Any experimentally derived number is inexact. Any number that comes from a measurement has an error associated with it. This brief note is an introduction to how to record and use quantities and their errors.
Suppose you measure the distance between two dots on a ticker-tape. If the ruler that you used is ruled in cm spacings then you will have to estimate the separation to the nearest 0.5 mm. Thus, you would record the separation, for example as 12.3 cm +/- 0.05 cm. The +/- 0.05 cm represents the worst that the error is likely to be. That is, its the limiting bound of the error - hence limit error.
Try It! Measure the separation between the dots:
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Sometimes you can get a much "finer" estimate of a quantity and its error if you can make repeated measurements of the quantity. Intuition probably tells you that averaging many measurements is a better representation of the actual quantity than is any one measurement. Mathematics will support this claim - especially if the errors that occur are random. As an example, Suppose that the following numbers are the timings for a cart to travel 12.3 cm +/- 0.05 cm along an incline:
Dt = 2.34 s, 2,29 s, 2.40 s, 2.36 s, 2.38 s
The average time is found to be 2.35 s. Notice that 2.35 s is NOT one of our measured times. If 2.35 s is the average time what is the error in the measurement? Answer: Not obvious but the standard deviation is a good measure of the spread in the values around the average (mean) The standard deviation for this set of numbers is 0.04 s. So, the time taken to travel 12.3 +/- 0.05 cm was 2.35 +/- 0.04 s.
On EXCEL you can easily calculate averages and standard deviations by using the =average(<block of numbers>) and =stdev(<block of numbers>) functions. Try this!
Recapping:
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How fast was the cart used in the previous example moving?
but both Dx and Dt have errors attached to them:
The trouble here is that the errors are of different types (cm and s). We canfigure out the error by looking at the extreme range of possible speeds:
The average velcoity is then 5.23 cm/s +/- 0.11 cm/s. BUT! there is a simpler way!!
Absolute and Fractional Error Forms
Up to now we have expressed the errors in their absolute forms. A much easier way to deal with errors is to convert them to a fractional or percentage form. Then all errors are of the same type (compare apples with apples!). So:
and
NOW! To calculate the total error just ADD the percentage errors of each quantity. This means that the error in average speed will be 0.4% + 1.7% or about 2.1%. What is 2.1% of 5.23 cm/s? Answer: 0.11 cm/s.
In Abstract Looking Form:
d X= error in quantity, dX/X = fractional error (multiply by 100 to convert to % form)
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You make 5 timings of an object rolling from rest down a 1.20 m ramp whose length was measured to the nearest 1 mm. The timings are:
0.45 s, 0.46 s, 0.51 s, 0.47 s, 0.42 s
What acceleration should you record?
If we insert numbers we get:
and conclude that a = 11.34 m/s2 +/- 0.13
To convert the fractional error +/1 0.13 into m/s2 we multiply by 'a' and get: a = 11.34 m/s2 +/- 1.5 m/s2
About Significant Digits? Notice that I was not particularly careful about sig digs. Although you may have learned elaborate rules for reporting sig digs the most important thing to note is that you need to be able to justify the number of sig digs that you display. It is important not to include more digits than you could possibly justify. If in doubt ask!