Measurement & Experimental Error: When interacting with the world around us and more particularly while doing experiments, we make and record observations. One can make both qualitative observations (sensory based, categorical) and quantitative observations (numerical). e.g. qualitative - blue, cloudy, acrid odour, shiny, heavy quantitative - 5.43 g, 12.1 M, 6.022×1023 atoms, 10.23 cm A variable is a property or factor that can exist in differing quantities. Measurement is the determination of a quantitative value of a variable in a particular instance through comparison to a standard (i.e. unit). A measurement must consist of both a quantity and a unit. There are many such units, but scientists generally use a system of measurement called Le Système International d'Unités (SI). The SI units are the most modern version of the metric units. There are only seven base SI units (see below). variable name length mass time temperature amount electric current luminous intensity variable symbol l m t T n I IV unit name metre kilogram second Kelvin mole ampere candela unit symbol m kg s K mol A cd All other variables can be measured using combinations of these base units and sometimes metric prefixes. For example: variable name volume density energy variable symbol V ρ various symbols possible unit symbol dm3 g cm-3 kg m2 s-2 A derived unit is a unit defined as a mathematical combination of other units, usually the seven base SI units. Derived units are used when stating measurements using the seven base SI units becomes inconvenient. For example, the Joule (J) and the Watt (W) are derived units: J ≡ kg m2 s-2 and W ≡ kg m2 s-3 ________________________________________________________________________ Mathematical statements are made relating variables to measurements of them. These contain four parts: the variable symbol, the equal sign, the quantity, and the unit symbol. length l = 11.42 m mass m = 22.4 kg time t = 4.5×104 s temperature T = 308 K amount n = 3.1 mol volume V = 0.56 dm3 density ρ = 8.95 g cm-3 energy E = 345.5 kJ Equations can be stated to relate variables: ρ= ! ! (density equals mass of material ÷ volume of material) Equations can also be stated to relate units: J = kg m2 s-2 It is not correct, however, to mix variables and units in equations without quantities: ! ρ ≠ !"! is incorrect, but ρ= ! ! ! = 2.70 !"! is correct As you can see in the examples provided, variables and units can get somewhat complicated as the same symbol can have different meanings (e.g. m for the variable mass, the unit metre, and also the metric prefix milli). ________________________________________________________________________ Sample Question #1: Give one example of each of the following: an SI base unit, an SI derived unit, and a non-SI unit. SI base unit: m (or 6 others) SI derived unit: J, W non-SI unit: inch, pound, °F Sample Question #2: The variable molarity (c) is equal to the amount of substance (n) in moles divided by the volume of solution (V) in dm3. Molarity has the units mol dm-3 which equals the derived unit M. Which equation is definitely not correct? M = mol dm-3 c = mol dm-3 ! c=! c = 0.0034 mol dm-3 c = mol dm-3 is incorrect as it equates a variable and units without a quantity. ________________________________________________________________________ Measurement of Temperature: Temperature is a measure of the average kinetic energy of a group of particles. The Celsius scale is most commonly used for the measurement of temperature. It is based on the arbitrary assignment of temperature values to the normal melting point of water (0°C) and the normal boiling point of water (100°C). The Kelvin scale is the accepted SI scale for temperature. The Kelvin scale is based on the Celsius scale as well as the concept of absolute zero. Absolute zero is the coldest possible temperature (i.e. 0 K) and thus the temperature at which there is no atomic or molecular motion (i.e. kinetic energy = 0). The Kelvin scale is most commonly used within the sciences when temperature values are used in calculations because Kelvin temperatures are a proportional measure of kinetic energy. TK ! TC subtract 273.15 absolute zero -273.15°C 0K mp water 0°C bp water +100°C +273.15 K +373.15 K Note: (a) Kelvin temperatures are always positive, (b) the value for a Kelvin temperature is always greater than the value for the corresponding Celsius temperature, and (c) a difference in temperatures is the same number in either unit (e.g. ΔT = 15C° = 15 K). TC ! TK add 273.15 Measurement Uncertainty: Part of every measurement is certain (no doubt) while part is uncertain (an estimate). This estimate is made between the smallest divisions on the “scale” of the measuring instrument. It is incorrect to not make an estimate. Consider the cm ruler shown below being used to make the indicated measurement: Note: From Reading a Metric Ruler, n.d., Craftsmanspace. • We are certain that the measurement is between 2 and 3 cm (∴ 2._ cm). • We are certain that the measurement is between 2.5 and 2.6 cm (∴ 2.5_ cm). • As there are no more lines between 2.5 and 2.6 cm, we have to make our best estimate of the next digit (∴ 2.54 cm). No more digits are allowed as the last digit was estimated. • In 2.54 cm, the “2” and the “.5” are certain digits, but the “.04” is the uncertain digit. Correctly made measurements can have any number of certain digits with 0 to 14 being the practically possible range and 1 to 4 being the common range. Correctly made measurements always have 1, and only 1, uncertain digit. To show more than one uncertain digit is incorrect as it would imply that the measuring instrument had divisions on its scale that were smaller than they really were. Measurement uncertainty is reflected in the way a measurement is written. This is done with the placement of the uncertain digit (e.g. 2.54 cm) or by the statement of the range of measurement uncertainty (e.g. 2.54 cm ± 0.01 cm). The stated range of measurement uncertainty describes the possible variation in the estimated digit. In this example, what has actually been determined is that the value being measured is between 2.53 cm and 2.55 cm. The range of uncertainty and the measurement to which it applies must always have the same number of decimal places, but the variance in the uncertain digit does not have to be “± 1”. There are also two different ways the units can be shown. e.g. 2.54 cm ± 0.01 cm and (2.54 ± 0.01) cm both correct 0.050 00 L ± 0.000 01 L and (0.050 00 ± 0.000 01) L both correct depending on discretion of measurer, 3.66 cm ± 0.03 cm could be correct 2.5 cm ± 0.01 cm and (2.5 ± 0.01) cm both incorrect 42.50 g ± 0.10 g incorrect 4200 J ± 212.54 J incorrect Examples of Specific Cases in Measurement: (A) simplest case (10 lines shown for the smallest order of magnitude): If a ruler has lines for every 0.1 cm, then measurements would be estimated to the nearest 0.01 cm and would be stated with an uncertainty range of ± 0.01 cm. Estimating between the lines gives one more decimal place than the decimal place indicated by the smallest divisions. e.g. 2.54 cm ± 0.01 cm (see example diagram from the previous page) (B) the measurement is “right on the line”: The placement of the uncertain digit and the range of uncertainty are determined by the instrument. Measurements are not “exact” just because they appear to be “right on the line”. e.g. 2.00 cm ± 0.01 cm as this ruler allows for measurements estimated to ± 0.01 cm Note: From Reading a Metric Ruler, n.d., Craftsmanspace. (C) digital scales: Read the value from the screen, but be aware that a component of the device is still making an estimate of the last digit. The range of uncertainty can be obtained from the operating manual of the device or sometimes by observing the visible waver in the last digit. e.g. reading of 12.43 g really means (12.43 ± 0.01) g (D) liquids: Liquids in containers have what is called a meniscus which is a curvature in the surface of the liquid due to the particles of the liquid adhering to the walls of the container with a greater or lesser strength than they do to themselves. To correctly read liquid volumes, the measurement is taken from the centre of the meniscus (i.e. the bottom of water’s concave meniscus and the top of mercury’s convex meniscus). The measurer’s line of sight must be parallel to the ground. 43.0 mL ± 0.1 mL -measure from centre of meniscus -line for every 1 mL -estimate to nearest 0.1 mL -appears “right on” 43 mL line Note: From ChemWiki: The Dynamic Chemistry E-textbook, n.d., Delmar Larsen (director). (E) less than 10 divisions of the smallest order of magnitude: If a graduated cylinder has lines for every 2 mL, then measurements would be estimated to the nearest 1 mL and would be stated with an uncertainty of ± 1 mL. Measurements can not be stated to ± 0.1 mL unless there is a line for every 1 mL. Such instruments are usually of poor quality and should likely not be used if an accurate measurement is required. -since there are not lines for every 1 mL, the units place is the estimated digit -it definitely between 12 mL and 14 mL -measurement is 12 mL ± 1 mL -one could possibly record 12.5 mL ± 0.5 mL, but given the poor quality of the instrument the more pessimistic 12 mL ± 1 mL would be advised Note: From Graduated Cylinder, n.d., Science Class Online, Chemistry and Physics Lessons. (F) what is being measured is less distinct than lines on instrument: If a stopwatch that records to ± 0.01 seconds is being used, but the users reaction time is slower than 0.01 seconds, which it definitely is, times should be recorded as ± 1s or perhaps ± 0.3 s. 31.5 s ± 0.3 s If a rough edge is measured with ruler with fine gradations, a similar situation can arise. Note: From Reading a Metric Ruler, n.d., Craftsmanspace. This measurement could be quickly recorded as 7.6 cm ± 0.1 cm. This indicates a value between 7.5 cm and 7.7 cm which makes sense. Another approach is to use the ruler to its limit and estimate a minimum and maximum possible reading: 7.51 cm and 7.65 cm. The middle of this range is 7.58 cm and the minimum and maximum estimate are respectively 0.07 cm below and above the middle of the range. This gives an overall measurement of 7.58 cm ± 0.07 cm. _______________________________________________________________________ Note that while measurements are never exact, “numbers” can be exact (e.g. I have exactly 10 toes). Types of Experimental Error: When doing quantitative experiments, there are two types of experimental “error”: random uncertainty and systematic error. These both contribute to the degree of error in a measured or calculated value. Random uncertainty is a type of experimental error involving positive and negative fluctuations that cause about half of the measured values to be too high and half to be too low. Random uncertainty can arise due to observational and environmental factors. e.g. Measurement uncertainty, the estimation on the part of the observer during measurement, is always a source of random uncertainty. e.g. Mechanical vibrations of the equipment and unpredictable fluctuations in the power line voltage or temperature can lead to random uncertainty in measurements in addition to any estimating done by the observer. Systematic error is a type of experimental error resulting in measured values that are consistently too high or too low. Possible sources of systematic error include instrumental, observational, environmental, and theoretical factors. e.g. A poorly calibrated thermometer might always read 2 C° too high. Any temperature measured using this thermometer would be too high. e.g. Parallax error in reading a scale due to improper positioning of the observer’s eye in relation to the scale will cause error. This is of particular note when recording volumes of liquids. This could also be a random error if the observer’s eyes were randomly positioned incorrectly. e.g. The temperature might be lower throughout the experiment than the calibration temperature for the measuring instrument used. e.g. Due to simplifications in the model system or approximations in the equations describing it, the theoretical and experimental results might consistently disagree. ________________________________________________________________________ Precision and Accuracy: Precision refers to the degree of agreement among several measurements of the same quantity. Random uncertainty affects the precision of a measured value. When multiple measurements of the same quantity have been made, the range of measurement uncertainty can be determined from the spread of the different measurements. • A precise measurement would have a relatively low uncertainty range. • An imprecise measurement would have a relatively high uncertainty range. • For single measurements, precision is generally considered to refer to the range of measurement uncertainty as estimated based on “the line spacing on the instrument.” Accuracy refers to the closeness of the measured value to the “accepted value”. Systematic errors affect the accuracy of a measured value. Such errors are not indicated in the statement of a measured value. • Accurate values are close to the accepted value. For our purposes, an accurate measurement contains the accepted value within its random uncertainty range. • Inaccurate values are far from the accepted value. Percent difference is a measure of the relative error in an experimental value compared to the accepted value. It is generally a measure of the accuracy of an experimental value. % difference = (accepted value - experimental value) × 100% accepted value Precision and Accuracy Overview / Relation to Types of Experimental Error: Consider the classic target description of accuracy and precision. Hitting the centre of the target is analogous to experimentally determining the accepted value. Suppose that the measurement of a physical quantity is repeated several times under experimental conditions. Case C is obviously the most desirable case. Case A is accurate but only in an average sense and potentially only due to the imprecision of the technique. random uncertainty systematic error A B C D significant minimal minimal significant minimal minimal significant significant Note: From Accuracy and Precision, 2013, Pharmacelsus, Contract Research Organization. Reducing Experimental Error: The experimenter should try to identify as many experimental errors as possible and then eliminate the systematic errors and try to minimize random uncertainty in the process of designing/perfecting an experiment. Measurement uncertainty can be reduced by acquiring more precise instruments (i.e. instruments that allow measurements with more decimal places). The effects of random uncertainty on an experimentally determined value can be generally minimized by taking multiple readings and averaging them. As the number of readings increases, the statistical likelihood that the randomly high and low values will cancel out increases thereby making the average value more accurate. Systematic errors are not minimized by averaging multiple readings. Measurement & Experimental Error Review Questions: #1. Which one of the following is not a base SI unit? A. h B. kg C. m D. K #2. Which one measurement is definitely incorrect? A. 25.00 mL ± 0.01 mL B. 24.40 g ± 0.02 g C. 25.12 cm ± 0.10 cm D. (9.8 ± 0.1) kg #3. Which one of the following recurring errors is a systematic error? A. measurement uncertainty B. heat loss from an insulated device C. vibrations of the table on which an electronic balance is sitting D. inconsistent homogeneity of an aqueous chemical reactant #4. An experiment was done five times and yielded the following results: 8.344, 8.346, 8.343, 8.345, and 8.344 The accepted value for this experiment was 8.314. Which option below properly describes the experiment’s accuracy, precision, and predominant type of error? A. inaccurate, precise, random error B. inaccurate, precise, systematic error C. accurate, imprecise, random error D. accurate, imprecise, systematic error #5. Categorize each of the following numbers as exact or inexact. (a) (b) (c) (d) (e) The elevation of my house in Dartmouth is 246 ft above sea-level. There are 12 eggs in a dozen. One inch is equal to 2.54 cm. The ticket sales for a hockey game are 10 306. Canadian federal government debt as of the fiscal year 2013 was $602.4 billion. #6. Convert the Celsius temperatures into Kelvin and vice versa. (a) (b) (c) (d) 25.00°C 342.13°C 232.96 K 589.23 K #7. An experiment yielded a value of 2.06 g whereas the accepted value was 2.00 g. Calculate the percent difference. = = = = ______________ K ______________ K ______________ °C ______________ °C #8. For each diagram, record the correct measurement including units and a range of measurement uncertainty. (a) 1 (b - top) in dm, (c - middle) in cm, (d - bottom) in mm 2 (e) in mL (f) in mL 3 (g) in mL 3 (i) in cm (h) in mL 4 3 5 1 2 3 4 5 Note: From Ruler, ClipArt ETC, 2014, Educational Technology Clearinghouse. Note: From Uncertainty, 2010, Aaron Keller. Note: From Graduated Cylinder, n.d., Science Class Online, Chemistry and Physics Lessons. Note: From LC Chemistry Volumetric Analysis, 2014, QuizLet.com. Note: From Reading a Metric Ruler, n.d., Craftsmanspace.
