1.4 Significant figures

We have already discussed different types of errors and how to handle them. Surprisingly, however, we hardly ever mention about error while assigning values to different physical quantities. It is the general case. As a matter of fact, we should communicate error appropriately, as there is provision to link error with the values we write.

We can convey existence of error with the last significant digit of the numerical values that we assign. Implicitly, we assume certain acceptable level of error with the last significant digit. If we need to express the actual range of error, based on individual set of observations, then we should write specific range of error explicitly as explained in earlier module.

We are not always aware that while writing values, we are conveying the precision of measurement as well. Remember that random error is linked with the precision of measurement; and, therefore, to be precise we should follow rules that retain the precision of measurement through the mathematical operations that we carry out with the values.

Context of values

Before we go in details of the scheme, rules and such other aspects of writing values to quantities, we need to clarify the context of writing values.

We write values of a quantity on the assumption that there is no systematic error involved. This assumption is, though not realized fully in practice, but is required; as otherwise how can we write value, if we are not sure of its accuracy. If we have doubt on this count, there is no alternative other than to improve measurement quality by eliminating reasons for systematic error. Once, we are satisfied with the measurement, we are only limited to reporting the extent of random error.

Another question that needs to be answered is that why to entertain “uncertain” data (error) at all. Why not we ignore doubtful digit altogether? We have seen that eliminating doubtful digit results in greater inaccuracy (refer module on “errors in measurement”). Measuring value with suspect digit is more “accurate” even though it carries the notion of error. This is the reason why we prefer to live with error rather than without it.

We should also realize that error is associated with the smallest division of the scale i.e. its least count. Error is about reading the smallest division – not about estimating value between two consecutive markings of smallest divisions.

Significant figures

Significant figures comprises of digits, which are known reliably and one last digit in the sequence, which is not known reliably. We take an example of the measurement of length by a vernier scale. The measurement of a piece of rod is reported as 5.37 cm. This value comprises of three digits, “5”, “3” and “7”. All three digits are significant as the same are measured by the instrument. The value indicates, however, that last digit is “uncertain”. We know that least count of vernier scale is ${10}^{-4}$ m i.e ${10}^{-2}$ cm i.e 0.01 cm. There is a possibility of error, which is equal to half the least count i.e. 0.005 cm. The reported value may, therefore, lie between 5.365 cm and 5.375 cm.