L1 - Uncertainties and Errors

objectives

Understandings

• Qualitative data includes all non-numerical information obtained from observations not from measurement.
• Quantitative data are obtained from measurements, and are always associated with random errors/uncertainties, determined by the apparatus, and by human limitations such as reaction times.
• Propagation of random errors in data processing shows the impact of the uncertainties on the final result.
• Experimental design and procedure usually lead to systematic errors in measurement, which cause a deviation in a particular direction.
• Repeat trials and measurements will reduce random errors but not systematic errors.

Applications and Skills

• Distinction between random errors and systematic errors.
• Record uncertainties in all measurements as a range (+) to an appropriate precision.
• Discussion of ways to reduce uncertainties in an experiment.
• Propagation of uncertainties in processed data, including the use of percentage uncertainties.
• Discussion of systematic errors in all experimental work, their impact on the results and how they can be reduced.
• Estimation of whether a particular source of error is likely to have a major or minor effect on the final result.
• Calculation of percentage error when the experimental result can be compared with a theoretical or accepted result.
• Distinction between accuracy and precision in evaluating results.

Guidance

• The number of significant figures in a result is based on the figures given in the data. When adding or subtracting, the final answer should be given to the least number of decimal places. When multiplying or dividing the final answer is given to the least number of significant figures.
• Note that the data value must be recorded to the same precision as the random error.
• SI units should be used throughout the programme.

MEASUREMENT UNCERTAINTY, ACCURACY, AND PRECISION

• Counting is the only type of measurement that is free from uncertainty, provided the number of objects being counted does not change while the counting process is underway.
• The result of such a counting measurement is an example of an exact number.
• If we count eggs in a carton, we know exactly how many eggs the carton contains.
• The numbers of defined quantities are also exact.
• By definition, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilogram.
• Quantities derived from measurements other than counting, however, are uncertain to varying extents due to practical limitations of the measurement process used.

SIGNIFICANT FIGURES IN MEASUREMENT

• The numbers of measured quantities, unlike defined or directly counted quantities, are not exact.
• To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid.

• Refer to the illustration in Figure 1.
• The bottom of the meniscus in this case clearly lies between the 21 and 22 markings, meaning the liquid volume is certainly greater than 21 mL but less than 22 mL.
• The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid’s volume would be 21.6 mL.
• In the number 21.6, then, the digits 2 and 1 are certain, but the 6 is an estimate.
• Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 22-mL mark and estimate this digit to be 7.
• Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain.
• In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division.
• The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.

• This concept holds true for all measurements, even if you do not actively make an estimate.
• If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g.
• The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams.
• The quarter weighs about 6.72 grams, with a nominal uncertainty in the measurement of ± 0.01 gram.
• If we weigh the quarter on a more sensitive balance, we may find that its mass is 6.723 g.
• This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram.
• Every measurement has some uncertainty, which depends on the device used (and the user’s ability).
• All of the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits.
• Note that zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows “120,” then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.

• Whenever you make a measurement properly, all the digits in the result are significant.
• But what if you were analyzing a reported value and trying to determine what is significant and what is not?
• Well, for starters, all nonzero digits are significant, and it is only zeros that require some thought.
• We will use the terms “leading,” “trailing,” and “captive” for the zeros and will consider how to deal with them.

• Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right.
• This is the number of significant figures in the measurement unless the last digit is a trailing zero lying to the left of the decimal point.

• Captive zeros result from measurement and are therefore always significant.
• Leading zeros, however, are never significant—they merely tell us where the decimal point is located.

• The leading zeros in this example are not significant.
• We could use exponential notation (as described in Appendix B) and express the number as 8.32407 × 10−3; then the number 8.32407 contains all of the significant figures, and 10−3 locates the decimal point.

• The number of significant figures is uncertain in a number that ends with a zero to the left of the decimal point location.
• The zeros in the measurement 1,300 grams could be significant or they could simply indicate where the decimal point is located.
• The ambiguity can be resolved with the use of exponential notation: 1.3 × 103 (two significant figures), 1.30 × 103 (three significant figures, if the tens place was measured), or 1.300 × 103 (four significant figures, if the ones place was also measured).
• In cases where only the decimal-formatted number is available, it is prudent to assume that all trailing zeros are not significant.

• When determining significant figures, be sure to pay attention to reported values and think about the measurement and significant figures in terms of what is reasonable or likely when evaluating whether the value makes sense.
• For example, the official January 2014 census reported the resident population of the US as 317,297,725. Do you think the US population was correctly determined to the reported nine significant figures, that is, to the exact number of people?
• People are constantly being born, dying, or moving into or out of the country, and assumptions are made to account for the large number of people who are not actually counted.
• Because of these uncertainties, it might be more reasonable to expect that we know the population to within perhaps a million or so, in which case the population should be reported as 3.17 × 108 people.

SIGNIFICANT FIGURES IN CALCULATIONS

• A second important principle of uncertainty is that results calculated from a measurement are at least as uncertain as the measurement itself.
• We must take the uncertainty in our measurements into account to avoid misrepresenting the uncertainty in calculated results.
• One way to do this is to report the result of a calculation with the correct number of significant figures, which is determined by the following three rules for rounding numbers:

1. When we add or subtract numbers, we should round the result to the same number of decimal places as the number with the least number of decimal places (the least precise value in terms of addition and subtraction).
2. When we multiply or divide numbers, we should round the result to the same number of digits as the number with the least number of significant figures (the least precise value in terms of multiplication and division).
3. If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, we “round down” and leave the retained digit unchanged; if it is more than 5, we “round up” and increase the retained digit by 1; if the dropped digit is 5, we round up or down, whichever yields an even value for the retained digit. (The last part of this rule may strike you as a bit odd, but it’s based on reliable statistics and is aimed at avoiding any bias when dropping the digit “5,” since it is equally close to both possible values of the retained digit.)

The following examples illustrate the application of this rule in rounding a few different numbers to three significant figures:

• 0.028675 rounds “up” to 0.0287 (the dropped digit, 7, is greater than 5)
• 18.3384 rounds “down” to 18.3 (the dropped digit, 3, is less than 5)
• 6.8752 rounds “up” to 6.88 (the dropped digit is 5, and the retained digit is even)
• 92.85 rounds “down” to 92.8 (the dropped digit is 5, and the retained digit is even)

ACCURACY AND PRECISION

• Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to know both the precision and the accuracy of their results.
• Measurements are said to be precise if they yield very similar results when repeated in the same manner.
• A measurement is considered accurate if it yields a result that is very close to the true or accepted value.
• Precise values agree with each other; accurate values agree with a true value.
• These characterizations can be extended to other contexts, such as the results of an archery competition

• Suppose a quality control chemist at a pharmaceutical company is tasked with checking the accuracy and precision of three different machines that are meant to dispense 10 ounces (296 mL) of cough syrup into storage bottles.
• She proceeds to use each machine to fill five bottles and then carefully determines the actual volume dispensed, obtaining the results tabulated in Table 5.

• Considering these results, she will report that dispenser #1 is precise (values all close to one another, within a few tenths of a milliliter) but not accurate (none of the values are close to the target value of 296 mL, each being more than 10 mL too low).
• Results for dispenser #2 represent improved accuracy (each volume is less than 3 mL away from 296 mL) but worse precision (volumes vary by more than 4 mL).
• Finally, she can report that dispenser #3 is working well, dispensing cough syrup both accurately (all volumes within 0.1 mL of the target volume) and precisely (volumes differing from each other by no more than 0.2 mL).

Further Explanation of Uncertainty

• Even electronic instrumentation has to 'decide' on the final decimal place. If an electronic balance measures to two decimal places, it has to choose the second decimal place by considering the (unseen) third decimal.
• If the third is 5 or greater than the second decimal is 'rounded up' If the third decimal is 4 or less, the second is left unchanged.
• We don't see this operation in practice, but it means that there is always an uncertainty of +/- 0.005 in a two decimal place measure.
• Glassware used to measure volume of solutions may be measuring cylinders, pipettes or burettes. All of these instruments require human judgement to gauge exactly at which level the solution lies. A solution is measured to a given 'mark' on the glassware. This requires our senses to be as accurate as possible, but everybody has human limitations. We just have to accept that there is an uncertainty when we record our results.
• Similarly the manufacturer of the glassware is conditioned by the accuracy of the manufacturing process; the machines made to manufacture instrumentation also have their own inaccuracy.
• And the conditions affect the measurements. Pipettes, for example are calibrated to measure solutions at 20ºC. This is rarely the exact temperature of a solution, once again introducing an inaccuracy.
• So what do we do with all of this inaccuracy? The answer is that we have to accept and record it as part of our experimentation, to understand that it is ever-present and to try and quantify it so that we know the upper and lower limits of each measurement taken.

Human Error

• This is not what students usually understand by the term. Human error does not necessarily indicate that a mistake has been made.
• To differentiate between mistakes in experimentation and normal human limitation, some authors use the term 'blunders' instead of error to mean mistakes.
• This gives a sense that the experimenter has done something that he/she should not have, ie spilled some solution to be titrated, or dropped something onto the floor.
• Unfortunately the term is often used in everyday life to suggest an incorrect action taken, or mistake made. "The crash was caused by human error"
• In scientific terms, human error means the limitations inherent in measurements made by human agency, such as timing the disappearance of a cross drawn onto the side of a beaker due to the formation of a precipitate, judging the end-point of a titration etc.

Systematic Errors

• These are errors that are consistently produced in the course of an experiment by poor design, or some inherent fault or limitation in the apparatus. They may also be due to poor experimental techniques.
• These errors can never be quantified completely, nor can the experiments be made more reliable by repetition. However, they can be improved by changing the experimental design, by improving the measuring techniques etc.

Typical systematic errors include

• Reading the position of the meniscus of a liquid incorrectly
• Losing heat to the environment in a thermochemistry experiment
• Using dirty pipettes, which retain drops of solution, reducing the volume delivered.

Random Errors

• Random errors are produced when measurements give readings that are either higher or lower indiscriminately and with equal probability.
• The effect of random error can be reduced by repetition.
• One example of random error is gauging the end-point of a titration.
• When using methyl orange as an indicator the colour change is from yellow to pink, with the endpoint being a shade of orange.
• It is difficult to be consistent about the exact end-point in these circumstances.

Reducing Random Error

• If the inaccuracy is truly random there is an equal possibility of the measurement being greater than or less than the accepted accurate value.
• Repetition in this case would be likely to give an average closer to the 'actual' value than any specific reading.

Inaccuracy Limits

• The accepted method for recording inaccuracy is to use the plus/minus sign, ±, to indicate the possible range of uncertainty of the reading shown.
• This may be expressed as a percentage of the whole measurement.
• The sign means that there is an equal possibility of the reading having either a higher limit or a lower limit.
• Certain instruments and pieces of apparatus have the inaccuracy range, also called the tolerance, written on the apparatus, along with the conditions under which the inaccuracy has been determined.

For example: A grade B 250 ml volumetric flask may have the following legend.

250 ml ± 0.30ml @ 20ºC.

• This means that when the apparatus is used exactly in the correct way, there is still an uncertainty of plus 0.30 ml to minus 0.30 ml in the actual volume of solution contained.
• In other words the actual volume of the solution could be anywhere between 250.30 and 249.70 ml.

Recording Inaccuracy

• When recording raw data it is important to also record the degree of accuracy of the measurements taken.
• If a standard piece of apparatus is used it may have the inaccuracy written on it.
• However, in many cases a value judgement must be applied to give a reasonable assessment of the degree of accuracy involved.
• A simple rule of thumb is to say use the idea that we are able to read a measurement to within half of the value of the smallest graduation of a visual measuring instrument, such as a ruler, burette or measuring cylinder.
• We must also remember that if the difference between two readings is being used then effectively two readings are taken and the overall inaccuracy is doubled.
• This is the case in measuring volumes of liquid delivered by a burette.

Absolute Uncertainty

• This is the actual uncertainty in a reading taken using a specific piece of apparatus.
• The value depends on the apparatus used.

• Some pieces of apparatus do not have an inaccuracy written on them, in which case you must make an assessment based on how accurately you believe you can measure using the apparatus.
• You should err on the side of caution.

• Electronic digital measuring apparatus such as electronic balances calculate the measurement for the operator.
• BUT they have to electronically round up the decimal place after the last decimal on the readout.
• An electronic balance that measures to 2 decimal places calculates the second decimal place according to the value of the third (unseen) decimal.
• If the third decimal is 5, or greater, the balance rounds up the second decimal place. If the third decimal is 4 or less than the second decimal stays the same.

• In summary there are three ways that you can obtain an absolute value for an uncertainty:

1. Directly from the measuring instrument
2. By judgement of activity undertaken
3. From the 'missing' decimal place of digital readouts

Percentage Uncertainty

• A percentage, by definition, is a value out of a potential hundred.
• The percentage is calculated by taking the absolute error in a measurement and dividing by the value of the measurement itself. This is then multiplied by one hundred.
• A single reading cannot have a percentage uncertainty, but a measured value such as volume, time or mass does.
• In other words the single reading from a burette cannot be expressed as a percentage uncertainty, while the absolute uncertainty of the volume measured from a burette does have a percentage uncertainty.

Propagation of Errors

• Whenever experiments are carried out involving more than one step, the uncertainties of each step must accumulate.
• It is not usually appropriate to add uncertainties in measurements of different types as absolute errors.
• Instead, the percentage error accrued at each measurement should be added together.
• Once all of the percentage errors have been added together, this percentage can be applied to any final answer to determine the absolute uncertainty in the final value.
• The best way to understand the process is to see how the uncertainties are propagated through a simple procedure, such as the preparation of a standard solution of potassium hydrogen phthalate (relative mass 204.23).

Approximation of Errors

• In situations where the errors arising from one measurement are much larger than from another, it is a safe approximation to ignore the smaller value.
• In the example given above both of the error margins were fairly similar and carry equal importance.
• However, if the expermiment is continued by extracting a 25 ml aliquot using a pipette and timing a reaction, such as the time taken for a sulfur precipitate to appear and obscure a mark placed below the reaction vessel when 1.0 ml of 2 mol dm-3 hydrochloric acid is added.
  • The uncertainty in the pipette = 25 ± 0.04 ml
  • The uncertainty in HCl addition = 1.0 ± 0.1 ml
  • The uncertainty in time taken = 32 ± 2 s
  • The percentage uncertainty in the pipette = 0.04/25 x 100 = 0.16%
  • The percentage uncertainty in the HCl volume = 0.1/2 x 100 = 5%
  • The percentage uncertainty in the time = 2/32 x 100 = 6.25%
• We can see that the uncertainly in the pipette measurement is far less than that of either the HCl volume or the time.
• It would be a reasonable approximation to ignore the pipette uncertainty when calculating the overall uncertainty in the final value.
• It is important to realise that errors arise from all sources, but that when one error is much larger than the others they become relatively insignificant.