Quantitative Discussions of Errors

Most of the graphical analysis we have been discussing can be done without giving very much thought to the way in which the experimental numbers vary.  Quick graphs of data are often plotted in a laboratory situation as experimenters try to get first impressions of what is happening in an experiment and whether or not things are working as expected.  Some quick estimates of uncertainties in measurements are also made using the average and sample standard deviation of a sequence of measurements.  However, these quick checks do not constitute the final analysis of the experimental uncertainties in the experiment.  Ultimately the researcher must do a thorough analysis of the entire experiment to determine what errors are encroaching on the data and how severely they affect any computations that use the measured quantities as input.  Errors in the measurements can propagate through these calculations producing an uncertainty in a final answer being sought from the experiment.

In the discussions so far, we have not dealt with the nature of experimental errors and their effects on computations.  We have noted only that uncertainties occur, and that these show up as a scatter of experimental data about some central value or about some trend line in a graph.  We have shown how to report a number and its uncertainty.  We must now address the more quantitative issues of experimental errors; their types, their behaviors, where they might arise, and how they affect computations and experimental conclusions.  Knowledge of how errors behave can help us in analyzing the probability that we are correct in our conclusions.  We will start with the methods of handling numbers in calculations.

Scientific Notation, Significant Figures, and Round off

Scientific Notation

Scientific notation is a way of writing numbers as a decimal followed by a power of ten.  Expressing numbers this way makes calculations a little easier, and also emphasizes the sizes of numbers more clearly.  With scientific notation, a number is normally written with one digit to the left of the decimal and the others to the right.  The decimal is then said to be in the standard position.  The number is then multiplied by a suitable power of ten to bring the it back to its original size.  For example, 136 is written as 1.36 ´ 10 ², 0.00136 is written as 1.36 ´ 10 ^{- 3}, 654,000,000 = 6.54 ´ 10 ⁸,  0.0000000000654 = 6.54 ´ 10 ^{- 11}.  One method for finding the power of ten that multiplies the number is to count the number of moves required to put the decimal point into the standard position.  If the move is to the left, the power of ten is positive, if the move is to the right, the power of ten is negative.

Scientific notation makes multiplication, division, raising to powers, and taking roots much easier while keeping track of the size of the result.  When multiplying two numbers expressed in scientific notation, multiply the numbers and follow this by a power of ten whose exponent is the sum of the exponents of the individual powers of ten.  Thus

 1.36  ´ 10  ² ´ 6.54 ´ 10 ^{- 11} = 8.89 ´ 10 ^{- 9}.

When dividing, divide the numbers and subtract the exponent of the power of ten in the denominator from the exponent of the power of ten in the numerator.  Thus

1.36  ´ 10 ² / 6.54 ´ 10 ^{- 11}  =  0.208 ´ 10 ^{2 - (- 11)}  =  0.208 ´ 10 ¹³  = 2.08 ´ 10 ¹².

Raising a number to a power is easy.

(6.54 ´ 10 ^{- 11} ) ⁵  =  6.54 ⁵ ´ (10 ^{- 11}) ⁵  =  11,964.34 ´ 10 ^{- 55}  =  1.20 ´ 10 ^{- 51}

To take a square root, express the number so that it is multiplied by an even power of ten.  Then take the square root of the number and multiply it by half the power of ten.  For example

(6.54 ´ 10 ^{- 11 ) 1/2}   =  (65.4 ´ 10 ^{- 12}) ^1/2  =  8.09 ´ 10 ^{- 6}.

To find other roots, make the power of ten divisible by the root.  For example, in finding the cube root of  1.36  ´ 10 ^{- 2}  we get (13.6 ´ 10 ^{- 3}) ^1/3  = 2.39 ´ 10 ^{- 1}.

When adding or subtracting two numbers in scientific notation we must be sure that both numbers have the same power of ten.  Otherwise the decimal points of the actual numbers are not properly aligned.  When adding 1.36 ´ 10 ^{- 2  +}  5.93 ´ 10 ^{- 4}, we note that the two number are not really of the same size.  However we can express this sum as 136. ´ 10 ^{- 4} + 5.93 ´ 10 ^{- 4}.  The result is 141.93 ´ 10 ^{- 4  =}  1.42 ´ 10 ^{- 2}.  We have rounded this result as discussed in the following section.

Precision and Roundoff

The precision of an experiment is implied by the number of significant digits recorded in the result.  As we mentioned above, there is also an uncertainty quoted along with this number.  The number of significant figures in a result is defined as follows:

1.   The leftmost nonzero digit is the most significant digit.

2.   If there is no decimal point, the rightmost nonzero digit is the least significant digit.

3.   If there is a decimal point, the rightmost digit is the least significant digit, even if it is a      zero.

4.   All digits between the least and most significant digits are counted as significant digits.

For example, the following numbers each have four significant digits: 1234, 123400, 123.4, 1001, 1000., 10.10, 0.0001010, 100.0.  If there is no decimal point, there are ambiguities when the rightmost digit is 0.  Thus the number 1010 is considered to have only three significant digits even though the last digit might be significant.  To avoid this ambiguity, it is better to supply decimal points or to write the number in scientific notation.  Thus, if we wish to claim four significant digits in the last example,  we could write it as 1010. or as 1.010 ´ 10 ^{- 3}.

If significant digits are dropped from a number, the last digit retained must be adjusted in such a way as to preserve the best accuracy in further computations involving the number.  To round a number to fewer significant digits than were originally specified, we truncate the number as follows:

1.   If the digit being dropped is greater than 5, increment the preceding digit by 1.

2.   If the digit being dropped is less than 5, do not increment the preceding digit.

3.   If the digit being dropped is 5, increment the preceding digit only if it is odd.

Rules 1 and 2 make sense in that the digit being dropped tells whether or not the preceding digit is more than half way to the next digit.  If the digit being dropped is a 5, however, there is some ambiguity.  The reason for rule 3 is that there is an equal probability of rounding a preceding digit up or truncating it when the digit being dropped is 5.  This avoids a systematic bias in rounding or truncating which, in turn, reduces the errors introduced when a lot of rounded numbers are averaged together.

When using computers or calculators, it is advisable to retain all available digits in the intermediate calculations, and then round only the final results.  You may also notice that some of the more recent graphing calculators do not use the rounding rule 3 but instead increment the last digit when the digit being dropped is greater than or equal to 5.  This is not a serious problem because the calculators and computers usually retain many more significant digits in their computations than are required in the final answer.  However, it is important to note the precision of the numbers that are entering into the calculations.  In general, the final answer in a computation is often no more precise that the least precise number entering into the calculation, but this depends somewhat on the specific kind of calculation and whether or not the errors in the individual measurements that enter into the calculation are independent of each other.  We must now consider how errors propagate through calculations.