I have heard students say that trailing zeros are significant because there is a decimal point in the number. That's not correct; or at least, that’s not the reason that the trailing zeroes are significant. Trailing zeros are significant if you measured them. The problem is, I need a way to tell you whether or not I measured them. If I write the number 100.0, I don't need that last zero to preserve the value of the number. Since I am an honest person, you can safely assume that I would only write that zero in the tenths place if I had measured to the tenths place. If I write the number 100, you don't know if I measured that number to the hundreds place, to the tens place, or all the way to the units place. Let’s say I only measured to the tenths. I cannot simply remove the 0 in the units place; that would change the value. I have to leave that last 0 there even though I didn’t measure it. That really chafes at me, since I’m an honest man. So if I don't give you any more information, you have to assume that the number written as 100 was measured only to the 100s place. That is I'm only sure that it's not 200, but I'm not sure that it's not 110, or even 90. The question then is what to do when I need the trailing zeros to hold the place (as I do for the leading zeros in the case of the number 0.016), but didn't measure them, or measured only some of them? The simplest thing to do is to use scientific notation. If I measured 100 to the tens place (two significant figures), I write 1.0x102. If I measured to the units place (three sig figs), I write 1.00x102. So, every digit in the first part, called the “coefficient,” is always significant. The exponent just tells you how many other places you need to hold. For example, in the number 6.02x1023, all three digits in the coefficient are significant. The 23 has no bearing on the the significant figures, but tells you would need to write 21 trailing zeroes after the 602. In the number 6.626x10-34, there are 4 sig. figs. The -36 just tells you that you would need 35 leading zeroes. In addition to being much easier to write than 602000000000000000000000, 6.02x1023 makes easily apparent the number of significant figures.
Exact numbers: these are definitions, like 12 in a dozen, or 1000 mm in a meter. These numbers have an infinite number of sig. figs.
Mathematical operations and Sig Figs. When adding and subtracting, you line the numbers up so that you are adding the right columns (10s place to 10s place etc.). This is one of the few things that scientific notation makes more difficult. Suppose you add 1.3 cm to 0.24 cm. Line it up. What happened to the 4 in the bottom number? Well, what am I adding it to? The top number is not 1.30; I didn’t measure it that well. It could be 1.32, or 1.37...I don’t know. So, I don’t know what to put in the hundredths place. Since I don’t know what the value is, and I’m an honest man, I leave that place blank. The story for multiplication is harder to explain but easier to execute. You simply have to provide the answer to the number of significant digits of the least precise number. So, 1.38 (3 sig. fig) times 0.10 (2 sig. figs) would yield the result 0.14 (2 significant figures).
Order of operation: This is important: whenever you complete an addition or subtraction step, adjust the significant figures before you move on. You may do runs of multiplication or division and just report the answer to as many sig. figs you have in your “worst” number. So The answer in the numerator is 11.1 (3 sig. fig), so the final answer will be in 3 sig fig. If the denominator were 5.4, the answer would be 2.0, only two sig figs.
We will talk more about significant figures in person. See you soon.
Add Questions and Comments here:
