Is That Last Digit Really Significant? |

Quality Digest112 day(s) ago

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Is That Last Digit Really Significant? |

Quality Digest112 day(s) ago

Whenever we make a measurement we have to decide how many digits to record Traditional answers for this question are often little more than guesswork glorified by time And with digital read-outs are all the displayed digits real This paper provides a sound and practical answer to these perennial questions ADVERTISEMENT There are two uncertainties that affect every measurement These uncertainties are the round-off and measurement error The round-off has to do with the uncertainty introduced by the choice of measurement increment Measurement error is the inherent uncertainty introduced by the measurement system itself To understand the impact of each of these uncertainties we begin by looking at them separately Round-off uncertainty Assume that the only uncertainty in our measurements is the round-off Since we only record observations to a finite number of decimal places there will only be a finite number of possible values for our measurements These possible values will always end up being some integer multiple of the measurement increment MI If we denote our observed value as X then X will be some multiple of MI Figure 1 Round-off uncertainty Now the product value P does not suffer from the constraint that it has to be some multiple of MI It can take on any value in the continuum along the line in figure 1 So when we make a measurement our observed value will err by some amount and the difference between X and P will be the round-off error of that measurement E All of the product values P that fall between X 05 MI and X 05 MI will be represented by the value X If we assume that these product values are all equally likely then we have a uniform distribution for those product values that will be represented by the observed value X With this uniform distribution half of the product values will fall in the interval This means that half the time the round-off error of our observed value will be less than 025 MI and half the time the error of our observed value will be greater than 025 MI Figure 2 The error of a measurement due to round-off Thus 025 MI is the median round-off error of a measurement The relationship between measurement increments and the median round-off error is shown in figure 3 Here the use of smaller measurement increments will result in smaller median errors for our observed values and using more digits is always good When round-off is the dominant source of uncertainty in a measurement you will always want to use as many digits as possible when recording the observed values But round-off is not always dominant Measurement system error As we use smaller and smaller measurement increments the uncertainty in the measurement system itself will eventually become the dominant source of measurement error As before we consider a product measurement X to be the sum of two parts the product value P and the measurement error E Further let us assume that the value for E is independent of the value for P Ever since 1810 when Laplace published what we call the central limit theorem the standard probability model for the measurement errors E has been a normal distribution This means that if we measured the same thing thousands of times using very small measurement increments we would end up with a histogram like figure 4 Figure 4 Repeated measurements of the same thing do not yield identical values Here the best estimate of the value for the thing being measured would be the average of all the readings The estimated standard deviation of these repeated measurements Sigma E would characterize the uncertainty due to the measurement system Now the normal distribution has the property that the middle 50 percent is defined by the mean plus or minus 0675 times the standard deviation Since the early 19th century the name given to 0675 Sigma E has been the Probable Error Figure 5 The median error of a measurement due to measurement uncertainty Half the time the observed value X will fall within the central portion of figure 5 and half the time it will fall in one of the tails of figure 5 Thus the Probable Error PE defines the median error of a measurement due to uncertainty in the measurement system Regardless of the size of measurement increment used a measurement will err by one probable error or more at least half the time If you are going to err by this amount or more half the time it does not pay to interpret a number more precisely than plus or minus one probable error Thus the probable error defines an effective lower bound on the resolution of a measurement and it does this regardless of how small we make the measurement increment While measurement error is commonly reported using the estimated standard deviation Sigma E of the measurement system we prefer to use the probable error PE We do this because it is much easier to get people to understand that the probable error is the median error introduced by the measurement system than it is to get them to understand that Sigma E is the radius of gyration for figure 5 Moreover since the PE is a simple multiple of Sigma E they both contain the same essential information Combining the two sources of uncertainty If we divide MI by PE we can express the measurement increment as a multiple of the probable error These values will define the horizontal scale for figure 7 If we divide the median error by PE we express the median error as a multiple of the probable error These values will define the vertical scale for figure 7 The median error due to round-off alone will be always be one-quarter of the size of the measurement increment and this relationship creates the sloping line in figure 7 The median error due to measurement uncertainty alone will be equal to the probable error Since this is independent of the measurement increment used it is shown in figure 7 as a horizontal line When we combine both sources of uncertainty we end up with the median error curve that is tabled in figure 6 and shown in figure 7 Figure 6 Measurement increments and corresponding median errors Figure 7 The median error of a measurement On the left where measurement error is the dominant source of uncertainty the median error curve is nearly horizontal As the measurement increment increases round-off begins to affect the measurements and the uncertainty increases On the right as round-off becomes the dominant source of uncertainty the median error curve becomes asymptotic to the sloping line The median error curve in figure 7 was independently obtained by each author using simulation studies and was later verified by direct computation The uncertainty in the last digit But how does all this stack up against the traditional way of characterizing the quality of a measurement the uncertainty in the last digit To answer this question consider what happens when for example a measurement has an increment equal to 22 PE units According to the table in figure 6 when the measurement increment is 22 PE units the median error of a measurement will be 1094 times the probable error When we divide 1094 PE units by 22 PE units per measurement increment we find that the median error of these measurements will be equal to one-half measurement increment Thus when the measurement increment equals 22 PE units the measurements will err by one-half of a measurement increment or less at least half the time Such measurements will be good to the last increment over half the time Inspection of the formula above will show that in order to express the median error in measurement increments we simply divide a value from the second column of figure 6 by the corresponding value in the first column The results of this exercise are summarized in figure 8 Figure 9 shows the third column of figure 8 as a function of the first column of figure 8 Figure 8 Median errors expressed in measurement increments Figure 9 The median uncertainty in the last digit For the usual case where the measurement increment is some power of 10 the curve in figure 9 and the values in figure 8 define the uncertainty in the last digit Specifically whenever a measurement increment exceeds 22 probable errors it will err by zero digits in the last place at least half the time Measurement increments that fall between 067 PE units and 22 PE units will err by one or fewer digits in the last place at least half the time Such values might be said to be good to within one digit in the last place Measurement increments that fall between 040 PE units and 067 PE units will err by two or fewer digits in the last place at least half the time Such values might be said to be good to within two digits in the last place Measurement increments that fall between 028 PE units and 040 PE units will err by three or fewer digits in the last place at least half the time Such values might be said to be good to within three digits in the last place Measurement increments that fall between 022 PE units and 028 PE units will err by four or fewer digits in the last place at least half the time Such values might be said to be good to within four digits in the last place As the measurement increment drops from 022 PE units down to 010 PE units the median error rapidly climbs from five increments to 10 increments Measurement increments that are powers of ten which fall below 010 PE units have no useful information in the last recorded digit since they will err by more than 10 digits in the last place at least half the time The twin uncertainties affecting every measurement In figures 7 and 9 we have the twin uncertainties that affect all measurements Since both graphs have the same horizontal axis we combine the two curves to get figure 10 There the vertical scale on the left shows the number of measurement increments that correspond to the median error of a measurement while the scale on the right shows the actual median error in PE units Figure 10 The twin uncertainties for every measurement Figure 10 shows the twin uncertainties that affect every measurement Large measurement increments will have less uncertainty in the last digit but they will have unnecessarily large median errors Very small measurement increments will have the minimum median error but they will be exceedingly soft in the last digit Thus the choice of a measurement increment requires a trade-off When the measurement increment falls between 067 PE units and 10 PE unit the measurements will be good to within one digit in the last place and they will have a median error that is within 2 percent of the minimum This is the zone marked as Best in figure 10 The zone marked Good in figure 10 extends from measurement increments of 040 PE units up to 15 PE units On the left side of this zone the measurements will be good to within two digits in the last place and will have a median error of less than 1009 PE On the right side they will be good to within one digit in the last place and will have a median error below 1044 PE The zone marked Acceptable in figure 10 extends from measurement increments of 022 PE units up to 22 PE units On the left side the measurements will be good to within four digits in the last place and will have a median error of less than 1003 PE On the right side they will be good to within one digit in the last place and will have a median error below 1094 PE Since this zone gives us an order of magnitude to play with it effectively defines the range of appropriately sized measurement increments to use Finally since the error of recording too many digits is less severe than that of recording too few digits we could classify measurements with increments in the range of 010 PE unit to 022 PE units as very soft but usable These measurements will err by five units or more in the last place at least half the time but will have a median error of 1001 PE or less Using the right number of digits When we combine all of the information above we end up with the guidelines shown in figure 11 There the horizontal scales show the ranges for the size of the measurement increment in terms of both the probable error and the standard deviation of measurement error Sigma E Depending on where your measurement increment falls the diagram describes the dominant uncertainty in your measurements and the appropriate course of action you should take Figure 11 Using the right number of digits Large measurement increments Since the median error will increase rapidly when the measurement increment exceeds 22 PE units and since recording one additional digit will convert such a measurement into one that has a measurement increment that is in the acceptable zone you should consider adding a digit to your recorded values whenever the measurement increment exceeds 22 PE units Example one Say for example that you have measurements recorded to the hundredth of a millimeter and that the probable error of the measurement system is PE 00014 mm Then when we divide the measurement increment by the probable error we discover that the measurement increment is equal to 001 00014 71 probable errors and we are in the region where round-off is the dominant source of uncertainty Figure 12 Example one From figure 6 we can compute the median error of our measurements to be approximately 185 PE 00026 mm From figure 9 we see that with a measurement increment of 7 PE our measurements will err by zero digits in the last place at least half the time Thus these measurements will be good to the nearest hundredth of a millimeter but they are not as good as the measurement system will allow them to be If it was economically feasible to add one extra digit to these measurements so that the measurement increment would become 0001 mm then the measurement increment would be equal to 071 probable errors From figure 6 the median error of a measurement would be about 1010 PE 00014 mm From figure 9 these measurements would be good to within one digit in the last place at least half the time Recording one extra digit here reduces the median error from 00026 mm to 00014 mm effectively cutting the uncertainty in the measurements in half Small measurement increments When the measurement increment is less than 010 PE units the measurements will err by ten digits or more in the last place at least half the time Clearly this is a zone where we should drop the last recorded digit By doing so we will not appreciably change the precision of the measurements But I prefer to write more digits Writing down random numbers does not make the measurements more precise It just makes them look better than they actually are When we insist on using a measurement increment that is smaller than 010 probable errors we are misrepresenting the precision of the measurements to ourselves and others Just because some extra digits show up on a digital read-out does not make the extra digits real Example two Say for example that you have a scale that records weight to the nearest microgram MI 1 microgram However let us assume that repeated measurements of a standard weight show that the standard deviation of the measurement system is Sigma E 17 micrograms Then the probable error for these measurements is PE 115 micrograms This means that while you may be recording weights to the nearest microgram they are going to err by 12 micrograms or more at least half the time and your last recorded digit will be nothing but noise When we divide the measurement increment by the probable error we discover that the measurement increment is equal to 1 115 0087 PE Since this is less than 010 PE we are in the left hand box of figure 11 and we need to drop a digit from our recorded values Figure 13 Example two By rounding our weights to the nearest 10 micrograms our measurement increment becomes MI 10 micrograms or 087 PE Interpolating in figure 6 we find that our median error will increase slightly to 1017 PE 117 micrograms and our recorded values will be good to within one in the last place at least half the time Rounding these values to the nearest multiple of ten micrograms prevents these measurements from looking better than they really are makes the values good to within one digit in the last place and does not appreciably increase the uncertainty in the measurements Soft and very soft measurements Very soft measurements have an increment between 010 PE and 022 PE and will have a median error of 5 to 10 in the last digit at least half the time While these measurements may occasionally be of use our recommendation is to make the measurement increment five or 10 times larger This will reduce the median uncertainty to one or two measurement increments without an undue increase in the median error of a measurement Soft measurements have an increment between 022 PE and 040 PE and will err by 3 to 4 in the last digit at least half the time While these soft measurements may be used in this form doubling the size of the measurement increment will reduce the median uncertainty to one or two increments without an undue increase in the median error of a measurement Summary Using a measurement increment that is too large will throw away useful information Using a measurement increment that is too small will add noise to your measurements With the information provided here you no longer need to guess about how many digits you need to record You can now determine exactly how many digits are appropriate for any measurement system Once you have an estimate for the standard deviation for the measurement system you can easily obtain the probable error Once you know the probable error you know the effective resolution of the measurements You can then make your own trade-offs to determine how many digits to record Acceptable measurement increments will fall between 022 probable errors and 22 probable errors In practice these cut-offs may be rounded to 02 PE and 2 PE Moreover since you can determine these values for your own measurement system operated by your own personnel in your plant you no longer have to depend upon the assurances of some salesman about the precision of a measurement system Neither are tradition and guesswork needed Now you have a way to know how good your measurements are and how many digits to record About The Authors Dr Donald J Wheeler is a Fellow of both the American Statistical Association and the American Society for Quality and is the recipient of the 2010 Deming Medal As the author of 25 books and hundreds of articles he is one of the leading authorities on statistical process control and applied data analysis Find out more about Dr Wheelers books at wwwspcpresscom Dr Wheeler welcomes your questions You can contact him at djwheeler spcpresscom James Beagle III is a lapsed CQE CRE and C6SBB asking the wrong questions and obsessing about impractical concepts before realizing what is important with 35 years experience in process quality and reliability engineering He primarily works with a large manufacturing base supplying data storage component products where he develops tools for data analysis designs experiments for product improvements and works on product and process qualification In his free time he works on numerical modeling and is also known to have guitars at home He can be reached at james idealstatesnet

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