Within the limits of the precision of the ruler, 1 mm, we both are correct. We should then state the length of the ruler as 21 ± 1 mm, and the other guy should state the length as 22 ± 1 mm. A scientist looking at our data now knows that the precision of our measurement is 1 mm. Some people call the ± 1 mm an error or probable error in our measurement. It is not an error, but an uncertainty. We took the best measurement we could with the equipment available. If we had made an error, we would have taken the measurement again!

To minimize the uncertainty in a quantity, scientists measure it several times. This also helps to eliminate true errors in a measurement, because the value in error will most likely be substantially different from the others. The value stated is the average value from all the measurements. For example, let's say we took five length measurements of an object:

The average value would be:

The uncertainty is found by first taking the difference between each measurement and the average value. This is called the deviation and is stated as a positive number. Deviation:

The mean deviation, dm, is the sum of the deviations divided by the number of measurements: Mean deviation:

The probable uncertainty is then found by dividing the mean deviation by the square root of the number of measurements:

The measurements of the length would then be reported as

Sometimes uncertainties are given in percents. The percent uncertainty is found by dividing the probable uncertainty by the average value and multiplying by 100.

What if we add, subtract, multiply, or divide several numbers with uncertainties? Let's say we have three numbers:

For addition and subtraction, the numerical uncertainty is equal to the square root of the sum of the squares of the probable uncertainties. If we add our three numbers, we get 24.41 with an uncertainty of

So the result would be written as

For multiplication and division, the numerical percent uncertainty is equal to the square root of the sum of the squares of the percent uncertainties. If we multiply our three numbers, we get 510 with an percent uncertainty of

First, turn probable uncertainties into percent uncertainties:

Then,

So the result would be written as

If many measurements are taken, the uncertainty is sometimes measured by the standard deviation. One standard deviation means that 68% of all measurements will be within this value. Two standard deviations mean 95% of all measurements will be within this value. Calculation of the standard deviation is beyond the scope of this note.

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