Uncertainty in Measurement

Uncertainty in Measurement

1.         Any measurement is only as good as the “skill of the person” doing the work and the “reliability of the equipment used”. In scientific work, we recognize two kinds of numbers; exact numbers and inexact numbers.

2.         Exact numbers are those that are known with exactly or are integers that result from counting numbers of objects. For example; by definition there are exactly 12 eggs in a dozen, exactly 1000 g in a kilogram and exactly 2.54 cm in an inch.

3.         Inexact numbers are those whose values have same uncertainty. Numbers obtained by measurement are always inexact i.e. uncertainties always exist in measured quantities. This is due to inherent limitations in the equipment used to measure quantities (equipment error) and there are differences in how different peoples make the same measurement (human error).

4.         Every physical measurement involves error and every physical measurement is an approximation. The terms precision and accuracy relate to how good an approximation is. We usually think of precision and accuracy as pretty much the same thing. But in science, these words are used in significantly different meanings. Similarly, the margin of error in a measurement is reflected by the number of significant figures in a measured quantity.

Significant Figures

Definition

It is often impossible to obtain the exact value of the quantity under investigation. The numerical value of every observed measurement is an approximation. No physical measurement (such as length, mass, time, volume etc) is ever absolutely correct. The accuracy (reliability) of every measurement is limited by the reliability of the measuring instrument, which is never absolutely reliable. The number of significant figures indicates margin of error in a measurement. A significant figure is one which is known to be reasonably reliable. In elementary measurements in chemistry and physics, the last digit is estimated and is also considered as a significant figure i.e. when significant figures are counted, the last digit is understood to be uncertain.

The Statistically Significant digits (the term digit denotes any one of the ten numerals, including zero) or Meaningful or Reliable digits of a number known with certainty in a measured or calculated quantity which are needed to express the precision of the measurement are known as significant figures. The certain digits of a measured quantity plus one uncertain rightmost last digit are counted as significant figures.

Significance  

The number of significant figures is directly related to the accuracy of measurements that are made in scientific work. Greater the number of significant figures in a quantity, greater the certainty and precision in that quantity. The measured quantity with greater significant figures is more precise than quantity having less significant figures. Hence less significant figures mean low precision.

Summary of Guideline Rules for determining the significant figures

Guideline Rules for determining the significant figures

1. non-zero digits (any digit that is not zero i.e. 1-9 integers) are all significant. e.g

    (i)         72 g has two significant figures.              

   (ii)        12.4 ml has three significant figures.

2. Captive Zeros (i.e. zeros placed between two non-zero digits) always count as significant. (i.e. zeros in the middle of a number are always significant) e.g.

(i)         2.001 g has four significant figures.        

(ii)        16.051 m has five significant figures.

3. Leading Zeros (i.e. zeros precede the nonzero digits) are not significant. (i.e. zeros at         the beginning of a number to the left of the first non-zero digit relating to the value less than    one are non-significant.) e.g.

  (i)         0.0004 ml has 1 significant figures.         

 (ii)        0.00021 kg has 2 significant figures.

4. Trailing zeros (i.e. zeros at the right end of the number) right of decimal point are always significant (i.e. Terminal or Final zeros to the right of the decimal point in a number less than or greater than one are significant.  The assumption is that these zeros would not be shown unless they were significant.) e.g. 55.220 has 5 significant figures (If the value were known to have only 4 significant figures, we would write 55.22). Similarly, 0.21000 has 5 significant figures

     (i)         3.00 m has three significant figures.        

     (ii)  6.900 g has four significant figures.

5.       Trailing zeros in numbers with no decimal point are not significant Zeros (zeros at the end of a number and before the decimal point relating to the value greater than one i.e. trailing zeros left of the decimal point are ambiguous. They may or may not be significant. e.g. 5200 may have 2, 3 or 4 significant figures but we cannot tell from the way is written. It is very poor form to report numbers with an ambiguous degree of uncertainty. This ambiguity can be avoided by using scientific notation. For example, 5.2 x 10³ has 2 significant figures, 5.20 x 10³ has 3 while 5.200 x 10³ has 4 significant figures).

         (i)         720 g has 2 significant figures.                

         (ii) 472000 m has 3 significant figures.

6.        Exact numbers i.e “Counts” and “Defined numbers” have no uncertain digits and they can be assumed to have an infinite number of significant figures and all digits in them are counted as significant. Exact numbers are of two types:

         (a)    Cardinal Numbers (Counting numbers); are exact by definition. A dozen of egg contains exactly 12 eggs, not 12.000001 eggs. Eggs only come in whole numbers. A week has exactly 7 days no 6.9 or 7.0 or 7.1 days.

         (b)     Constants or Mathematical relationships; are exact by definition. e.g. the speed of  light is defined as exactly 299, 792, 458 m/s , and there are exactly 1000 g in a     kilogram.