This section discusses uncertainty and how to calculate uncertainty in the lab. The required content can vary depending on the preferred teaching method and the student’s grade level.
Uncertainty is the margin of error associated with the values measured or determined in an experiment.
Every experiment involves some degree of inaccuracy in the measurement. The measurement read in the lab is always the most accurate one under the circumstances. Nevertheless, uncertainty is used to describe the range in which the exact value is found.
Uncertainty can arise from the measuring instrument, carelessness on the part of the person taking the measurement or difficulty involved in interpreting a measurement on a given scale.
Whether absolute or relative, uncertainty is always expressed as a single significant figure.
Absolute uncertainty is the maximum error that can be expected when taking a measurement on an instrument or device.
Every experimental result lies somewhere between a minimum and a maximum value. This result |x| lies between a minimum value |x_{min}| and a maximum value |x_{max}.| The range of possible values for the measurement |x| can be expressed as |[x_{min}, \: x_{max}].|
A simpler way to write down the measurement with its uncertainty is: |x \pm \Delta x,| where |\Delta{x}| is how far the measurement value |x| is from the minimum value |(x - \Delta{x} = x_{min})| or the maximum value |x + \Delta{x} = x_{max}.|
A ruler is used to measure a book. The measurement with its absolute uncertainty is |(21.90 \pm 0.05)\ \text {cm}.| This means that the smallest value the book could have |(x_{min})| is |21.85 \ \text {cm},| while the largest value |( x_{max})| is |21.95 \ \text {cm}.|
The experimental result gives a value |x,| which is the best possible estimate of the result of the reading, and a value |\Delta x,| which represents the absolute uncertainty associated with this value.
Case 1: Analogue measuring instruments
Analogue measuring instruments are devices equipped with a pointer that indicates the value of the measured quantity on a scale or with graduation marks that display the value of the measured quantity.
Needle voltmeters, graduated cylinders and alcohol thermometers are analogue instruments. They all have scales that are read to obtain the value of the quantity being measured.
The uncertainty associated with an analogue measuring instrument is half the smallest scale on the instrument.
For example, the absolute uncertainty of a ruler graduated in millimetres is: |\dfrac {1\ \text{mm}}{2} = 0.5\ \text{mm}.|
The uncertainty could also be calculated in centimetres: |\dfrac {0.1\ \text {cm}}{2} = 0.05\ \text {cm}.|
The absolute uncertainty of an alcohol thermometer with the smallest scale in degrees is: |\dfrac {1^\circ\!\text{C}}{2} = 0.5^\circ\!\text{C}.|
Case 2: Digital measuring instruments
Digital measuring instruments are devices that give the reading directly as a numerical value that appears on a screen.
Multimeters and stopwatches are examples of digital instruments. These devices give a reading directly on the device.
The uncertainty associated with a digital measuring instrument is the equivalent of one unit of the smallest scale of the instrument.
The absolute uncertainty of a stopwatch accurate to one one-hundredth of a second is one one-hundredth of a second |({0.01\ \text {s}}).|
The absolute uncertainty of a multimeter measuring the resistance to the nearest unit is one ohm |( {1\ \Omega}).|
Case 3: Theoretical values
The uncertainty associated with a theoretical value is the equivalent of one unit of the last digit.
The boiling temperature of water is |100^{\circ}\text {C},| so the uncertainty is |\pm 1^{\circ}\text{C}.|
The density of water is |1.00\ \text{g/mL},| so the uncertainty is |\pm 0.01\ \text {g/mL}.|
Sometimes the uncertainty from an observer’s measurement must be added to the absolute uncertainty from an instrument. In these cases, the measurement uncertainty is often equal to the sum of the uncertainties of each reading.
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The parallax effect: If two lines, such as the pointer of an analogue instrument and the graduated lines under it, have to be matched to take a measurement, the reading recorded by different observers can differ depending on where their eyes are positioned relative to these lines.
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Reflex time: There is an uncertainty related to the observer’s reflexes. For example, if a person is timing how long it takes for an object to fall, the delay between the moment the object hits the ground and the moment the stopwatch button is pressed by the observer must be considered.
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The meniscus: A particular phenomenon that must always be taken into account when measuring the volume of a liquid is the curved line formed by the liquid in the graduated cylinder. This curvature, called the meniscus, can be concave or convex. This means there is some uncertainty when reading the volume. To reduce the uncertainty, the reading should be performed at eye level.
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Zero readings: There is uncertainty in zero readings, because they must be taken in the same way as readings would be taken if they were not at the zero point.
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Measurements that require two readings: When using a ruler, the uncertainty at the point where the measurement is taken on the ruler must be considered, as well as the uncertainty at the zero point, where the ruler was placed to take the measurement. In these cases, it is best to double the uncertainty of the reading.
Relative uncertainty is the ratio of absolute uncertainty to the measurement. This ratio is expressed as a percentage.
To calculate relative uncertainty, the absolute uncertainty of the device must be determined first. Calculating the relative uncertainty makes it possible to compare the accuracy of different measurements. The most accurate measurement is the one with the lowest relative uncertainty.
|\text{Relative uncertainty} = \dfrac{\text{Absolute uncertainty}}{\text{Measured value}} \times \text{100}|
What is the relative uncertainty of a measurement taken with a ruler if the length of the object being measured is |21.3\ \text {cm}|?
Since the smallest unit of measurement of a ruler is |0.1\ \text {cm},| the absolute uncertainty associated with this measuring instrument is |\pm 0.05\ \text {cm}.| The relative uncertainty is expressed as follows:
|\begin{align}\text {Relative uncertainty} &= \dfrac{{0.05\ \text {cm}}}{{21.3\ \text {cm}}}\times \text {100}\\ \text {Relative uncertainty} &= 0.23471...\% \end{align}|
Since uncertainties are always expressed with a single significant figure, the uncertainty obtained must be rounded to conform to this rule. The relative uncertainty is therefore |\pm 0.2\%.|
The measurement taken by the ruler can be expressed as follows: |21.3\ \text {cm} \pm 0.2\%.|
Uncertainty in addition and subtraction
To calculate uncertainty in addition or subtraction, the sum of the absolute uncertainties is determined to give the absolute uncertainty of the result of the addition or subtraction.
What is the total volume of water if |25.0\ \text {mL} \pm 0.3\ \text {mL} | of water is added to a 50.0 mL graduated cylinder already containing |10.0\ \text {mL} \pm 0.4\ \text {mL}|?
To find the total volume, add the volumes together: |25.0 + 10.0 = 35.0\ \text {mL}.|
To find the uncertainty, add the uncertainties: |0.3 + 0.4 = \pm 0.7\ \text {mL}.|
The final measurement is |(35.0 \pm 0.7)\ \text {mL}.|
What total volume of acid is left in a burette if it initially contained |50.00\ \text {mL} \pm 0.05\ \text {mL},| and |18.50\ \text {mL} \pm 0.05\ \text {mL}| were used in a neutralization?
To find the remaining volume, subtract the given volumes: |50.00 – 18.50 = 31.50\ \text {mL}.|
To find the uncertainty, add the uncertainties: |0.05 + 0.05 = \pm 0.1\ \text {mL}.|
The final measurement is |(31.5 \pm 0.1)\ \text {mL}.|
Uncertainty in multiplication and division
There are two ways to calculate uncertainty. These methods are used to calculate the uncertainty of data obtained as a result of a mathematical calculation.
Calculating uncertainty using relative uncertainty
To calculate uncertainty when multiplying or dividing, add the relative uncertainties from the original data and multiply the sum by the final answer.
What is the area of a rectangle with a length of |20.0\ \text {m} \pm 0.5\ \text {m}| and a width of |12.0\ \text {m} \pm 0.4\ \text {m}|?
To find the total area, multiply the length and width: |20.0 \times 12.0 = 240.0\ \text {m}^2.|
To find the uncertainty, use the relative uncertainties.
|\Delta \text {x} =\left( \left(\dfrac{{0.5\ \text {m}}}{{20.0\ \text {m}}}\right) + \left(\dfrac{{0.4\ \text {m}}}{{12.0\ \text {m}}}\right) \right) \times 240.0\ \text {m}^2 \approx 14\ \text {m}^2 = 1 \times 10^1 \ \text {m}^2|
The final measurement is |(24 \pm 1 \times 10^1)\ \text {m}^2.|
What is the density of an object with a mass of |109.47\ \text {g} \pm 0.05\ \text {g}| and a volume of |12.3\ \text {mL} \pm 0.3\ \text {mL}|?
To find the density, divide the mass by the volume: |\dfrac{109.47\ \text{g}} {12.3\ \text{mL}}= 8.90\ \text {g/mL}.|
To find the uncertainty, use the relative uncertainties.
|\Delta \text {x} =\left( \left( \displaystyle \frac{{0.05\ \text {g}}}{{109.47\ \text {g}}} \right) + \left( \displaystyle \frac{{0.3\ \text {mL}}}{{12.3\ \text {mL}}} \right) \right) \times 8.90\ \text {g/mL} \approx 0.2\ \text {g/mL}|
The final measurement is |(8.9 \pm 0.2)\ \text {g/mL}.|
Calculating uncertainty using extremes
To calculate the uncertainty when multiplying or dividing, find the difference between the maximum value and the minimum value that can be obtained by the uncertainties, then divide this difference by 2.
What is the area of a rectangle with a length of |20.0\ \text {m} \pm 0.5\ \text {m}| and a width of |12.0\ \text {m} \pm 0.4\ \text {m}|?
To find the total area, multiply the length and width: |20.0 \times 12.0 = 240.0\ \text {m}^2.|
To find the uncertainty, determine the minimum and maximum values of each given variable.
|\text{x}_{min} = (20.0 - 0.5)\times (12.0 - 0.4) = 226.2\ \text {m}^2|
|\text{x}_{max} = (20.0 + 0.5)\times (12.0 + 0.4) = 254.2\ \text {m}^2|
The uncertainty is then determined.
|\begin{align}\Delta \text {x} &= \left(\dfrac{\text{x}_{max}-\text{x}_{min}}{{2}} \right) \\ \Delta \text {x} &= \left(\dfrac{254.2\ \text {m}^2-226.2\ \text {m}^2}{{2}} \right) \\ \Delta \text {x} &= 14\ \text {m}^2 \\ \Delta \text {x}&= 1 \times 10^1 \ \text {m}^2 \end{align}|
The final measurement is |(24 \pm 1) \times 10^1\ \text {m}^2.|
When dividing, pay special attention to the data used to find the maximum and minimum values.
What is the density of an object with a mass of |109.47\ \text {g} \pm 0.05\ \text {g} | and a volume of |12.3\ \text {mL} \pm 0.3\ \text {mL}|?
To find the density, divide the mass by the volume: |\dfrac{109.47\ \text{g}}{12.3\ \text{mL}} = 8.90\ \text {g/mL}.|
To find the uncertainty, determine the minimum and maximum values of each given variable.
|\text{x}_{min} = \dfrac{(109.47 \color{#EC0000} - 0.05)}{(12.3 \color{#EC0000}+ 0.3)} = 8.68\ \text {g/mL}|
|\text{x}_{max} = \dfrac{(109.47 \color{#EC0000}+ 0.05)}{(12.3 \color{#EC0000}- 0.3)} = 9.13\ \text {g/mL}|
The uncertainty is then determined.
|\begin{align} \Delta \text {x} &= \left(\dfrac{\text{x}_{max}-\text{x}_{min}}{{2}} \right)\\ \Delta \text {x} &= \left(\dfrac{9.13\ \text {g/mL}-8.68\ \text {g/mL}}{{2}} \right)\\ \Delta \text {x} &= 0.225\ \text {g/mL} \approx 0.2\ \text {g/mL}\end{align}|
The final measurement is |(8.9 \pm 0.2)\ \text {g/mL}.|