Measurements - Errors and Uncertainty Oakton Community College (revised 11/11/22 J) Introduction Research in science requires making and reporting measurements in such a way as to make known to the reader the accuracy and precision of the measurements. The following discussion does not attempt to show you how to make measurements with various devices, but rather how to report those measurements. To begin, we need to discuss the di ff erence between precision and accuracy With every measurement there is assumed to be a true or accepted value of the quantity being measured. The true value might be the result of highly precise and accurate experiments that have already been done, or the true value might be defined to have a certain value. For example, the true value for the speed of light in a vacuum is 299,792,458 m/s. This is exactly what the speed of light is in a vacuum because of how scientists have defined the meter and the second. It’s quite possible that if a quantity is not a defined quantity, the true value can never be known to 100% accuracy. The accuracy of a measurement refers to how close the measurement, or the average (or mean) of several measurements, is to the true or accepted value. Precision refers to how close together several measurements of the same quantity are regardless of how close they are to the true value. For example, suppose we are trying to measure a distance that has a true value of 3.255 m. Using two di ff erent techniques, or measuring devices, we obtain the following five measurements for each technique: Technique A Technique B 3.135 m 3.262 m 3.137 m 3.242 m 3.138 m 3.257 m 3.134 m 3.258 m 3.136 m 3.251 m Mean = 3.136 m Mean = 3.254 m Technique A is clearly more precise because the five measurements are identical down to the hundredths place. Precision therefore has something to do with the number of meaningful digits, or significant figures , in the measurement. Technique B is clearly more accurate since each measurement agrees with the true value to the tenths place, and since the mean agrees even better with the true value. The best measurement is one that is both precise and accurate. Compared to one another, techniques A and B are either precise or accurate, but not both. Thus, each technique has its problems. The device used in technique A was a fairly precise device able to measure definitely to the nearest hundredth of a meter (to the nearest centimeter) with uncertainty in the thousandths place. The thousands place, or the millimeter place, is estimated by the person doing the measuring. The smallest reading of a measuring device from which someone can determine a definite number, and not an estimate, is called the device's least count So device in technique A has a 0.01 m, or a 1 cm, least count. When recording a measurement from a device it is customary to write down numbers for decimal places down to the least count, and then one more estimated decimal place. These numbers are then referred to as significant figures because they all are precisely known except for the last digit which is a good estimate. The measurements in technique A have four significant figures. A problem with technique A is that all the measurements are too small, by almost equal amounts, compared to the true value. Thus, there was some error in the measurements of technique A. The error could have been an improperly calibrated meter stick, a meter stick that had been worn down on one end, or a whole host of other possibilities. The point is that whatever the error was it shi ft ed all the measurements in one direction relative to the true value. This kind of error is called a systematic error Systematic errors o ft en lead to poor accuracy and may not a ff ect precision. The device used in technique B is clearly less precise since there is uncertainty in the hundredths place rather than the thousandths place. This device's least count is in the tenths place and so we really shouldn't even record a digit in the thousandths place. Technique B should really show only three significant figures. Technique B is more accurate because of a lack of systematic errors. However, there are other errors that make technique B less precise. The values recorded in technique B are not really very close to each other, but they surround the true value. Errors that cause the measurements to be sort of randomly scattered around the true value are called random errors Random errors can not be avoided, but can be dealt with by taking many measure- ments and averaging. Some measurements are larger than the true value while other measurements are smaller than the true value. Averaging tends to cancel out these di ff erences. To summarize, the presence of systematic errors a ff ects accuracy while the presence of random errors a ff ects precision. Ideally we would like to have neither, but there will always be some random errors Determining and Reporting Accuracy and Precision 2 Measurements-ErrorsAndUncertainty.nb There are several methods that can be used to determine the precision and accuracy of a set of measurements. Remember that accuracy refers to how close your measurements are to the true value, whereas precision refers to how close your measurements are to one another. One way of quantifying accuracy is by calculating the percent error which compares the average of your experimental value, E, with the true value, T, using the following equation: percent error = E - T T * 100 If you do not know the true value, but you have two experimental average values you can use the percent di ff erence : percent di ff erence = E 1 - E 2 ( E 1 + E 2 )/ 2 * 100 Precision can be quantified by considering things such as estimated uncertainty, deviation, standard deviation, and standard deviation of the mean Estimated Uncertainty of a Single Measurement When you make a measurement you write down precise digits down to the least count of the instrument and then you estimate the last digit. Whether you estimate the last digit as a 5, a 3, an 8, etc., depends on how easy it is for you to estimate between the smallest divisions of the instrument. What- ever value you estimate, that estimated digit has uncertainty. Suppose you make a measurement with a meter stick and you write down 35.4 cm. The digit 4 has some uncertainty. If, in your opinion, you think that digit 4 is a very good estimate you might say your uncertainty is ± 0.1 cm, but if you think the 4 has more uncertainty you might say the uncertainty is ± 0.3 cm. Your measured value would be written as ( 35.4 ± 0.1 ) cm, or maybe ( 35.4 ± 0.3 ) cm. Your estimated uncertainty simply tells the reader how precise you think the last digit is. Whenever you make measurements you should always write down your estimated uncertainty. If you are making a bunch of measurements using the same measuring device you would usually write down the same uncertainty for all values you measured. Deviation of a Single Measurement Once you have measured a set of data you first calculate the average or mean. Usually your individual data will be di ff erent from the average. The deviation of the i th data point, x i , is its di ff erence between the value and the average, x : deviation of x i = xi - x Here is the table of data presented earlier with columns for the deviation: Measurements-ErrorsAndUncertainty.nb 3 Technique A Deviation Technique B Deviation 3.135 m - 0.001 m 3.262 m 0.008 m 3.137 m 0.001 m 3.242 m - 0.012 m 3.138 m 0.002 m 3.257 m 0.003 m 3.134 m - 0.002 m 3.258 m 0.004 m 3.136 m 0 m 3.251 m - 0.003 m Average = 3.136 m Average Deviation = 0 m Average = 3.254 m Average Deviation = 0 m What we’d like to do is give an estimate of the uncertainty of the average or mean. This would be a number that tells the reader how spread out the data is around the mean. As you can see from the deviation columns, the average of the deviation is zero. This will be true of any data set. So, deviation tells us how far an individual data point is from the average, but the average of the deviation tells us nothing We need something else. Standard Deviation of a Set of Measurements Since the average of the deviation doesn’t help us because some deviation values are positive and others are negative, how about we square them? If we consider the squares of the deviations and then average those we’ll certainly get something positive, and thus useful. But because the quantity we are measuring certainly has some units, when we square the deviations we get something that has the units squared. So to fix the units the last thing we do is take the square root. This square root of the average of the deviations squared (say that 10 times fast!) is called the standard deviation σ x = ∑ i = 1 N ( x i - x ) 2 N - 1 You might ask why we’re dividing by N-1 when there are N data points. There are good statistical reasons for this, but a simplified explanation is that if we had only one data point so that the average was the same as its value, the standard deviation would be zero if there was a N in the denominator (be sure you see why this is so). That one value certainly has some uncertainty so the standard deviation wouldn’t give a reasonable estimate of that uncertainty. What does the standard deviation of a set of data tell us? There’s a lot of fun statistics behind this, but all you need to know is that if your data has what’s called a normal distribution , then about 68% of your data lies within one standard deviation of the average. A normal distribution means that if you have enough data points you would find most of them near the average and fewer and fewer as you looked farther from the average. The distribu- tion of data around the average is called a bell curve or gaussian and looks something like this: 4 Measurements-ErrorsAndUncertainty.nb Plots of PDF for typical parameters: - 4 - 2 0 2 4 6 0.2 0.4 0.6 0.8 min max μ 0 | σ 1 μ 0 | σ 0.447214 μ 1 | σ 2.23607 The three curves show di ff erent distributions of data with di ff erent averages (given by μ ) and standard deviations. Also, we can say that 95% of your data is within two standard deviations of the average, and 99% is within three. The table below again shows the data from before but now the standard deviations are calculated. Measurements-ErrorsAndUncertainty.nb 5 Technique A Deviation Technique B Deviation 3.135 m - 0.001 m 3.262 m 0.008 m 3.137 m 0.001 m 3.242 m - 0.012 m 3.138 m 0.002 m 3.257 m 0.003 m 3.134 m - 0.002 m 3.258 m 0.004 m 3.136 m 0 m 3.251 m - 0.003 m Average = 3.136 m Standard Deviation = 0.0016 m Average = 3.254 m Standard Deviation = 0.0078 m The standard deviation of technique A is much smaller suggesting greater precision. For technique A we could report our measurement result as: ( 3.136 ± 0.002 ) m Whereas for technique B we would write: ( 3.254 ± 0.008 ) m Standard Deviation of the Mean In our ongoing discussion of comparing techniques A and B we’ve talked about the fact that technique A is clearly more precise but less accurate due to some sort of systematic error. Technique B is less precise due to some sort of random error. We discussed how you can reduce random error by taking more measurements and averaging, but there’s no way to reduce systematic error without changing your measuring device or procedure. To reflect the fact that the imprecision due to random error should decrease as the number of measurements increases we calculate what is called the standard deviation of the mean For reasons that we won’t worry about, the standard deviation of the mean (denoted by σ x ) is given by: σ x = σ x N As the number of measurements, N, increases, the standard deviation of the mean decreases. Comparing Measurements In many experiments we will be measuring a quantity in two di ff erent situations to see if di ff erences in the two situations have an e ff ect of the quantity. For example, we might want to know whether or not the amount of time it takes for a pendulum to swing back and forth depends of the size of the swing. We would make time measurements of the swing for two di ff erent swing sizes, or angles, and compare the results. One way of determining whether the two results are statistically in agreement or disagreement is by using what’s called a t-test. There are a variety of ways to do this and we won’t go into why the following equation is appropriate for our purposes, but here it is: 6 Measurements-ErrorsAndUncertainty.nb t = A - B σ A 2 + σ B 2 where A and B are the averages of the two measurements and the denominator contains the standard deviations of the means squared. In other words, you measured the period of a pendulum that swings with a 10-degree initial angle and you measured this period, say, 10 times. A is those ten period measurements added together and then divided by 10. You then measure the period of a pendulum that had an initial angle of 20-degrees and you measured this period also 10 times. B is those ten period measurements added together and then divided by 10. For the set of 10-degree period mea- surements σ A 2 is calculated and then for the set of 20-degree measurements σ B 2 is calculated. If the t-score is less than one then it is very likely that the two measurements are statistically equivalent. If t is greater than 3 or so then the two measure- ments are most likely di ff erent. If t is between 1 and 3 it’s hard to say. More precise measurements may need to be taken. A separate document gives examples of using the standard deviation and t-score to compare measurements. More on Significant Figures We need to say a few more things about significant figures. Significant figures reflect the precision of the measurement and say nothing about accu- racy. They refer to the numbers that can be read directly from an instrument scale plus one extra number that is estimated. A measurement such as 6.23 m has three significant figures and suggests that the actual length being measured might fall between 6.225 m and 6.235 m, but the thousandths place is certainly not known so we don’t report it. Zeros in numbers may or may not be significant depending on where they are relative to the decimal point and other significant figures. For example, the number 0.0034 m has only two significant figures because the three zeros serve only to tell the reader where the decimal point is. To make this clear we use scientific notation to show only the significant figures: 0.0034 m = 3.4 x 10 - 3 m. The number 2.40 m has three significant figures because the last zero is our estimated digit and we are fairly certain that it should be 0 and not 1 or 2. If you made a measurement where the tenths place was your estimated digit then you would write your measurement as 2.4 m and leave out the zero. Here are some rules: 1. Zeros to the le ft of the first non-zero digits are NOT significant. They merely serve to locate the decimal point. Measurements-ErrorsAndUncertainty.nb 7 2. Zeros between non-zero digits ARE significant. 3. Zeros to the right of the last non-zero digit ARE significant IF they are also to the right of the decimal point. The number 2.300 m has four signifi- cant figures, but it is not clear how many significant figures the number 3400 m has. 4. Zeros to the right of the last non-zero digit may or may not be significant if there is no decimal point. It is not clear how many significant figures the number 300 m has. As a rule, if there is no decimal point then 300 m has only one significant figure. If both zeros are actually meaningful, i.e. the second zero is our estimated digit, then we would write the measurement as 300. m (notice that we included a decimal point, but there is no zero to its right), or 3.00 x 102 m. Finally, if the first zero is significant, but the second is not, we would write the number in scientific notation as 3.0 x 10 2 m. 5. Exact numbers have an infinite number of significant figures. The number of centimeters in an inch (2.54) is an exact number. So is the speed of light in a vacuum, c = 299,792,458 m/s. When combining measurements in a calculation the final result can not have more precision that the least precise measurement. This is similar to the idea that a chain is only as strong as its weakest link. How then do you determine the correct number of significant figures of a number that was calcu- lated from other numbers with various significant figures? There are well defined statistical techniques to handle this, but for now we’ll simply state two simple, but approximate, rules: 1. When multiplying, dividing, or taking roots of quantities the result has as many significant figures as the quantity with the least. 2. When adding or subtracting quantities the number of decimal places in the result is equal to the number of decimal places in the quantity with the least. For example, if you multiplied the numbers 3.23 m and 4.2 m your calculator would give you: 13.566 m 2 , but you would write the answer so that it had only two significant figures, and you would round if necessary. The actual answer would be written as 14 m 2 This makes sense if you consider that the number 4.2 m really represents a length that is somewhere between 4.15 m and 4.25 m, and 3.23 m represents a length that is really between 3.225 m and 3.235 m. Multiplying 3.225 m by 4.15 m, and 3.235 m by 4.25 m gives us a range of numbers between which we know the real answer lies. We get 13.38375 m 2 and 13.74875 m 2 suggesting that the actual answer is closer to 14 m 2 than it is to 13 m 2 , and the tenths place is not at all precisely known. That’s all for now. Happy measuring! References 8 Measurements-ErrorsAndUncertainty.nb J. R. Taylor, An Introduction to Error Analysis , 2nd ed. (University Science Books, 1997) 2. O. L. Lacy, Statistical Methods in Experimentation (Macmillan, 1953) Measurements-ErrorsAndUncertainty.nb 9