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# Microsoft Corporation Message Board

• cfuryurself cfuryurself Jan 13, 2013 6:59 PM Flag

## the first uncertain digit

creeky
Your source on sig digs which was the frostberg.edu site right ??

""Why should the rules for propagating significant digits not be applied to averages?""

******************
IIf you dutifully follow the guidelines, your reasoning goes something like this:

Adding up the weights gives a total weight of 763.6 lbs, which has 4 significant figures.
To get the average, divide by 5. The 5 is an exact number, so you don't lose any significant figures by doing this. The exact answer is 152.72 lbs
The answer rounded to 4 digits is 152.7 lbs.

That implies that you know the weight to the nearest tenth of a pound.

But do you really know the weight this precisely? Is the tenths place really the first uncertain digit? This is one of those cases where you have to keep the definition of significant figures in mind.

You have a series of replicated measurements, so you can see that the precision implied by the scale isn't the same as the precision you're actually getting, because the person is balancing themselves slightly differently every time. Even though the measuring device has a scale that allows you to read to the nearest 10th of a pound, the measurements are changing in pounds place. Pounds are the first uncertain digit, so the reported weight should be 153 lbs.
*****************

Do you agree with this??
I certainly would not as it leads to the result being dependent on the most erroneous measurement.

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• I couldn't get the frostberg.edu site to load. However, I was thinking about this all day as a matter of magnitude. Each digit to the right introduces the possibility of error that is an order of magnitude 10 higher than the last digit.

In your example above where we are apparently averaging weights of a person (or persons.)

*** Adding up the weights gives a total weight of 763.6 lbs, which has 4 significant figures.
*** To get the average, divide by 5. The 5 is an exact number, so you don't lose any significant figures by doing this. The *** exact answer is 152.72 lbs
*** The answer rounded to 4 digits is 152.7 lbs.

We can be pretty certain that the first digit of 1, for 100 pounds is accurate. We are surely certain that the person (or persons) we weighed, weighed AT LEAST 100 pounds!

For the same reason, we are pretty certain about that 5 being at least 50 pounds.

Thus we have a high degree of certainty that we measured and average weight of at least 150 pounds!

Now we get to that 2. Well, we're pretty sure it was at least 2 pounds. So with a reasonable degree of certainty we can claim the average was at least 152 pounds.

Then comes that decimal point and that 7 and 2 following the decimal point. And we wonder, "Should we round up that 0.7 to the next 1?" Well, now that would make our average 153 instead of 152. Just how accurate is that 0.7? We study our measuring equipment, our measuring technique and even as you point out weather the subject being weighed was still, unbalanced or wiggling while being measured. And we find that we are not all that comfortable with the accuracy of that 0.7.

What it looks like is our "true" average is between 152 pounds and 153 pounds. Something like:
152.0 pounds

Sentiment: Hold

• 3 Replies to bullwinkle.themoose
• Sorry, Yahoo is breaking up over "less than" and "less than or equal" math symbols! CHEEZE! ON RUBBER CRUTCHES!

What it looks like is our "true" average is between 152 pounds and 153 pounds. Something like:
152.0 pounds 'less than' "True" Average 'less than' 153.0 pounds

It is true that:
152.0 'less than' 152.7 'less than' 153.0
It is also true that:
152.0 'less than' 152.72 'less than' 153.0

If we REALLY believe soundly in that 7, then it would also be true that:
152.7 pounds 'less than or equal to' "True" Average 'less than' 153.0 pounds
or:
152.7 'less than or equal to' 152.72 'less than' 153.0

[ASIDE: I am not a student of calculus. But this is looking very "calculsie" to me! X is approaching some limit but never getting there! Much like that classic Greek example: If Achilles, the fastest human ever, runs half the distance to some place; then half the remaining distance; then half the still remaining distance, and so on...he will NEVER get there but will continue to approach getting there! ]

But as you (or the example) pointed out we DON'T really believe too soundly in the accuracy of that 0.7 and a whole magnitude of 10 less of that 0.02! Now just suppose our average calculation had come out to 152.72625. . . (PS that's a 5/8 after the 0.02...)

Now cfury's original theorem claimed that the extra digits would be good significant digits. But since we DON'T even believe much in the 0.7 and even less in the 0.02 how much faith can we really have in the 0.00625? I would say whole magnitudes of 10, 100 and 1000 times less faith!

Once again, I come back to the practical application of our average. I don't know, in the example given, weather we weighed one person five times (to get five numbers to average to a "true" weight) or weather we weighed five different people one time each, and thus have an average of five people. But it is clear our average is between 152 and 153 pounds. Does it really NEED to be more accurate than that?

Sentiment: Strong Sell

• 152.0 pounds

Sentiment: Hold

• #\$%\$! Yahoo broke up my post. . . Continued:

What it looks like is our "true" average is between 152 pounds and 153 pounds. Something like:
152.0 pounds

Sentiment: Hold

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