This section presents the uncertainty and the calculations of uncertainty to be performed in a laboratory. However, this content may vary depending on the teaching method used or the student's level of education.
Uncertainty represents the margin of error associated with the values that are measured or determined during an experiment.
In any experiment, there is a component of inaccuracy in the measurements taken. A measure taken in a laboratory is always the most accurate under the circumstances. However, uncertainty describes the dispersion of the value, or the interval within which the exact value lies.
Uncertainty may be a result of the measuring instrument used, the lack of rigour shown by the person measuring, or the interpretation of a measure on a given scale.
Uncertainty, whether absolute or relative, is always written with one single significant figure.
Absolute uncertainty is the maximum error that can be made when determining a measurement from a device.
Any experimental outcome lies between a minimum value and a maximum value. This result, which we can call |x,| therefore lies between a minimum value called |x_{min}| and a maximum value |x_{max}.| The range of possible values for the measurement |x| can therefore be described as |\left[x_{min}, \: x_{max}\right].|
To simplify writing the uncertainty, write the measurement with its uncertainty as follows: |x \pm \Delta x.|
A ruler is used to measure a book. The measurement obtained, with its absolute uncertainty, is |(21.90 \pm 0.05) \: \text {cm}.| This means that the smallest possible value of the book |\left( x_{min} \right)| is |21.85 \: \text {cm},| while the largest possible value |\left( x_{max} \right)| is |21.95 \: \text {cm}.|
The experimental outcome gives a value |x,| which is the best possible estimate of the measure of the book, and a value |\Delta x,| which represents the absolute uncertainty associated with this value.
Case 1: Analogue Measurement Instruments
Analog measuring instruments are devices fitted with a needle indicating the value of the quantity measured on a scale, or with a graduated scale that indicates the value of the quantity measured.
A voltmeter with a pointer, a graduated cylinder or an alcohol thermometer are all analogue instruments, as they all consist of scales which require a reading to be taken to find the value of the quantity being measured.
The uncertainty of reading a measurment from an analog measuring instrument is equal to half the smallest graduation of the instrument.
The absolute uncertainty of a ruler graduated in millimetres is therefore: |\displaystyle \frac {1 \: \text {mm}}{2} = 0.5 \: \text {mm}|.
The uncertainty could also be calculated in centimetres: |\displaystyle \frac {0.1 \: \text {cm}}{2} = 0.05 \: \text {cm}.|
The absolute uncertainty of an alcohol thermometer whose smallest graduation is the degree would be: |\displaystyle \frac {1 \: ^{\circ}\text {C}}{2} = 0.5 \: ^{\circ}\text {C}.|
Case 2: Digital Measurement Instruments
Digital measurement instruments are devices that give a direct reading in the form of a digital value.
A multimeter and a stopwatch are examples of digital instruments, since these devices provide you with a reading directly simply by observing the device.
The reading uncertainty associated with a digital measurement instrument is equal to the equivalent of one unit of the smallest graduation of the instrument.
The absolute uncertainty of a stopwatch which reads to the one hundredth of a second will be one hundredth of a second |\left( {0.01 \: \text {s}}\right).|
The absolute uncertainty of a multimeter measuring the resistance of a resistor to within one unit will be one ohm |\left( {1 \: \Omega} \right).|
Case 3: Theoretical Values
The uncertainty associated with a theoretical value is equal to the equivalent of one unit on the last digit.
Since the boiling temperature of water is |100 \: ^{\circ} \text {C},| the uncertainty will be |\pm\ 1 \: ^{\circ} \text {C}.|
Given that the density of water is |1.00\ \text {g/ml},| the uncertainty will be |\pm\ 0.01 \: \text {g/ml}.|
Sometimes, in addition to the absolute uncertainty of an instrument, the uncertainty of the measurement by the observer must also be added. In these cases, the uncertainty related to the measurement is often equal to the sum of the uncertainties of each reading.
- The Parallax Effect: When two lines must be matched to interpret a measurement, such as the pointer on an analogue device and the scale below it, the reading may vary from one observer to another depending on the position of the eye in relation to these lines.
- Reflex time: There is an uncertainty linked to the observer's reflexes. For example, if a person has to time the time it takes for an object to fall, you have to consider the delay between the object actually landing on the ground and the moment your thumb presses the stopwatch button.
- The meniscus: When measuring the volume of a liquid, a particular phenomenon must always be taken into account, namely the presence of a curved line formed by the liquid in the graduated cylinder. This curve, known as the meniscus, may be concave or convex. Reading the volume therefore involves a degree of uncertainty. To reduce this uncertainty, it is important to align the eye with the meniscus, placing it at the same height.
- Measurements given by two readings: When using a ruler, you need to consider the uncertainty at the point where the measurement is taken on the ruler, as well as the uncertainty at zero, or the point where the ruler was placed to measure from. In these cases, we should multiply the uncertainty in the reading by two.
- Zero readings: There is an uncertainty in zero readings, since they have to be taken in the same way as those that would be taken if they were not at zero.
Relative uncertainty is the ratio between the absolute uncertainty and the measurement. This ratio is expressed as a percentage.
To calculate the relative uncertainty, it is important to determine the absolute uncertainty of the device. Calculating the relative uncertainty allows you to compare the accuracy of different measurements. The most accurate measure is that with the lowest relative uncertainty.
|\text {Relative uncertainty} = \displaystyle \frac{\text {Absolute uncertainty}}{\text {Measured value}}\times \text {100}|
What is the relative uncertainty of a measure taken with a ruler, given that the length of the object to be measured is |21.3 \: \text {cm}|?
Since the smallest unit of measure on a ruler is |0.1 \: \text {cm}|, the absolute uncertainty associated with this measurement instrument is |\pm 0.05 \: \text {cm}.| The relative uncertainty is therefore expressed as follows:
|\text {Relative uncertainty} = \displaystyle \frac{{0.05 \: \text {cm}}}{{21.3 \: \text {cm}}}\times \text {100}|
|\text {Relative uncertainty} = 0.23471... \%|
Since uncertainties are always expressed using only one significant figure, the uncertainty must be rounded to comply with this rule. The relative uncertainty will be |\pm\ 0.2 \: \%.|
The measurement taken by the ruler is then expressed as follows: |21.3 \: \text {cm} \pm 0.2 \: \%.|
Uncertainty in Addition or Subtraction
To calculate the uncertainty when adding or subtracting, the absolute uncertainties are added together to give the absolute uncertainty of the sum or difference.
What is the total volume of water if we add |25.0 \: \text {ml} \pm 0.3 \: \text {ml} | of water into a |50.0\ \text {ml}| graduated cylinder containing |10.0 \: \text {ml} \pm\ 0.4\ \text {ml} |?
To find the total volume, the volumes must be added together: |25.0 + 10.0 = 35.0 \: \text {ml}|.
To find the uncertainty, add up the uncertainties: |0.3 + 0.4 = \pm\ 0.7 \: \text {ml}|.
The final measurement is therefore |\left( 35.0 \pm 0.7 \right) \: \text {ml}.|
What is the total volume of acid remaining in a burette if it contained |50.00\ \text {ml} \pm 0.05 \: \text {ml} | and |18.50 \: \text {ml} \pm 0.05 \: \text {ml} | was used for neutralisation?
To find the remaining volume, subtract the 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 therefore |\left( 31.5 \pm 0.1 \right) \: \text {ml}.|
Uncertainty in Multiplication or Division
Here are two proposed methods of calculating uncertainty. These methods represent ways of calculating the uncertainty in data or values obtained following a mathematical operation.
- Calculation of Uncertainty using Relative Uncertainty
- Calculation of Uncertainty by the Method of Extremes
To calculate uncertainty when multiplying or dividing, the relative uncertainties of the initial data values must be added together and the sum multiplied by the final answer.
What is the area of a rectangle whose length measures |20.0 \: \text {m} \pm 0.5 \: \text {m} | and whose width measures |12.0 \: \text {m} \pm 0.4 \: \text {m} |?
To find the total area, multiply the length by the width: |20.0 \times 12.0 = 240.0 \: \text {m}^2.|
To find the uncertainty, use the relative uncertainties.
|\Delta \text {x} =\left( \left( \displaystyle \frac{{0.5 \: \text {m}}}{{20.0 \: \text {m}}} \right) + \left( \displaystyle \frac{{0.4 \: \text {m}}}{{12.0 \: \text {m}}} \right) \right) \times 240.0 \: \text {m}^2 = 14\: \text {m}^2 = 1 \times 10^1 \: \text {m}^2|
The final measurement is therefore |\left( 24 \pm 1 \right) \times 10^1 \: \text {m}^2.|
What is the density of an object whose mass is |109.47? \text {g} \pm 0.05 \: \text {g} | and whose volume is |12.3 \: \text {ml} \pm 0.3 \: \text {ml} |?
To find the density, divide the mass by the volume: |109.47 \div 12.3 = 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} = 0.2\: \text {g/ml}|
The final measurement is therefore |\left( 8.9 \pm 0.2 \right) \: \text {g/ml}.|
To calculate the uncertainty when multiplying or dividing, the difference between the possible maximum value and minimum value that can be obtained using the uncertainties must be halved.
What is the area of a rectangle whose length measures |20.0 \: \text {m} \pm 0.5 \: \text {m} | and whose width measures |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, we need to determine the minimum and maximum values.
|\text{x}_{\text {min}} = (20.0 - 0.5)\times (12.0 - 0.4) = 226.2 \: \text{m}^2|
|\text{x}_{\text {max}} = (20.0 + 0.5)\times (12.0 + 0.4) = 254.2 \: \text {m}^2|
The uncertainty is then found.
|\Delta \text {x} = \left( \displaystyle \frac{\text{x}_{\text {max}}-\text{x}_{\text {min}}}{{2}} \right)|
|\Delta \text {x} = \left( \displaystyle \frac{254.2\: \text {m}^2-226.2\: \text {m}^2}{{2}} \right)|
|\Delta \text {x} = 14\: \text {m}^2 = 1 \times 10^1 \: \text {m}^2|
The final measurement is therefore |\left( 24 \pm 1 \right) \times 10^1 \: \text {m}^2.|
When dividing, be careful with the values that are used to find the maximum and minimum values.
What is the density of an object whose mass is |109.47 \text {g} \pm 0.05 \: \text {g} | and whose volume is |12.3 \: \text {ml} \pm 0.3 \: \text {ml} |?
To find the density, divide the mass by the volume: |109.47 \div 12.3 = 8.90 \: \text {g/ml}|.
To find the uncertainty, determine the minimum and maximum values.
|\text{{x}_{\text {min}}} = (109.47 - 0.05)\div (12.3 + 0.3) = 8.68 \: \text {g/ml}|
|\text{{x}_{\text {max}}} = (109.47 + 0.05)\div (12.3 - 0.3) = 9.13 \: \text {g/ml}|
The uncertainty is then found.
|\Delta \text {x} = \left( \displaystyle \frac{\text{x}_{\text {max}}-\text{x}_{\text {min}}}{{2}} \right)|
|\Delta \text {x} = \left( \displaystyle \frac{9.13 \: \text {g/ml}-8.68 \: \text {g/ml}}{{2}} \right)|
|\Delta \text {x} = 0.225 \: \text {g/ml} = 0.2 \: \text {g/ml}|
The final measurement is therefore |\left( 8.9 \pm 0.2 \right) \: \text {g/ml}.|