Significant Digits for Uncertainty Values

And with the last number, I don't know what to do. I understand that you have to apply the uncertainty formula to the given data, but you cannot apply it if you do not know if the clock has recorded the time with a precision of milliseconds or a precision of hundredths of a second.In summary, the conversation discusses finding the random uncertainty in a set of measurements for dropping a pencil from a height of one meter using a stopwatch that measures to the millisecond. The given data has a precision of hundredths of a second, and the teacher suggests using the instrumental uncertainty of ±0.01 seconds to determine the last significant digit in the result. However, there is confusion about the precision of the stopwatch and whether the uncertainty should
  • #1
AllisonW4

Homework Statement


Question on worksheet: What is the random uncertainty in this set of measurements? Use the instrumental uncertainty to limit the number of digits in your random uncertainty result.

Data for dropping a pencil from a height of one meter using a stopwatch that measures to the millisecond:
0.23 s 0.28 s 0.26 s 0.26 s 0.24 s 0.26 s 0.30 s

Homework Equations


We are given that σ = (high value - low value) / (number of measurements)^1/2
(I know that this formula is a really, really rough estimation - I think the teacher is just trying to keep it a bit simpler for us instead of making us find the standard deviation)

The Attempt at a Solution


σ = (high value - low value) / (number of measurements)^1/2
σ = (0.30 s - 0.23 s) / (7)^1/2
σ = 0.02646 s

This is where I am stuck - how many digits should I round this to? I believe that the instrumental uncertainty of our stopwatch was ±0.01 s because it measured to the nearest millisecond. In the question it says to "use the instrumental uncertainty to limit the number of digits in your random uncertainty result" - should I round the uncertainty to the nearest millisecond because the watch was only accurate to the millisecond? Or do something else? Our teacher also wrote on the worksheet to "use the instrumental uncertainty to determine the last significant digit"... I am not really clear on what she means by that.
 
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  • #2
AllisonW4 said:
±0.01 s because it measured to the nearest millisecond.
0.01s is not a millisecond. Also, if it is to the nearest millisecond, what is the maximum error?
AllisonW4 said:
should I round the uncertainty to the nearest millisecond because the watch was only accurate to the millisecond?
That appears to be the instruction. Seems reasonable.
 
  • #3
haruspex said:
0.01s is not a millisecond. Also, if it is to the nearest millisecond, what is the maximum error?

That appears to be the instruction. Seems reasonable.
Thank you! And oops I was being dumb, of course 0.01 is not a millisecond but one hundredth of a second. So would you agree that I should round it to ±0.03 s then?
 
  • #4
AllisonW4 said:
So would you agree that I should round it to ±0.03 s then?
No, you are still looking at the wrong decimal place.
Express 0.02646s in msec.
 
  • #5
AllisonW4 said:
Data for dropping a pencil from a height of one meter using a stopwatch that measures to the millisecond:
0.23 s 0.28 s 0.26 s 0.26 s 0.24 s 0.26 s 0.30 s
First of all: the units of measure you have in the data are seconds.

There is a problem in the statement. The statement says that the clock measures milliseconds. But the data is given with a precision of hundredths of a second.
If the chronometer measures milliseconds, why not appear 0.230s, 0.280s, etc? (add a zero at the end, and round to three decimal places)

If you use two decimals, rounding 0.03 is fine. But if you use 3 decimals then you have to round it with the next number.

I do not understand why in the numbers of the measures of time, there are only two decimal places. If the clock shows milliseconds, three should appear.
 

FAQ: Significant Digits for Uncertainty Values

What are significant digits and why are they important in representing uncertainty values?

Significant digits are the digits in a numerical value that are known with certainty. They are important in representing uncertainty values because they indicate the precision and accuracy of a measurement or calculation. The more significant digits there are, the more precise the value is.

How do you determine the number of significant digits in a measurement or calculation?

The general rule for determining the number of significant digits is to count all non-zero digits and any zeros between them. For example, the number 10.5 has three significant digits, while the number 0.025 has two significant digits. Zeros at the beginning or end of a number may or may not be significant, depending on the context.

What is the significance of rounding when dealing with significant digits?

Rounding is important when dealing with significant digits because it ensures that the final value is presented with the appropriate level of precision based on the number of significant digits. When rounding, the last digit should be rounded up if the next digit is 5 or above, and rounded down if the next digit is below 5.

How are significant digits used in scientific notation?

In scientific notation, the number is written in the form of a decimal number between 1 and 10 multiplied by a power of 10. The significant digits are represented by the decimal number, while the power of 10 indicates the number of times the decimal point needs to be moved to get the original number. For example, the number 0.0056 can be written as 5.6 x 10^-3, where 5.6 has two significant digits.

Can significant digits be applied to all types of measurements and calculations?

Yes, significant digits can be applied to any type of measurement or calculation. This includes measurements using instruments, calculations involving mathematical operations, and even conversions between units. It is important to pay attention to significant digits in order to accurately represent the uncertainty of a value in scientific work.

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