PHYSICS 30S/40S - GUIDE TO MEASUREMENT ERROR AND SIGNIFICANT FIGURES ACCURACY AND PRECISION An important rule in science is that there is always some degree of uncertainty in measurement. The last digit recorded in any measurement made in science is an estimate and it is therefore uncertain. This uncertainty arises from two major sources: (a) the limitations of the instrument and/or (b) the skill of the observer using the instrument. [Procedural errors usually result in completely inaccurate measurements. Example: Completely inaccurate measurements result when you spill a bit of a solution while you are measuring the total volume of a solution.] Accuracy is defined as a measure of how close a measurement is to an accepted standard (true value) for that measurement. The absolutely true value for a measurement can never be known since it must always be measured (which always involves at least some error). Standard values for many of the constants of science are accepted by the worldwide scientific community after repeated reproducible results are found by appropriate measurement techniques. The smaller the difference between a measured value and the accepted standard value, the greater the accuracy of that measurement. Precision is defined as a measure of how closely two or more measurements of the same thing agree. All of the compared measurements must be made with equal care using the same procedure and instruments. The lower the uncertainty in measuring, the greater the precision. Highly precise measurements (grouped closely together) may still be completely inaccurate (significantly different from the true value or accepted standard). Precision is increased (and uncertainty is decreased) by using a measurement device which determines the measurement to a smaller place value. For example, a meterstick marked off in millimetres (±0.001 m) is more precise than a meterstick only marked off in centimetres (±0.01 m). This means that the mm marked meterstick will yield length measurements which are more reproducible. Furthermore, the estimated digit on the mm meterstick will be for a smaller amount of length, and so the significance of the estimation to the overall value will be reduced. (i.e. a poorly estimated mm is less significant to the overall value than a poorly estimated cm). The relative uncertainty (aka percent uncertainty) indicates the degree to which the precision of the measurement instrument used influences the overall value for the measurement. relative uncertainty = uncertainty in the measurement x 100 % measurement value Example: A length of plastic is measured twice, using two different metersticks, one marked in mm, the other in cm. Calculate the relative uncertainty for each measurement: mm divisions meterstick = 3.456 m cm divisions meterstick = 3.46 m relative uncertainty = 0.001 m X 100 % relative uncertainty = 0.01 m X 100 % 3.456 m 3.46 m = 0.03 % uncertain = 0.3 % uncertain (lower uncertainty = higher precision of mm vs. cm ruler divisions)
Page 2 of 6 SIGNIFICANT FIGURES (aka Significant Digits) Significant figures are used to express the precision to which a measurement was made. Any measured value consists of a number of digits of whose value you are absolutely certain, and one (the last one written) for which there is some doubt. For measured values, the last digit written is an estimate and must be considered uncertain. Example: The mass of a stone was measured as 2.45 g on a sensitive classroom balance. This measurement indicates that the 2 and the 4 are accurately known, but there is uncertainty in the 5. The actual mass of the stone could lie anywhere between 2.44 and 2.46 g, with the 5 hundreths number being an estimated digit. RULES FOR WRITING SIGNIFICANT FIGURES The following rules are for writing measurements to the correct number of significant figures (sig. figs.). The rules must be used whenever any measured values are reported. These rules also apply for the results of any calculations involving measured values. 1. All non-zero numbers are significant. Example: 1234 has 4 sig. figs. 2. All zeros between non-zero digits are significant. Example: 10204 has 5 sig. figs. 3. All zeros to the right of the decimal place, but to the left of a non-zero digit are not significant. These zeros serve only to locate the decimal point. Example: 0.012 has 2 sig. figs. (range: 0.011-0.013, ±0.001) 4. All zeros to the right of the decimal place and to the right of a non-zero digit are significant. These zeros help indicate the precision of the measurement. Example 1: 0.0120 has 3 sig. figs. (range: 0.0119-0.0121, ±0.0001) Example 2: 2.000 has 4 sig. figs. (range: 1.999-2.001, ±0.001) The next two rules (a and b below) can be ambiguous as there are alternative methods of assigning sig. figs. in such cases. The problem can be avoided entirely by writing the values in scientific notation and applying the preceding four sig. fig. rules (#1-4). a) All zeros to the right of a non-zero digit, but to the left of an understood decimal point are not significant. Example: 2300 has 2 sig. figs. (range: 2200-2400, ±100) b) All zeros to the left of the decimal place and to the right of a non-zero digit are significant. Example 1: 2300. has 4 sig. figs. (range: 2299-2301, ±1) An alternate rule to statements (a) and (b) above indicates that the zeros in the example are not significant if there is no unit associated with the measurement (eg. 2300 has only 2 sig figs), but that the zeros are significant if the measurement unit follows the value (eg. 2300 metres has 4 sig figs). The best way to avoid this issue is to represent the values in scientific notation, as below.
Page 3 of 6 NOTE: Students will be expected to use scientific notation to avoid the ambiguous case noted above. Consult the teacher in advance if a question from the text or data from a lab involves values which are ambiguous in this way. Tip: If the value is a whole number with a 0 in the ones column, you MUST write the value in scientific notation to the correct number of sig. figs. All numbers may be conveniently written to the correct number of significant figures B using scientific notation. The exponential term ( x 10 ) is not at all considered for significant figures. The standard and scientific forms of the same measured number must have the same number of sig. figs. Example: If a value such as two thousand metres must be reported with only three 3 significant figures, it must be written as 2.00 x 10 metres. Significant figures are not applied to counted things. For example, a count of 26 people is exactly accurate. There is no doubt about any digit. When performing calculations involving measured values, counted values are not considered when determining the correct number of significant figures for the final answer. The number of significant figures involved in a unitary rate used for conversions between SI units of the same base may be ignored. There is no uncertainty about any digit because these values are theoretical, not measured. The number of significant figures is thus derived only from the original value on which the conversion was performed. Example: 3 123 cm x 10 mm = 1230 mm = 1.23 x 10 mm 1 cm (must be rounded to 3 sig. figs. from the value 123) When determining the correct number of significant figures for a final answer, consider only the measured values actually used for that particular calculation. Usually this means going back to examine the number of sig figs in the original data (as measured in the lab or given in the question). Measured constants introduced into your calculations from other data tables must be considered for their sig figs as well. The final answer to a calculation can never be more precise than the least precise value involved. For all calculations, the answer should be rounded off to the correct number of significant figures only after all calculations have been performed. Otherwise the compounding of the rounded off amounts will result in an inaccurate final result. Note: For a many-part answer, you must use the unrounded answer from question part (a) if its value is needed to calculate for question part (b), etc. Obtain the number of sig figs needed for the final answer from the original data used. When showing your work on an assignment or test, be sure to write the unrounded answer ( all digits from your calculator) and then the rounded answer. It could save you mark deductions: -1/2 for a sig. fig. error, -1 for a calculation error, -1 if the teacher can't tell if the error is due to calculation or rounding. A zero must be placed to the left of the decimal if no other digit occupies that position. This will help ensure that the decimal point is noted by the reader. Example: Write 0.99 cm rather than.99 cm.
Page 4 of 6 RULES FOR ADDITION AND SUBTRACTION OF SIGNIFICANT FIGURES Assume that all values on this page are measured. The units have been left off for clarity. In addition and subtraction, the absolute uncertainty in the final answer must be the same as the absolute uncertainty of the least precise factor. eg. Addition Absolute Uncertainty eg. Subtraction Absolute Uncertainty 13.132 0.001 4.76 0.01 ** + 2.3 0.1 ** - 1.324 0.001 15.432 3.436 Rounded answer: 15.4 Rounded answer: 3.44 ** absolute uncertainty of the least precise factor eg. 0.1 eg. 8.8-0.049 + 3.20 0.051 12 (answer value on a calculator) Rounded answer: 0.1 Rounded answer: 12.0 (Note how 0.049 was insignificant (you must add the tenth's 0) when subtracted from 0.1) RULES FOR MULTIPLICATION AND DIVISION OF SIGNIFICANT FIGURES When multiplying two numbers A x B or dividing A/B the final answer will have the same number of significant figures as does the value which has the least number of sig figs. This rule also applies to exponents (repeated multiplication) and radical functions. eg. Multiplication #Sig. Figs. eg. Division #Sig. Figs. 2 3.021 x 10 4 14.263 5 3 x 4.2 x 10 2** 1.2 2** 6 1.2688 x 10 = 11.88 6 Rounded answer = 1.3 x 10 (2 sig figs) Rounded answer = 12 ** least number of significant figures NOTE: Students may completely ignore the addition/subtraction rule for significant figures whenever the calculation also involves multiplication and/or division functions.
Page 5 of 6 MEASURING AND COMPARING ERROR There are a variety of techniques for determining the accuracy of a measurement or the precision of a set of measurements. The most common calculation formats are given below. These calculations are often used to provide evidence for the accuracy and/or precision of the values measured during a lab activity. (Hint: use these calculations correctly to write more effective Lab Report Conclusions!) Experimental error gives a measure of the accuracy of a measured value. a) Absolute error is the actual difference between a measured value and the accepted value for that measurement. The absolute error value is always positive since the calculation is performed in absolute values. absolute error = *observed value - accepted value* b) Relative error is expressed as a percentage (also called percent error). relative error = absolute error x 100% accepted value Deviation gives a measure of the precision of a group of measured values. The values must be for the measurement of the same quantity. a) Absolute deviation is the difference between a single measured value and the average (mean) of several measurements made in the same way. The absolute deviation value is always positive since the calculation is performed in absolute values. absolute deviation = *observed value - mean of the values* b) Relative deviation is the percentage average deviation of a set of measurements. average absolute deviation relative deviation = of a set of observed values x 100% mean of the values Consider the examples on the reverse of this page for two measurement scenarios involving error calculations in example 1, and deviation calculations in example 2.
Page 6 of 6 Example 1: Comment upon the accuracy of the data obtained in Experiment A as opposed to Experiment B. Experiment A: The observed value was recorded as 4.0 units. The accepted standard for this measurement was 3.0 units. Absolute error = 1.0 unit Relative error = 33% Experiment B: The observed value was recorded as 52 units. The accepted standard for this measurement was 49 units. Absolute error = 3 units Relative error = 6.1% Comments: Though the absolute error in Experiment A was lower, the observed value for Experiment B must be considered more accurate, as its relative error was lower. In general, relative error is a more useful measurement since it reflects the error independently of the magnitude of the accepted standard s value. Example 2: Determine the absolute deviation for each of the values collected into the data table below. Also determine the relative deviation for the entire set of values. Group or Trial Data Value Absolute Deviation 1 3 units 2.4 2 6 units 0.6 ** most precise group or trial 3 7 units 1.6 4 9 units 3.6 5 2 units 3.4 Calculated Average of Data = 5.4 units Average of Abs. Dev. = 2.32 Note: This calculation may be used to determine how precise a particular experimental trial was in a case of repeated trials. This calculation may also be used to determine how precise your lab group s data was when compared to the rest of the data values collected by all of the lab groups in the class. The relative deviation for this group of data = 2.32 x 100% = 42.9% 5.4 Relative deviation values are difficult to draw conclusions from. Other measurements of deviation are often substituted. For example, the standard deviation of the set of data values may be determined and conclusions may be inferred from that value.