X (mm) From Mean From Mean

0.73 + 0.01 0.0001

0.71 - 0.01 0.0001

0.75 + 0.03 0.0009

0.71 - 0.01 0.0001

0.70 - 0.02 0.0004

0.72 0.00 0.0000

0.74 + 0.02 0.0004

0.73 + 0.01 0.0001

0.71 - 0.01 0.0001

0.73 + 0.01 0.0001

For the moment we will only be interested in the first two columns above. A glance at the deviations shows the random nature of the scattering.

The formula for the mean yields:

The mean is calculated as 0.723 mm but since there are only two significant figures in the readings, we can only allow two significant figures in the mean. So, the mean is 0.72 mm. Once we have the mean, we can calculate the figures in the 2^nd column of the Table above. These are the deviation of each reading from the mean.

We can use the maximum deviation from the mean, 0.03 mm, as the “maximum probable error (MPE)” in the diameter measurements. So, we can state the diameter of the copper wire as 0.72 ± 0.03 mm (a 4% error). This means that the diameter lies between 0.69 mm and 0.75mm.

An interesting thought occurs: What if all the readings of the diameter of the wire had worked out to be the same? What would we use as an estimate of the error then?

In that case, we would look at the limit of reading of the measuring instrument and use half of that limit as an estimate of the probable error. So, as stated above, our micrometer screw gauge had a limit of reading of 0.01mm. Half the limit of reading is therefore 0.005mm. The diameter would then be reported as 0.72 ± 0.005 mm (a 0.7% error). This means that the diameter lies between 0.715 mm and 0.725 mm. Note that we still only quote a maximum of two significant figures in reporting the diameter.

It is also worth emphasizing that in the stated value of any measurement only the last digit should be subject to error. For example, you would not state the diameter of the wire above as 0.723 ± 0.030 mm because the error is in the 2^nd decimal place. This makes the 3^rd decimal place meaningless. If you do not know the 2^nd decimal place for certain, there is no point stating a 3^rd decimal place in the value of the quantity. So, do not write an answer to 5 decimal places just because your calculator says so. Think about how many figures are really significant.

We can now complete our answer to the question: How do we take account of the effects of random errors in analysing and reporting our experimental results? At high school level, it is sufficient to:

Take a large number of readings – at least 10, where time and practicality permit.Calculate the mean of the readings as a reasonable estimate of the “true” value of the quantity. Use the largest deviation of any of the readings from the mean as the maximum probable error in the mean value.If all the readings are the same, use half the limit of reading of the measuring instrument asthe MPE in the result.

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