Uncertainty in Measurement (Significant Figures, Exponential Notation, Rounding off, Presision, Accuracy, Errors and its types)

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.

All measurements of physical quantities are subject to uncertainties in the measurements. Variability in the results of repeated measurements arises because variables that can affect the measurement result are impossible to hold constant. Even if the "circumstances," could be precisely controlled, the result would still have an error associated with it. This is because the scale was manufactured with a certain level of quality, it is often difficult to read the scale perfectly, fractional estimations between scale marking may be made and etc.  Of course, steps can be taken to limit the amount of uncertainty but it is always there. 

In order to interpret data correctly and draw valid conclusions the uncertainty must be indicated and dealt with properly. For the result of a measurement to have clear meaning, the value cannot consist of the measured value alone.  An indication of how precise and accurate the result is must also be included.  

Thus, the result of any physical measurement has two essential components:

(1)    A numerical value (in a specified system of units) giving the best estimate possible of the quantity measured, and

(2)    the degree of uncertainty associated with this estimated value. 

Uncertainty is a parameter characterizing the range of values within which the value of the measurand can be said to lie within a specified level of confidence. 

For example, a measurement of the width of a table might yield a result such as 95.3 +/- 0.1 cm.  This result is basically communicating that the person making the measurement believe the value to be closest to 95.3cm but it could have been 95.2 or 95.4 cm. The uncertainty is a quantitative indication of the quality of the result.  It gives an answer to the question, "how well does the result represent the value of the quantity being measured?"

 

 Significant Figures/Significant Digits 

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 term digit denotes any one of the ten numerals, including zero.

The Statistically Significant digits or Meaningful or Reliable digits of a number known with certainty in a measured (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.

 

Significant figures are those meaningful digits that are known with certainty. They indicate uncertainty in an experiment or calculated value. For example, if 15.6 mL is the result of an experiment, then 15 is certain while 6 is uncertain, and the total number of significant figures are 3.  Hence, significant figures are defined as the total number of digits in a number including the last digit that represents the uncertainty of the result.

 

Significance                                                          

The number of significant figures is directly related to the precision and 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 figure. Hence less significant figure means low precision.

Summary of Guideline Rules for determining the significant figures

Guideline Rules for determining the significant figures                 

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

(i) 67 m has two significant figures.
(ii) 435 kg 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)     4090 mg has three significant figures.    

(ii)    28.073 g 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. Such zeroes merely indicate the position of the decimal point) 

e.g.
(i) 0.0006 has 1 significant figures.
(ii) 0.00027 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 has three significant figures.    

(ii)    5.900 has four significant figures.

5.     Trailing zeros in numbers with no decimal point are not significant Zeros

(i)     140 miles has 2 significant figures.         

(ii)    786000 calories has 3 significant figures.

Correct Rule

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 and are not necessarily significant. They may or may not be significant. e.g.

(i)     5200 miles 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).

(ii)    50,600 calories may have 3,4 or 5 significant figures.

This potential ambiguity can be avoided by using standard exponential or scientific notation. For example,

(i)     Depending upon whether the number of significant figures is 2,3 or 4, we would write 5200 calories   as

5.2 x 10³ calories     (2 significant figures)

5.20 x 10³ calories   (3 significant figures)

5.200 x 10³ calories (4 significant figures)

6.     Exact numbers i.e. “Counts” and “Defined numbers” have infinite number of significant digits. They 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. 

 

Explanation of Rule # 1 … Non-zero Digits are always significant

If we measure something by using a device (like ruler, thermometer, triple beam balance etc.), we get a number, then we have made a measurement decision and act of measuring gives significance to that particular numeral or digit in the overall value we obtain. Hence a number like 58.97 g would have four significant figures and 3.64 m would have three. (The problem comes with numbers like o.0008740 or 45.002.

Explanation of Rule # 2 …. Captive or Captured zeros are always significant

Any zeros between two significant digits are significant. These zeroes are sometimes called captured zeroes.

Suppose you had a number like 406. By the first rule, the 4 and the 6 are significant. However, to make a measurement decision on the 4 (in the hundred’s place) and the 6 (in the unit’s place), you had to have made a decision on the ten’s place. The scale for this number would have hundreds and tens marked with estimation made in the unit’s place. Like this,

Explanation of Rule # 3, 4…. A final or trailing zeros in the decimal portion only are significant. 

0.00500

0.03040

2.30 x 10^-5

4.500 x 10¹²

Use of significant digits in addition and subtraction

The result of addition or subtraction must carry the same number of digits to the right of the decimal as any of the original number with fewest digits to the right of the decimal. (i.e. the result has the same number of decimal places as the term having smallest number of decimal places or the least precise measurement used in the calculation). This means that only as many digits are to be retained to the right of the decimal point in the answer as the number with fewest digits to the right of the decimal. 

Note for addition and subtraction, the decimal places are counted. 

 Example # 1:  Let us add 15.020, 9986.0 and 3.518

The result should be rounded off to one decimal place because 9986.0 has smallest number of decimal place (only one).

Example # 2:  Let us add 12.12, 18.0 and 1.014

Use of significant digits in multiplication and division

The result of multiplication and division cannot have more significant figures than the least precise term of known quantity in the calculation (i.e. the number of significant figures in the final product or quotient contains the same number of significant figures as the original number with the fewest significant figures called least precise factor; LPF) This means that number obtained as a result of multiplication or division of two or more numbers must have no more significant figures than the least number of significant figures in any of the multiplied or divided term. 

Note that for multiplication and division, significant figures are counted.

 

Example #1: Solve 1.32 x 4.421

The result is rounded off to three significant figures because least precise factor is 1.32 which has 3 significant figures.