3 1/2 digit and 4 1/2 digit displays

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Re: 3 1/2 digit and 4 1/2 digit displays
Coming at this question from a different angle 
A 31/2 digit display has three full digits as the three leastsignificant digits and a 0 or 1 as a fourth mostsignificant digit. The maximum display is then ±1999 before the display goes overrange.
A 41/2 digit display has four full digits as the four leastsignificant digits and a 0 or 1 as a fifth mostsignificant digit. The maximum display is then ±19999 before the display goes overrange. In other words, it has one more full digit that the 31/2 digit display.
Displays are cheap. The typical multimeter has a display that can resolve more than the meter is accurate. In other words, a 31/2 digit display can resolve down to 1 out of 2000 or 0.05% and usually you'll see a digital multimeter with specifications of ±0.5%±1 digit (or worse), making the last digit on the display worthless for actual data. Worse yet, when you go to an AC function, the meter error usually jumps to more like ±1%±1 digit, making the two leastsignificant digits questionable.
The leastsignificant digits are only good for watching trends, assuming that the meter is more stable over the trend period than the trend is.
By all rights, a manufacturer should blank out those extraneous digits, but no one does  Agilent, Fluke, Keithley. It would be horrid marketing, regardless of how honest it would be.
A 31/2 digit display has three full digits as the three leastsignificant digits and a 0 or 1 as a fourth mostsignificant digit. The maximum display is then ±1999 before the display goes overrange.
A 41/2 digit display has four full digits as the four leastsignificant digits and a 0 or 1 as a fifth mostsignificant digit. The maximum display is then ±19999 before the display goes overrange. In other words, it has one more full digit that the 31/2 digit display.
Displays are cheap. The typical multimeter has a display that can resolve more than the meter is accurate. In other words, a 31/2 digit display can resolve down to 1 out of 2000 or 0.05% and usually you'll see a digital multimeter with specifications of ±0.5%±1 digit (or worse), making the last digit on the display worthless for actual data. Worse yet, when you go to an AC function, the meter error usually jumps to more like ±1%±1 digit, making the two leastsignificant digits questionable.
The leastsignificant digits are only good for watching trends, assuming that the meter is more stable over the trend period than the trend is.
By all rights, a manufacturer should blank out those extraneous digits, but no one does  Agilent, Fluke, Keithley. It would be horrid marketing, regardless of how honest it would be.
Dean Huster, Electronics Curmudgeon
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.

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Re: 3 1/2 digit and 4 1/2 digit displays
Good post Dean
I've said for years that without stability and accuracy  resolution is meaningless!
(Any of these terms can be interchanged with the same result)
I've said for years that without stability and accuracy  resolution is meaningless!
(Any of these terms can be interchanged with the same result)

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Re: 3 1/2 digit and 4 1/2 digit displays
I used to illustrate the problem of false resolution to my students who would use a meter to measure a current of 135mA with an accuracy of ±3% and another meter to measure a voltage of 12.34 to an accuracy of ±0.1% and then multiply these two readings to get power and expect their calculation to be 1.6659 watts down to the last digit, not understanding that their answer has no better accuracy than the worst factor in the calculation.
My illustration was a person using a 5gallon bucket to fill an empty swimming pool to figure out how much water it holds. That "5 gallon" mark is at what point on the bucket? And how much slops out carrying the water to the pool? His final count may be 13,487 buckets, a very accurate and precise count, but the final answer can't be any more precise that the bucket measurement which may be off by as much as 20 or 25%. All those significant digits (67,435 gallons total) are totally worthless. From an engineering standpoint, you have to round your final answer off to a single significant digit  70,000 gallons  something that a person who has spent days hauling water and counting buckets is loathe to do!
My illustration was a person using a 5gallon bucket to fill an empty swimming pool to figure out how much water it holds. That "5 gallon" mark is at what point on the bucket? And how much slops out carrying the water to the pool? His final count may be 13,487 buckets, a very accurate and precise count, but the final answer can't be any more precise that the bucket measurement which may be off by as much as 20 or 25%. All those significant digits (67,435 gallons total) are totally worthless. From an engineering standpoint, you have to round your final answer off to a single significant digit  70,000 gallons  something that a person who has spent days hauling water and counting buckets is loathe to do!
Dean Huster, Electronics Curmudgeon
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.
Re: 3 1/2 digit and 4 1/2 digit displays
"All those significant digits (67,435 gallons total) are totally worthless."
Not quite. Though you are correct that a formula can not, in general, give an answer that is more precise than the least precise value in it**, some level of extra precision, even some that is beyond what is believable, is still of use. You should never decrease the precision of a measurement just because it will be used with another measurement that is less precise. You correct for the precision and accuracy of the numbers after you do the calculation. While doing the calculation you are supposed to proceed with at least one, and more generally 2 or 3 more digits in the calculation than what you use at the end. In other words, correction for precision is the last thing done, it should never be done before the actual calculation.
Furthermore, in this case the 13,487 trips is significant to 5 sig. figs. Assuming there wasn't any problem in the counting, it is a highly accurate number and may be used for other things besides multiplying by 5 to calculate the total volume of water. If you were to round the number of trips to say 10,000 (one sig. fig.) then it would be useless for a number of other possible calculations that it might be used in.
** there are cases where the result can be more accurate than any particular measurement. For example, when calculating averages. An average is often more precise than the individual numbers used to calculate it. That's pretty much what the whole concept of "average" is about.
Not quite. Though you are correct that a formula can not, in general, give an answer that is more precise than the least precise value in it**, some level of extra precision, even some that is beyond what is believable, is still of use. You should never decrease the precision of a measurement just because it will be used with another measurement that is less precise. You correct for the precision and accuracy of the numbers after you do the calculation. While doing the calculation you are supposed to proceed with at least one, and more generally 2 or 3 more digits in the calculation than what you use at the end. In other words, correction for precision is the last thing done, it should never be done before the actual calculation.
Furthermore, in this case the 13,487 trips is significant to 5 sig. figs. Assuming there wasn't any problem in the counting, it is a highly accurate number and may be used for other things besides multiplying by 5 to calculate the total volume of water. If you were to round the number of trips to say 10,000 (one sig. fig.) then it would be useless for a number of other possible calculations that it might be used in.
** there are cases where the result can be more accurate than any particular measurement. For example, when calculating averages. An average is often more precise than the individual numbers used to calculate it. That's pretty much what the whole concept of "average" is about.
Re: 3 1/2 digit and 4 1/2 digit displays
Hi Jimmy,
Perhaps you can give some examples of what you are talking about?
I think what Dean was saying was similar to the following example...
In a DC circuit, an exact current measurement would give 1.000000 amps and an exact voltage measurement would give 1.000000 volts exactly, but we are using meters that are only 1 percent accurate, so that means
the measurements of either or both can be off by 1 percent. Problem is, if they are both off by 1 percent
when we calculate the power (watts) by P=E*I we can get a result that is actually MORE inaccurate than the two readings taken by themselves. If the amps is 1.01 and the volts is 1.01 (both 1 percent high) then we would get a result of 1.0201, which is clearly 2 percent inaccurate. If we got lucky and one was right on and the other was off by 1 percent, we would get 1.01 watts, only 1 percent off. If we got even luckier and one was high and the other low we would get 0.99*1.01 which is 0.9999 which is right on (to three places).
If we had lots of volt meters and lots of current meters we could measure with all of them and average the total readings and get a statistically more accurate reading, but we dont, we only have two meters on hand.
That's really the point of this discussion i think, how to interpret results from measurements that have to be used together similar to this way when the individual readings can be off by some percentage. It depends a lot on what kind of operations are going to be used, as in this case it was multiplication. Yes, in sampling theory we can over sample and get a much better result, but even then there are other constraints about how the signal is varying with time that we have to consider, and if absent, dont do us any good anyway.
Perhaps you can give some examples of what you are talking about?
I think what Dean was saying was similar to the following example...
In a DC circuit, an exact current measurement would give 1.000000 amps and an exact voltage measurement would give 1.000000 volts exactly, but we are using meters that are only 1 percent accurate, so that means
the measurements of either or both can be off by 1 percent. Problem is, if they are both off by 1 percent
when we calculate the power (watts) by P=E*I we can get a result that is actually MORE inaccurate than the two readings taken by themselves. If the amps is 1.01 and the volts is 1.01 (both 1 percent high) then we would get a result of 1.0201, which is clearly 2 percent inaccurate. If we got lucky and one was right on and the other was off by 1 percent, we would get 1.01 watts, only 1 percent off. If we got even luckier and one was high and the other low we would get 0.99*1.01 which is 0.9999 which is right on (to three places).
If we had lots of volt meters and lots of current meters we could measure with all of them and average the total readings and get a statistically more accurate reading, but we dont, we only have two meters on hand.
That's really the point of this discussion i think, how to interpret results from measurements that have to be used together similar to this way when the individual readings can be off by some percentage. It depends a lot on what kind of operations are going to be used, as in this case it was multiplication. Yes, in sampling theory we can over sample and get a much better result, but even then there are other constraints about how the signal is varying with time that we have to consider, and if absent, dont do us any good anyway.
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Re: 3 1/2 digit and 4 1/2 digit displays
If we use 6 meters to measure the same voltage node and the meters are +/ 1% accurate, I don't see where averaging can insure success. The law of statistics says that half the meters should be off target by +1% and that the other half is 1%, thereby canceling out error and giving us an accurate reading. But wait  theres another law to consider here. The law of Murphy says that all meters will be off the mark in the same direction and by the maximum error. Depending on the type of test, they will add and show even a higher false reading
Re: 3 1/2 digit and 4 1/2 digit displays
Hi Robert,
Well, the theory might not really apply for something like 6 meters, it applies for something like 6 million meters.
Alternately, a smaller number of meters randomly chosen from a set of 6 million meters.
If we got unlucky and picked 6 meters that were all reading 1 percent high and we averaged the readings,
we would still get a reading that is 1 percent high. In other words, 6*1.01=6.06, and that averaged out
over 6 samples comes out again to 1.01.
On the other hand, if we found that there was a bit of noise in all of the readings, and that noise was completely random, we could actually INCREASE the accuracy by n bits of precision by taking 2^(2*n) samples and discarding some of the least significant bits. That's what over sampling is all about. I dont think this is that pertinent to the subject of this thread though, but it's still interesting to compare, right?
Well, the theory might not really apply for something like 6 meters, it applies for something like 6 million meters.
Alternately, a smaller number of meters randomly chosen from a set of 6 million meters.
If we got unlucky and picked 6 meters that were all reading 1 percent high and we averaged the readings,
we would still get a reading that is 1 percent high. In other words, 6*1.01=6.06, and that averaged out
over 6 samples comes out again to 1.01.
On the other hand, if we found that there was a bit of noise in all of the readings, and that noise was completely random, we could actually INCREASE the accuracy by n bits of precision by taking 2^(2*n) samples and discarding some of the least significant bits. That's what over sampling is all about. I dont think this is that pertinent to the subject of this thread though, but it's still interesting to compare, right?
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Re: 3 1/2 digit and 4 1/2 digit displays
MrAl
Yes I have to admit my example was somewhat extreme, but it is within the realm of possibility. As in your previous reference to multiplied errors such as computing wattage, my mention of increasing errors was along those lines. However I was thinking more in terms of many readings of various propertys that get plugged into a long equation and then accumulate error to give the worst case (Murphy) result. I do use averaging occasionally, but never feel 100% comfortable with it. This is one reason I would never buy a voltmeter with beyond 4 1/2 digit readout. Even the best of them can only be verified by periodic and expensive calibration.There are "Voltage Standards" that can be built quite cheaply and have little long term drift, and this might be a better way of getting more accurate readings with your voltmeter. Its not going to make the meter any more accurate, but now you know exactly what you are reading and can adjust for it.
Sorry about getting off topic Vinod, but there is some good food for thought in these replys when considering purchasing higher digit readouts.
Yes I have to admit my example was somewhat extreme, but it is within the realm of possibility. As in your previous reference to multiplied errors such as computing wattage, my mention of increasing errors was along those lines. However I was thinking more in terms of many readings of various propertys that get plugged into a long equation and then accumulate error to give the worst case (Murphy) result. I do use averaging occasionally, but never feel 100% comfortable with it. This is one reason I would never buy a voltmeter with beyond 4 1/2 digit readout. Even the best of them can only be verified by periodic and expensive calibration.There are "Voltage Standards" that can be built quite cheaply and have little long term drift, and this might be a better way of getting more accurate readings with your voltmeter. Its not going to make the meter any more accurate, but now you know exactly what you are reading and can adjust for it.
Sorry about getting off topic Vinod, but there is some good food for thought in these replys when considering purchasing higher digit readouts.

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Re: 3 1/2 digit and 4 1/2 digit displays
It's mathematical heresy/perjury to round off factors before calculation unless you can legitimately do so: Jimmy can't count worth squat and is always at least ±20% off. It would be conceivable to round then considering that the factor is worthless to an "exact" count.Furthermore, in this case the 13,487 trips is significant to 5 sig. figs. Assuming there wasn't any problem in the counting, it is a highly accurate number and may be used for other things besides multiplying by 5 to calculate the total volume of water. If you were to round the number of trips to say 10,000 (one sig. fig.) then it would be useless for a number of other possible calculations that it might be used in.
13,487 is a precise number, accurate to five significant digits. Of course it can be used for other purposes. Never said it couldn't. Each of Jimmy's trips to get water and dump it into the pool was 136 feet, you could use the count to figure out total distance travelled. A rounded number for that would be ignorant.
A 5gallon bucket for wallboard cement is a different size that that for paint, etc. Where is the exact 5gallon mark on each? How accurate can you fill to that mark? How much slops out on that 136foot trip to the pool? It's those questions that void placing a 5.00000 gallonpertrip capacity on that bucket. It's bucket slop that "ruins" the end calculation. If you only can guarantee buckettrip capacity by ±15%, your final answer can be no better than ±15% regardless of the precision of the trip count and must be rounded accordingly.
Dean Huster, Electronics Curmudgeon
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.
Re: 3 1/2 digit and 4 1/2 digit displays
Hello again,
Robert:
Oh yeah, i know what you mean there about long calculations like that. I found this out when i was working with large matrixes for doing linear regression. I found that when i tried to go over a matrix of 8 x 8 even the quite good precision of the home PC (around 16 digits) wasnt good enough. Not only that, but the solution equation that involved higher powers of 'x' (like x^8) quickly loose accuracy with even the smallest imprecision.
For illustration, that same error of 1 percent high in a sample that was really 1.000000 which makes it look like 1.01, taken that up to the power of 8, we get 1.0828567056280801 instead of a perfect 1.000000, which is a whopping 8 percent off! That's not very good when doing numerical calculations and it's amazing that this can happen starting with a sample that is only 1 percent off.
I needed much better precision for some things (the home PC usually rounds or truncates to 16 or 17 digits) so i ended up writing a program to handle very very very large and very very very precise numbers, with many digits of precision like 256 or 512, which results in very accurate calculations, but the big big drawback there is that these calculations take a heck of a lot more time to complete! Too bad i guess.
Robert:
Oh yeah, i know what you mean there about long calculations like that. I found this out when i was working with large matrixes for doing linear regression. I found that when i tried to go over a matrix of 8 x 8 even the quite good precision of the home PC (around 16 digits) wasnt good enough. Not only that, but the solution equation that involved higher powers of 'x' (like x^8) quickly loose accuracy with even the smallest imprecision.
For illustration, that same error of 1 percent high in a sample that was really 1.000000 which makes it look like 1.01, taken that up to the power of 8, we get 1.0828567056280801 instead of a perfect 1.000000, which is a whopping 8 percent off! That's not very good when doing numerical calculations and it's amazing that this can happen starting with a sample that is only 1 percent off.
I needed much better precision for some things (the home PC usually rounds or truncates to 16 or 17 digits) so i ended up writing a program to handle very very very large and very very very precise numbers, with many digits of precision like 256 or 512, which results in very accurate calculations, but the big big drawback there is that these calculations take a heck of a lot more time to complete! Too bad i guess.
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Re: 3 1/2 digit and 4 1/2 digit displays
I suppose the biggest problem with my students is the fact that if that 10digit calculator gave 10 digits of answer, by gosh, it was correct and valid to the last digit! After all, it's a calculator and we're not doing it by hand!
I suppose all this significant figure and rounding stuff is fallout back to 1967 when the engineering profs drummed that into our heads night and day  not to mention that we were using slide rules where you had to work hard to suck out more than three significant digits for an answer. Back then, if you needed more precision, you were doing the math by hand, using log and trig tables or hoping to find one of those superduper MonroeMatic calculators that could actually divide or multiply. Fortunately, we at least had an IBM 360 on campus so if we were willing to sit at an 049 card punch, we could do some of the more distressing calculations that way.
I suppose all this significant figure and rounding stuff is fallout back to 1967 when the engineering profs drummed that into our heads night and day  not to mention that we were using slide rules where you had to work hard to suck out more than three significant digits for an answer. Back then, if you needed more precision, you were doing the math by hand, using log and trig tables or hoping to find one of those superduper MonroeMatic calculators that could actually divide or multiply. Fortunately, we at least had an IBM 360 on campus so if we were willing to sit at an 049 card punch, we could do some of the more distressing calculations that way.
Dean Huster, Electronics Curmudgeon
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.
Contributing Editor emeritus, "Q & A", of the former "Poptronics" magazine (formerly "Popular Electronics" and "Electronics Now" magazines).
R.I.P.
Re: 3 1/2 digit and 4 1/2 digit displays
Then by all means require them to use ALL the digits of Pi when calculating and don't let them divide anything by 3, make them multiply by 1/3. Finally have them price out bulk resistors at digikey (often priced in fractions of a cent). "the price for those is $3.567"I suppose the biggest problem with my students is the fact that if that 10digit calculator gave 10 digits of answer, by gosh, it was correct and valid to the last digit! After all, it's a calculator and we're not doing it by hand!
Speaking of rounding. The IRS adjusted my gross income by $1 this year which resulted in a whopping $0 net change in my tax refund. What a waste of a stamp.
Re: 3 1/2 digit and 4 1/2 digit displays
Mr. Al
The issue arises when two (or more) measurements have widely different precisions (by definition accuracy is basically always unknown).
Working with the given example of a precisely known bucket count but a bucket of unknown accuracy and precision. And, the bucket's precision may also be further thrown off by sloping water out of a bucket carried too fast.
Assume you suspect the 5 gallon bucket is only accurate (and precise) to +/ 0.5 gallon. So the "5" is a reasonably accurate description, though some may argue that it should be written as "5.0" since it is the usual method to include the last certain digit plus one more (uncertain) digit.
A naive user of math and measurement might then say "well if it is 5 gallons to one sig. fig. then the 13,487 should be rounded to 10,000, where the zeros are not significant. If you do the total gallons calc. as (5)(10,000) you get 50,000 gallons (duh). If you retain all the digits in the accurate number you get 67,435 which should then be rounded to one place (or two depending on how you want to treat the +/0.5 gallon error in bucket volume) to 70,000 or 67,000. There is a pretty big difference between the 50,000 result and the 70,000 (or 67,000) result. Mathematical it is generally considered that the 70,000 (or 67,000) value is correct and not the 50,000. If you look at the suspected error in the bucket volume (0.5 gal., 10%) then apply that to the 50,000 result you only get 55,000 gallons as the upper limit. If you apply it the other way to the 70,000 result you get 63,000. So even taking into account the suspected error the two calculations give different results.
There is an accepted (in science, engineering, mathematics...) way to do these calculations.
As an aside. In the 2000 presidential election the vote in Florida looks to me to have been a tie. The error in counting six million votes is probably more than the difference between the Gore and Bush final counts, IIRC it was less then 1,000 votes out of 6,000,000, a margin of error of less than 0.017%. Given the clear problems with the counting, statistically and mathematically it was a dead tie. But since proper sampling statistics and counting mathematics is not part of the US election process it basically came down to a "coin flip". 50% chance Gore wins the national election and 50% chance Bush does.
The issue arises when two (or more) measurements have widely different precisions (by definition accuracy is basically always unknown).
Working with the given example of a precisely known bucket count but a bucket of unknown accuracy and precision. And, the bucket's precision may also be further thrown off by sloping water out of a bucket carried too fast.
Assume you suspect the 5 gallon bucket is only accurate (and precise) to +/ 0.5 gallon. So the "5" is a reasonably accurate description, though some may argue that it should be written as "5.0" since it is the usual method to include the last certain digit plus one more (uncertain) digit.
A naive user of math and measurement might then say "well if it is 5 gallons to one sig. fig. then the 13,487 should be rounded to 10,000, where the zeros are not significant. If you do the total gallons calc. as (5)(10,000) you get 50,000 gallons (duh). If you retain all the digits in the accurate number you get 67,435 which should then be rounded to one place (or two depending on how you want to treat the +/0.5 gallon error in bucket volume) to 70,000 or 67,000. There is a pretty big difference between the 50,000 result and the 70,000 (or 67,000) result. Mathematical it is generally considered that the 70,000 (or 67,000) value is correct and not the 50,000. If you look at the suspected error in the bucket volume (0.5 gal., 10%) then apply that to the 50,000 result you only get 55,000 gallons as the upper limit. If you apply it the other way to the 70,000 result you get 63,000. So even taking into account the suspected error the two calculations give different results.
There is an accepted (in science, engineering, mathematics...) way to do these calculations.
As an aside. In the 2000 presidential election the vote in Florida looks to me to have been a tie. The error in counting six million votes is probably more than the difference between the Gore and Bush final counts, IIRC it was less then 1,000 votes out of 6,000,000, a margin of error of less than 0.017%. Given the clear problems with the counting, statistically and mathematically it was a dead tie. But since proper sampling statistics and counting mathematics is not part of the US election process it basically came down to a "coin flip". 50% chance Gore wins the national election and 50% chance Bush does.
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