# Greatest Possible Uncertainty and Sig Figs

• B
• Cardinalmont
In summary, the balance is accurate to 0.01g but provides 5.67g as the reading. The uncertainty is then +/- 0.005g.

#### Cardinalmont

Gold Member
There is something I seriously don't understand about uncertainty.

Suppose there is an electric balance that reads 5.67g
The limit of reading is 0.01g
The greatest possible error is half of the limit of reading and is thus 0.005g

By this logic, and assuming the very best possible situation, I would think one could record the mass of the coin as (5.67±0.005)g.

This makes sense to me because the scale shows 5.67g, but the actual mass of the coin can be anywhere from 5.665 to 5.675 and the scale had to round the number to just 2 decimal places.

The problem with the way I understand it is that a quantity's uncertainty is limited by the decimal place of the quantity. If 5.67 ends in the hundredths so must the 0.005, turning it into 0.01, thus ruining the logic established in the previous paragraph.

As I understand it:
There is 'precision' and there is 'accuracy.' If the accuracy exceeds the precision (as you're assuming for this balance), you don't really need to 'specify' +/- 0.005 - it's implied by the '5.67.' Where the precision exceeds the accuracy, the uncertainty should be conveyed with '+/-'. Worth noting: For many instruments, precision exceeds accuracy, over at least part of the measuring range.

Cardinalmont
If you used an electronic balance which read 5.67, how would you write the reading with the uncertainty?

Cardinalmont said:
If you used an electronic balance which read 5.67, how would you write the reading with the uncertainty?

5.67 +- 10%
or whatever percent your uncertainty is.

Unless your scale specifically says what its accuracy (uncertainty) is, you have no way of knowing for sure. But for a general guess, I have a postal scale that claims capacity of 55 pounds with 0.1 ounce accuracy; that is 0.01% for a 55 pound object, or 10% for a 1 ounce object.

To give a really good idea of Expected Accuracy, needs more than just one number. This is particularly true for an instrument that covers a wide range of values. Weiging Scales can easily have an Offset, which will be similar over the whole range of weights. The Zero adjust can help there. Then they will have mechanical stickiness which may be far worse for light objects. Then there may be levers inside which can introduce trigonometrical errors if someone has assumed linearity.
Basically if you need to rely your readings, you need a graph of likely errors (+ and - error lines over the whole range of use). There is no end to this so don't go further than you need for your particular application.
Go online and look at chemical balances that you can buy. The best ones have loads of information about their accuracy - just read the spec sheets. A good cure for insomnia unless you really want to know about it.

If your balance really is that odd combination you describe I don't think that it would be incorrect to write 5.670±0.005g. But it would be an odd design decision by the manufacturer to provide centigram resolution for a balance with milligram precision.

That said, I have an inexpensive gram resolution scale that is more precise than 1g, at least on the low end of the range. It's not precise to 0.1g but it is consistently sensitive to changes of around 0.2-0.3g. As it's intended for home use a wobbly tenths digit would probably just annoy the average user.

If your balance were accurate to 1mg you wouldn't have to settle for 5mg uncertainty. You could add a milligram at a time until the display changed. And 1mg/2mg/etc. weights of the precision needed for this could be cheaply constructed at home with a sufficient length of quality thin wire, scissors, a good tape measure, and your balance.

But it's most likely that it really isn't that accurate.

## 1. What is the greatest possible uncertainty in a measurement?

The greatest possible uncertainty in a measurement is half of the smallest division on the measuring instrument. For example, if a ruler is marked in millimeters, the greatest possible uncertainty would be 0.5 mm.

## 2. What is a significant figure?

A significant figure is a digit in a number that is known with certainty, plus one estimated digit. It is used to indicate the precision or uncertainty of a measurement.

## 3. How do you determine the number of significant figures in a measurement?

To determine the number of significant figures in a measurement, start counting from the left and include all non-zero digits. The last digit is the estimated digit, so it is also significant. Zeroes between non-zero digits are also significant. However, zeroes that are placeholders or trailing zeroes after a decimal point are not considered significant.

## 4. How do you perform calculations with significant figures?

When performing calculations with significant figures, the final answer should have the same number of significant figures as the number with the least amount of significant figures in the calculation. Intermediate calculations should be carried out to one more significant figure than the original numbers.

## 5. Can significant figures be used in non-numerical values?

Yes, significant figures can also be used in non-numerical values, such as in conversion factors. In this case, the significant figures indicate the precision of the conversion factor and should be carried through the calculation to maintain accuracy.