I.                   Uncertainty: Measuring Precision

The true value of a physical quantity is almost never known exactly. Several determinations of a quantity can be made using the same apparatus, with differing results.

Sometimes the accuracy with which a given measurement can be made is determined by variations in the thing being measured. For instance, a number of measurements of the diameter of a baseball would probably show that the ball is not a perfect sphere and consequently the measured values would be distributed over a range of values.

Sometimes the accuracy with which a measurement can be made is determined by the accuracy with which the scale on the instrument can be read. For example, it is hardly possible to read a meter stick more closely than + 0.5mm. The limits of accuracy may be set either by the precision of the scale of the instrument or by the ability and/or skill of the observer. But limits always exist.

It is also possible to have systematic error due to faulty instruments, for example, a meter stick which is not exactly one meter long. Then all measurements made with the instrument are in error, usually by a constant factor.

Uncertainty is not the failure of the observer to read the instruments correctly. If the observer records a 99.5 when the value should have been 89.5, this is not uncertainty, but is a mistake.

It is always of interest and usually necessary to know just how dependable are the results of an experiment and it is usually not the absolute uncertainty that is important but the percent uncertainty between the measured value and the ``true'' value (a.k.a. the ``accepted value'')

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For example, a 1000 km uncertainty in measuring the distance from Abilene to Moscow is much worse than a 1000 km uncertainty in measuring the distance from Abilene to the Sun.

 

When an accepted answer exists, the percent error is calculated from the difference divided by the accepted value:

If large enough number of measurements for the same physical quantity is performed then the average between all the measurements can be taken as the accepted value for this quantity. A satisfactory way to estimate absolute uncertainty of the final result would be by taking the maximum of absolute uncertainties for each of the measurements of this quantity. The precision of the measuring device and limitations on the scale reading also have to be taken into account. If it so happens that limitations of the scale and reading are larger than the uncertainty predicted based of the spread of your measurements then instrumental uncertainty due to the measuring device has to be taken as the final estimate of the absolute uncertainty.

 

II.                Propagation of Uncertainty

Along with knowing the percent error of experimental result, it is also necessary sometimes to know whether the experimental result and the true value are consistent, i.e., is it possible for the numbers to be equal to each other if the uncertainties on the numbers are taken into account.

 

For example, suppose the ``true value" is 20, but your experimental result is 17+3. At first glance we can say that 17 does not equal 20, but since the actual range of the result is from 14 to 20 (i.e., 17 plus or minus 3), it very well could be equal to 20. If this is the case, we say that the experimental result and the true value are consistent. If the experimental result was 15+3, we say it is inconsistent with the true value.

 

Therefore, it is essential to know the uncertainty range (A.K.A. margin of error, or error-bars) on your experimental results. This is how you tell whether your answer is ``good enough" or not.

The uncertainty range on an experimental result depends on the uncertainties of all the measurements that were made during the lab leading up to this result. Taking these various measurement uncertainties and determining the uncertainty range on the final answer requires a process known as Error Propagation. One result of error propagation is that the various experimental uncertainties always combine to increase the overall uncertainty.

 

For example: suppose measurements of the length of two pieces of string are made, with the goal of knowing their combined length. If the first piece is measured to be 10+1 cm and the second is measured to be 5+1 cm, the total length is 15 cm, but the overall uncertainty is +2 cm. The 1 cm uncertainty on each measurement added to give a combined uncertainty of 2 cm. Notice that the first string can be no shorter than 9cm and no longer than 11cm (10+1 cm). Similarly, the second string can be no shorter than 4cm and no longer than 6cm (5+1 cm). Therefore, the combination can be no shorter than 13cm and no longer than 17cm: 15+2cm.

 

The above simple example dealt with what we call absolute uncertainty. Another way to express uncertainty is the percent uncertainty. This is equal to the absolute uncertainty divided by the measurement, times 100%.

 

For example, the percent uncertainty from the above example would be  and  .

 

 In some cases of error propagation the uncertainties are used and in other cases, the percent uncertainties are used. The rules for determining which to use are given below:

1. When two measurements with associated absolute uncertainties are added or subtracted, the overall absolute uncertainty is equal to the sum of their absolute uncertainties.

2. When two measurements with associated percent uncertainties are multiplied or divided, the overall percent uncertainty is equal to the sum of their percent uncertainty.

 

Example:

The area and perimeter of a rectangular table are to be calculated. The table is measured to be 176.7 cm+0.2 cm along one side and 148.3 cm+0.3 cm along the other side. Because the perimeter is found by adding the sides, rule 1 is used:

The perimeter is. The area of the table is calculated to be (significant digits are underlined)

 

Since ,  we write the area in a variety of ways:

The reason some digits are called insignificant is that they are insignificant:

All of these round to , giving

 

 


Note that when calculating of the final result involves more complex mathematical operations than just simple addition or multiplication, the error propagation rules are becoming more complex as well. In such cases calculus has to be used in order to figure out the right way of calculation uncertainty or you can estimate the uppermost and the lowermost values of the result just as we did in the example about two pieces of the string.