## Measurement errors

Sources of uncertainties Students should know that every measurement has some inherent uncertainty. The important question to ask is whether an experimenter can be confident that the true value lies in the range that is predicted by the uncertainty that is quoted. Good experimental design will attempt to reduce the uncertainty in the outcome of an experiment. The experimenter will design experiments and procedures that produce the least uncertainty and to provide a realistic uncertainty for the outcome. In assessing uncertainty, there are a number of issues that have to be considered. These include: the resolution of the instrument used the manufacturer’s tolerance on instruments the judgments that are made by the experimenter the procedures adopted (eg repeated readings) the size of increments available (eg the size of drops from a pipette). Numerical questions will look at a number of these factors. Often, the resolution will be the guiding factor in assessing a numerical uncertainty. There may be further questions that would require candidate to evaluate arrangements and procedures. Students could be asked how particular procedures would affect uncertainties and how they could be reduced by different apparatus design or procedure A combination of the above factors means that there can be no hard and fast rules about the actual uncertainty in a measurement. What we can assess from an instrument’s resolution is the minimum possible uncertainty. Only the experimenter can assess the other factors based on the arrangement and use of the apparatus and a rigorous experimenter would draw attention to these factors and take them into account. Readings and measurements It is useful, when discussing uncertainties, to separate measurements into two forms: Readings Measurements the values found from a single judgement when using a piece of equipment the values taken as the difference between the judgements of two values. Examples When using a thermometer, a student only needs to make one judgement (the height of the liquid). This is a reading. It can be assumed that the zero value has been correctly set. For burettes and rulers, both the starting point and the end point of the measurement must be judged, leading to two uncertainties. The following list is not exhaustive, and the way that the instrument is used will determine whether the student is taking a reading or a measurement. Reading (one judgement only) Measurement (two judgements required) thermometer ruler pH meter stopwatch top pan balance analogue meter measuring cylinder burette volumetric flask The uncertainty in a reading when using a particular instrument is no smaller than plus or minus half of the smallest division or greater. For example, a temperature measured with a thermometer is likely to have an uncertainty of ±0.5 °C if the graduations are 1 °C apart. Students should be aware that readings are often written with the uncertainty. An example of this would be to write a voltage as (2.40 ± 0.01) V. It is usual for the uncertainty quoted to be the same number of significant figures as the value. Unless there are good reasons otherwise (eg an advanced statistical analysis), students at this level should quote the uncertainty in a measurement to the same number of decimal places as the value. Measurement example: length When measuring length, two uncertainties must be included: the uncertainty of the placement of the zero of the ruler and the uncertainty of the point the measurement is taken from. As both ends of the ruler have a ±0.5 scale division uncertainty, the measurement will have an uncertainty of ±1 division. ruler For most rulers, this will mean that the uncertainty in a measurement of length will be ±1 mm. This “initial value uncertainty” will apply to any instrument where the user can set the zero (incorrectly), but would not apply to equipment such as balances or thermometers where the zero is set at the point of manufacture. In summary The uncertainty of a reading (one judgement) is at least ±0.5 of the smallest scale reading. The uncertainty of a measurement (two judgements) is at least ±1 of the smallest scale reading. The way measurements are taken can also affect the uncertainty. Measurement example: the extension of a spring Measuring the extension of a spring using a metre ruler can be achieved in two ways. 1. Measuring the total length unloaded and then loaded. Four readings must be taken for this: The start and end point of the unloaded spring’s length and the start and end point of the loaded spring’s length. The minimum uncertainty in each measured length is 1 mm using a meter ruler with 1 mm divisions (the actual uncertainty is likely to be larger due to parallax in this instance). The extension would be the difference between the two readings so the minimum uncertainty would be 2 mm. 2. Fixing one end and taking a scale reading of the lower end. Two readings must be taken for this: the end point of the unloaded spring’s length and the end point of the loaded spring’s length. The start point is assumed to have zero uncertainty as it is fixed. The minimum uncertainty in each reading would be 0.5 mm, so the minimum extension uncertainty would be 1 mm. Even with other practical uncertainties this second approach would be better. Realistically, the uncertainty would be larger than this and an uncertainty in each reading of 1 mm or would be more sensible. This depends on factors such as how close the ruler can be mounted to the point as at which the reading is to be taken. Other factors There are some occasions where the resolution of the instrument is not the limiting factor in the uncertainty in a measurement. Best practice is to write down the full reading and then to write to fewer significant figures when the uncertainty has been estimated. Examples A stopwatch has a resolution of hundredths of a second, but the uncertainty in the measurement is more likely to be due to the reaction time of the experimenter. Here, the student should write the full reading on the stopwatch (eg 12.20 s), carry the significant figures through for all repeats, and reduce this to a more appropriate number of significant figures after an averaging process later. If a student measures the length of a piece of wire, it is very difficult to hold the wire completely straight against the ruler. The uncertainty in the measurement is likely to be higher than the ±1 mm uncertainty of the ruler. Depending on the number of “kinks” in the wire, the uncertainty could be reasonably judged to be nearer ± 2 or 3 mm. The uncertainty of the reading from digital meters depends on the electronics and is not strictly the last figure in the readout. Manufacturers usually quote the percentage uncertainties for the different ranges. Unless otherwise stated it may be assumed that 0.5 in the least significant digit is to be the uncertainty in the measurement. This would generally be rounded up to 1 of the least significant digit when quoting the value and the uncertainty together. For example (5.21 0.01) V. If the reading fluctuates, then it may be necessary to take a number of readings and do a mean and range calculation. Uncertainties in given values In written exams, students can assume the uncertainty to be 1 in the last significant digit. For example, if a boiling point is quoted as being 78 °C, the uncertainty could be assumed to be 1 °C. The uncertainty may be lower than this but without knowing the details of the experiment and procedure that lead to this value there is no evidence to assume otherwise. Repeated measurements Repeating a measurement is a method for reducing the uncertainty. With many readings one can also identify those that are exceptional (that are far away from a significant number of other measurements). Sometimes it will be appropriate to remove outliers from measurements before calculating a mean. On other occasions, particularly in Biology, outliers are important to include. For example, it is important to know that a particular drug produces side effects in one person in a thousand. If measurements are repeated, the uncertainty can be calculated by finding half the range of the measured values. For example: Repeat 1 2 3 4 Distance / m 1.23 1.32 1.27 1.22 1.32 – 1.22 = 0.10 therefore Mean distance: (1.26 ± 0.05) m Percentage uncertainties The percentage uncertainty in a measurement can be calculated using: = 100% The percentage uncertainty in a repeated measurement can also be calculated using: = 100% Further examples: Example 1. Some values for diameter of a wire Repeat 1 2 3 4 Diameter / mm 0.35 0.37 0.36 0.34 The exact values for the mean is 0.355 mm and for the uncertainty is 0.015 mm This could be quoted as such or recorded as 0.36 0.02 mm given that there is a wide range and only 4 readings. Given the simplistic nature of the analysis then giving the percentage uncertainty as 5% or 6% would be acceptable. Example 2. Different values for the diameter of a wire Repeat 1 2 3 Diameter / mm 0.35 0.36 0.35 The mean here is 0.3533 mm with uncertainty of 0.005 mm The percentage uncertainty is 1.41% so may be quoted as 1% but really it would be better to obtain further data. Titration Titration is a special case where a number of factors are involved in the uncertainties in the measurement. Students should carry out a rough titration to determine the amount of titrant needed. This is to speed up the process of carrying out multiple samples. The value of this titre should be ignored in subsequent calculations. In titrations one single titre is never sufficient. The experiment is usually done until there are at least two titres that are concordant ie within a certain allowable range, often 0.10 cm3 . These values are then averaged. For example: Titration Rough 1 2 3 Final reading 24.20 47.40 24.10 47.35 Initial reading 0.35 24.20 0.65 24.10 Titre / cm3 23.85 23.20 23.45 23.25 Here, titres 1 and 3 are within the allowable range of 0.10 cm3 so are averaged to 23.23 cm3 . Unlike in some Biology experiments (where anomalous results are always included unless there is good reason not to), in Chemistry it is assumed that repeats in a titration should be concordant. If they are not then there is likely to have been some experimental error. For example, the wrong volume of solution added from the burette, the wrong amount of solution measuring the pipette or the end point might have been misjudged. The total error in a titre is caused by three factors: Error Uncertainty Reading the burette at the start of the titration Half a division = ±0.05 cm3 Reading the burette at the end of the titration Half a division = ±0.05 cm3 Judging the end point to within one drop Volume of a drop = ± 0.05 cm3 Total ± 0.15 cm3 This will, of course, depend on the glassware used, as some burettes are calibrated to a higher accuracy than others. Uncertainties from gradients To find the uncertainty in a gradient, two lines should be drawn on the graph. One should be the “best” line of best fit. The second line should be the steepest or shallowest gradient line of best fit possible from the data. The gradient of each line should then be found. The uncertainty in the gradient is found by: percentage uncertainty = |best gradient−worst gradient| best gradient × 100% Note the modulus bars meaning that this percentage will always be positive. In the same way, the percentage uncertainty in the y-intercept can be found: percentage uncertainty = |best − worst | be × 100% 5 10 15 20 25 30 35 0 20 40 60 80 100 Best gradient Worst gradient could be either: Steepest gradient possible or Shallowest gradient possible. Combining uncertainties Percentage uncertainties should be combined using the following rules: Combination Operation Example Adding or subtracting values = + Add the absolute uncertainties Δa = Δb + Δc Initial volume in burette = 3.40 ± 0.05 cm3 Final volume in burette = 28.50 ± 0.05 cm3 Titre = 25.10 ± 0.10 cm3 Multiplying values = × Add the percentage uncertainties εa = εb + εc Mass = 50.0 ± 0.1 g Temperature rise (T) = 10.9 ± 0.1 °C Percentage uncertainty in mass = 0.20% Percentage uncertainty in T = 0.92% Heat change = 2278 J Percentage uncertainty in heat change = 1.12% Absolute uncertainty in heat change = ± 26 J (Note – the uncertainty in specific heat is taken to be zero) Dividing values = Add the percentage uncertainties εa = εb + εc Mass of salt in solution= 100 ± 0.1 g Volume of solution = 250 ± 0.5 cm3 Percentage uncertainty in mass = 0.1% Percentage uncertainty in volume = 0.2% Concentration of solution = 0.400 g cm–3 Percentage uncertainty of concentration = 0.3% Absolute uncertainty of concentration = ± 0.0012 g cm–3 Power rules = Multiply the percentage uncertainty by the power εa = c × εb Concentration of H+ ions = 0.150 ± 0.001 mol dm–3 rate of reaction = k[H+ ] 2 = 0.207 mol dm–3 s –1 (Note – the uncertainty in k is taken as zero and its value in this reaction is 0.920 dm6 mol–2 s –1 ) Percentage uncertainty in concentration = 0.67% Percentage uncertainty in rate = 1.33% Absolute uncertainty in rate = ± 0.003 mol dm–3 s –1 Note: Absolute uncertainties (denoted by Δ) have the same units as the quantity. Percentage uncertainties (denoted by ε) have no units. Uncertainties in trigonometric and logarithmic functions will not be tested in A-level exams.

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PS 2.3 Evaluate results and draw conclusions with reference to measurement uncertainties and errors

PS 3.3 Consider margins of error, accuracy and precision of data

## Variables

control variables A control variable is one which may, in addition to the independent variable, affect the outcome of the investigation and therefore has to be kept constant or at least monitored. dependent variables The dependent variable is the variable of which the value is measured for each and every change in the independent variable. independent variables The independent variable is the variable for which values are changed or selected by the investigator.

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PS 2.4 Identify variables including those that must be controlled

## Graphing

Graphing skills can be assessed both in written papers for the A-level grade and by the teacher during the assessment of the endorsement. Students should recognise that the type of graph that they draw should be based on an understanding of the type of data they are using and the intended analysis of the data. The rules below are guidelines which will vary according to the specific circumstances. Labelling axes Axes should always be labelled with the quantity being measured and the units. These should be separated with a forward slash (solidus): time / seconds length / mm Axes should not be labelled with the units on each scale marking. Data points Data points should be marked with a cross. Both and marks are acceptable, but care should be taken that data points can be seen against the grid. Error bars can take the place of data points where appropriate. Scales and origins Students should attempt to spread the data points on a graph as far as possible without resorting to scales that are difficult to deal with. Students should consider: the maximum and minimum values of each variable the size of the graph paper whether 0.0 should be included as a data point how to draw the axes without using difficult scale markings (eg multiples of 3, 7, 11 etc) In exams, the plots should cover at least half of the grid supplied for the graph. Please note that in section M, many generic graphs are used to illustrate the points made. For example, the following three graphs are intended to illustrate the information above relating to the spread of data points on a graph. Students producing such graphs on the basis of real practical work or in examination questions would be expected to add in axes labels and units. 0 5 10 15 20 25 30 35 0 20 40 60 80 100 This graph has well-spaced marking points and the data fills the paper. Each point is marked with a cross (so points can be seen even when a line of best fit is drawn) This graph is on the limit of acceptability. The points do not quite fill the page, but to spread them further would result in the use of awkward scales. At first glance, this graph is well drawn and has spread the data out sensibly. However, if the graph were to later be used to calculate the equation of the line, the lack of a y-intercept could cause problems. Increasing the axes to ensure all points are spread out but the y-intercept is also included is a skill that requires practice and may take a couple of attempts. Lines of best fit Lines of best fit should be drawn when appropriate. Students should consider the following when deciding where to draw a line of best fit: Are the data likely to be following an underlying equation (for example, a relationship governed by a physical law)? This will help decide if the line should be straight or curved. Are there any anomalous results? There is no definitive way of determining where a line of best fit should be drawn. A good rule of thumb is to make sure that there are as many points on one side of the line as the other. Often the line should pass through, or very close to, the majority of plotted points. Graphing programs can sometimes help, but tend to use algorithms that make assumptions about the data that may not be appropriate. Lines of best fit should be continuous and drawn as a thin pencil line that does not obscure the points below and does not add uncertainty to the measurement of gradient of the line. Not all lines of best fit go through the origin. Students should ask themselves whether a 0 in the independent variable is likely to produce a 0 in the dependent variable. This can provide an extra and more certain point through which a line must pass. A line of best fit that is expected to pass through (0,0), but does not, would imply some systematic error in the experiment. This would be a good source of discussion in an evaluation. Dealing with anomalous results At GCSE, students are often taught to automatically ignore anomalous results. At A-level, students should think carefully about what could have caused the unexpected result and therefore whether it is anomalous. A student might be able to identify a reason for the unexpected result and so validly regard it as an anomaly. For example, an anomalous result might be explained by a different experimenter making the measurement, a different solution or a different measuring device being used. In the case where the reason for an anomalous result occurring can be identified, the result should be recorded and plotted but may then be ignored. Anomalous results should also be ignored where results are expected to be the same (for example, in a titration in chemistry). Where there is no obvious error and no expectation that results should be the same, anomalous results should be included. This will reduce the possibility that a key point is being overlooked. Please note: when recording results it is important that all data are included. Anomalous results should only be ignored at the data analysis stage. It is best practice whenever an anomalous result is identified for the experiment to be repeated. This highlights the need to tabulate and even graph results as an experiment is carried out. Measuring gradients When finding the gradient of a line of best fit, students should show their working by drawing a triangle on the line. The hypotenuse of the triangle should be at least half as big as the line of best fit. = ∆ ∆ When finding the gradient of a curve, eg, the rate of reaction at a time that was not sampled, students should draw a tangent to the curve at the relevant value of the independent variable (x-axis). Use of a set square to draw a triangle over this point on the curve can be helpful in drawing an appropriate tangent. 25 26 27 28 29 30 31 32 33 34 35 20 40 60 80 100 Δy The line of best fit here has an equal number of points on both sides. It is not too wide so points can be seen under it. The gradient triangle has been drawn so the hypotenuse includes more than half of the line. In addition, it starts and ends on points where the line of best fit crosses grid lines so the points can be read easily (this is not always possible).The equation of a straight line Students should understand that the following equation represents a linear relationship. y = mx + c Where y is the dependent variable, m is the gradient, x is the independent variable and c is the yintercept. Δy = 28 – 9 = 19 Δx = 90 – 10 = 80 gradient = 19 / 80 = 0.24 (2 sf) y-intercept = 7.0 equation of line: y = 0.24 x + 7.0. Testing relationships Sometimes it is not clear what the relationship between two variables is y 2 ∝ x. A quick way to find a possible relationship is to manipulate the data to form a straight line graph from the data by changing the variable plotted on each axis. For example: Raw data and graph This is clearly not a straight line graph. The relationship between x and y is not clear. A series of different graphs can be drawn from these data. The one that is closest to a straight line is a good candidate for the relationship between x and y. This is an idealised set of data to illustrate the point. The straightest graph is y2 against x, suggesting that the relationship between x and y is. More complex relationships Graphs can be used to analyse more complex relationships by rearranging the equation into a form similar to y=mx+c. Example one: testing power laws A relationship is known to be of the form y=Axn , but n is unknown. Measurements of y and x are taken. A graph is plotted with log(y) plotted against log(x). The gradient of this graph will be n, with the y intercept log(A), as log(y) = n(log(x)) + log(A) Example two The equation that relates the rate constant of a reaction to temperature is = − This can be rearranged into ln() = − ( 1 ) + So a graph of ln() against ( 1 )should be a straight line, with a gradient of− and a y-intercept of �

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PS 3.1 Plot and interpret graphs

Credits: Neil Goalby