ERRORS
&
Errors and Uncertainties
Lesson Objectives
At the end of the lesson, you should be able to:
●
●
●
understand and explain the effects of systematic errors (including zero errors) and
random errors in measurements.
understand the distinction between precision and accuracy.
assess the uncertainty in a derived quantity by simple addition of absolute or
percentage uncertainties.
• Most experiments require scientists to make
measurements.
• A measurement tells us about a property of
something
Measurements of quantities are made with the aim of
finding the true value of that quantity
Tell whether the given figures shows precision and/or accuracy:
Precision and Accuracy
Precision
● Precise measurements are ones in which there is very little spread
about the mean value, in other words, how close the measured
values are to each other
● If a measurement is repeated several times, it can be described as
precise when the values are very similar to, or the same as, each
other
● The precision of a measurement is reflected in the values recorded
- measurements to a greater number of decimal places are said to
be more precise than those to a whole number
Accuracy
● A measurement is considered accurate if it is close to the true
value
● The accuracy can be increased by repeating measurements and
finding a mean of the results
● Repeating measurements also helps to identify anomalies that can
be omitted from the final results
Precision and Accuracy
• In reality, it is impossible to obtain the true value of any quantity
as there will always be a degree of error.
• Measurements are rarely exactly the same.
• Measurements are always somewhat different from the “true
value.”
• These deviations from the true value are called errors.
• Errors lead to uncertainty in measurements, an estimate of the
difference between a measurement reading and the true value
All measurements in science
suffer from uncertainty which
results from unavoidable
errors.
Errors
Types of Error
Error is the difference between the measured value
and the ‘true value’ of the thing being measured.
Errors are often classified as:
1. Random Error
2. Systematic Error
Random Error
●
●
●
Random errors occur randomly, and sometimes have no
source/cause
Random errors cause unpredictable fluctuations in an
instrument’s readings as a result of uncontrollable factors,
such as environmental conditions
This affects the precision of the measurements taken,
causing a wider spread of results about the mean value
EXAMPLE:
1.
When the observer makes consistent observational mistakes (such not
reading the scale correctly and writing down values that are constantly too
low or too high)
2.
When unpredictable changes occur in the environment of the experiment
(such as students repeatedly opening and closing the door when the
pressure is being measured, causing fluctuations in the reading)
Random Error
Examples of random errors are:
» reading a scale, particularly if this involves the experimenter’s judgement about
interpolation between scale readings
» timing oscillations without the use of a reference marker, so that timings may not
always be made to the same point of the swing
» taking readings of a quantity that varies with time, involving the difficulty of
reading both a timer scale and another meter simultaneously
» reading a scale from different angles introduces a variable parallax error
» When unpredictable changes occur in the environment of the experiment (such as
students repeatedly opening and closing the door when the pressure is being
measured, causing fluctuations in the reading)
Random Error
● To reduce random error:
○ Repeat measurements several times
and calculate an average from them
Systematic Error
A systematic error, is an error which occurs at each reading. Sources
of systematic errors include:
➔ The observer being less than perfect in the same way every
time
➔ An instrument with a zero offset error
➔ An instrument that is improperly calibrated
Systematic Error
● Systematic errors arise from the use of faulty instruments used or
from flaws in the experimental method
● This type of error is repeated consistently every time the
instrument is used or the method is followed, which affects the
accuracy of all readings obtained
● To reduce systematic errors:
○ Instruments should be recalibrated, or different instruments
should be used
○ Corrections or adjustments should be made to the technique
Systematic Error
EXAMPLES
1.
When the instrument being used does not function properly causing error in
the experiment (such as a scale that reads 2g more than the actual weight of
the object, causing the measured values to read too high consistently)
2.
When the surrounding environment (such as a lab) causes errors in the
experiment (you are conducting an experiment that requires room
temperature, but you are working in an airconditioned room)
3.
When the scientist inaccurately reads a measurement wrong (such as when
not standing straight-on when reading the volume of a flask causing the
volume to be incorrectly measured)
4.
When the model system being used causes the results to be inaccurate
(such as being told that humidity does not affect the results of an experiment
when it actually does)
Systematic Error
Dealing with Errors
• Identify the errors and their magnitude.
• Try to reduce the magnitude of the error.
HOW?
• Better instruments
• Better experimental design
• Collect a lot of data
Check your understanding:
An ammeter has a zero offset error. This fault will affect
A.
neither the precision nor the accuracy of the readings.
B.
only the precision of the readings.
C.
only the accuracy of the readings.
D.
both the precision and the accuracy of the readings.
Uncertainty
● There is always a degree of uncertainty when measurements are taken;
the uncertainty can be thought of as the difference between the actual
reading taken (caused by the equipment or techniques used) and the
true value
● Uncertainties are not the same as errors
○ Errors can be thought of as issues with equipment or methodology
that cause a reading to be different from the true value
○ The uncertainty is a range of values around a measurement within
which the true value is expected to lie, and is an estimate
● For example, if the true value of the mass of a box is 950 g, but a
systematic error with a balance gives an actual reading of 952 g, the
uncertainty is ±2 g
Expressing Uncertainty of Measurements
Reporting Measurement Results:
Measured value ± uncertainty (unit of measurement)
Example:
In an experiment, a measurement of a certain quantity is:
5.07 ± 0.02 g
The experimenter is confident that the actual value for the quantity being
measured lies between 5.05 g and 5.09 g.
Expressing Uncertainty of Measurements
Measured value ± uncertainty (unit of measurement)
●
5.07 ± 0.02 g
Uncertainties can be represented in a number of ways:
○ Absolute Uncertainty: where uncertainty is given as a fixed quantity; the
actual value of the uncertainty
○
Fractional Uncertainty: where uncertainty is given as a fraction of the
measurement
○
Percentage Uncertainty: where uncertainty is given as a percentage of the
measurement
Expressing Uncertainty of Measurements
Express the given uncertainty in the measurements into Absolute, Fractional and
Percentage Uncertainties:
a)
b)
c)
d)
52.3 ± 0.1 cm
400.50 ± 0.05 N
100.250 m ± 0.005
2.8 s ± 5.1%
Expressing Uncertainty of Measurements
How many digits should be kept?
●
●
Absolute uncertainties should only have one significant digit
The number of decimals in the measured value should be the same to the number of
decimals in the uncertainty
Wrong:
52.3 ± 0.13 cm
Correct:
52.3 ± 0.1 cm
Wrong:
1.237 ± 0.12 s
Correct:
1.2 ± 0.1 s
Expressing Uncertainty of Measurements
To find uncertainties in different situations:
● The uncertainty in a reading: ± half the
smallest division
● The uncertainty in repeated data: half the
range i.e. ± ½ (largest - smallest value)
● The uncertainty in digital readings: ± the
last significant digit unless otherwise quoted
Expressing Uncertainty of Measurements
To find uncertainties in different situations:
● The uncertainty in a reading: ± half the
smallest division
● The uncertainty in repeated data: half the
range i.e. ± ½ (largest - smallest value)
● The uncertainty in digital readings: ± the
last significant digit unless otherwise quoted
Instruments
Meter Rule
Protractor
Thermometer
Precision
(Least Count (LC) of the instrument)
0.1 cm
1o
1oC
Uncertainty
(50% of the LC)
0.1 cm
0.5o
0.5oC
Check your understanding:
With a vertical meter stick (with 1 mm as the least count), you might
measure the height of a chair as 41.2 cm. How should you report the value
of your measurement?
41.2 ± 0.1 cm
Check your understanding:
The length of a rod is measured using part of a ruler that is graduated in
millimetres, as shown below. What is the measurement, with its
uncertainty, of the length of the rod? Express the uncertainties in terms of
absolute, fractional, and percentage uncertainties.
Combining Uncertainties
Adding / Subtracting Data
✔
If data are to be added
or subtracted, add the
absolute uncertainties.
Combining Uncertainties
Adding / Subtracting Data
✔
If data are to be added or subtracted, add the absolute uncertainty.
Add: 1.2 ± 0.1 g, 12.01 ± 0.01 g, 7.21 ± 0.01 g
20.4 ± 0.1 g
Check your understanding:
I1 and I2 are two currents coming into a junction in a circuit. The current I
going out of the junction is given by I = I1 + I2
In an experiment, the values of I1 and I2 are determined as 2.0 ± 0.1 A and
1.5 ± 0.2 A respectively. What is the value of I? What is the uncertainty in
this value?
Check your understanding:
In an experiment, a liquid is heated electrically, causing the temperature to
change from 20.0 ± 0.2°C to 21.5 ± 0.5°C. Find the change of temperature,
with its associated uncertainty.
Combining Uncertainties
Adding / Subtracting Data
What is the sum of the measurements below:
45.02 ± 0.05 cm, 10.5 ± 0.5 cm, 150.21 ± 0.02 cm
205.73 ± 0.57 cm
205.7 ± 0.6 cm
Combining Uncertainties
Multiplying / Dividing Data
✔
If data are to be multiplied or divided,
add the percentage or fractional
uncertainties.
Combining Uncertainties
Multiplying / Dividing Data
✔
Find the work done given the following data: (W = Fd)
F = 10.5 ± 0.2 N
d = 3.06 ± 0.05 m
✔
Find the momentum of a car given the following data:
m = 1000 ± 5 kg
v = 20.40 ± 0.05 m
✔
What is the percent and absolute uncertainty of the speed?
d = 10 ± 1 m
t = 5.0 ± 0.2 s
Combining Uncertainties
Multiplying by a Constant
✔
If you’re multiplying a number with an uncertainty by a constant factor,
the rule varies depending on the type of uncertainty. If you’re using a
percentage uncertainty, this stays the same:
✔
If you’re using absolute uncertainties, you multiply the uncertainty by the
same factor:
Combining Uncertainties
Multiplying by a Constant
✔
Find the circumference of a circle whose radius is 5.0 ± 0.2 cm.
31 ± 1 cm
Combining Uncertainties
Raising to a Power
✔
If you’re taking a power of a value with an uncertainty, you multiply the
percentage uncertainty by the number in the power.
Combining Uncertainties
Raising to a Power
✔
If you’re taking a power of a
value with an uncertainty,
you multiply the percentage
uncertainty by the number
in the power.
Combining Uncertainties
Raising to a Power
✔
If you’re taking a power of a value with an uncertainty, you multiply
the percentage uncertainty by the number in the power.
Practice:
What is the kinetic energy of a car whose mass is m = 1200 ± 3 kg and
speed of v = 18.0 ± 0.5 ms-1? (KE = ½ mv2)
194400 J ± 5.81%
Combining Uncertainties
Uncertainty in Multiple Measurements
✔
When recording the uncertainties of multiple measurements, get
the mean of the values and the uncertainty is determined by
dividing the range of the values by two.
Example:
Six students measure the resistance of a lamp. Their answers in Ω are: 509; 566; 539;
561; 554; 528. What should the students report as the resistance of the lamp?
Note: When taking several measurements, it should be clear if you have a value with a
large error. Do not be afraid to throw out any measurement that is clearly a mistake.
Combining Uncertainties
Uncertainty in Multiple Measurements
Example:
Six perpendicular measurements are made of the diameter of a wire.
What should be the reported value of the wire’s diameter?
Results, in mm, are:
45.7
44.9
46.2
60.8
45.4
46.0.
Check your understanding:
The power dissipated in a resistor of resistance R carrying a current I
is equal to I2R. The value of I has an uncertainty of ± 2% and the value
of R has an uncertainty of ± 10%. The value of the uncertainty in the
calculated power dissipation is
A. ±8%.
B. ±12%.
C. ±14%.
D. ±20%.
Check your understanding:
When a force F of (10.0 ± 0.2) N is applied to a mass m
of (2.0 ± 0.1) kg, the percentage uncertainty attached to
the value of the calculated acceleration is
A. 2%.
B. 5%.
C. 7%.
D. 10%.
Check your understanding:
A student measures a distance several times. The readings lie
between 49.8 cm and 50.2 cm. This measurement is best
recorded as
A. 49.8 ± 0.2 cm
B. 49.8 ± 0.4 cm
C. 50.0 ± 0.2 cm
D. 50.0 ± 0.4 cm
Check your understanding:
Below are the measurements of the diameter of a metal ball. What
is the best estimate result of the measurement with uncertainty?
Check your understanding:
1. A piece of string 1.000 ± 0.002 m is cut from a ball of string of
length 100.000 ± 0.002 m. Calculate the length of the remaining
string and the uncertainty in this length.
2. A runner completes 100 ± 0.02 m in 18.6 ± 0.2 s. Calculate his
average speed and the uncertainty in this value.
3. A car accelerates, with constant acceleration, from 24 ± 1 m s–1
to 31 ± 2 m s–1 in 9.5 ± 0.1 s. Calculate the acceleration. State your
answer with its absolute uncertainty.
4. A cube has a mass of 7.870 ± 0.001 kg and sides of length 10.0
± 0.1 cm. Give the value of the density of the cube.
Physics Worksheet
1) The mass of a marble is measured using a digital
balance. Suggest one possible source of:
(a) a systematic error
(b) A random error
In each case, suggest how the error may be reduced.
Physics Worksheet
2. Read and answer Question number 20 from page 16.
Two set-squares and a ruler are used to measure the
diameter of a cylinder.
The cylinder is placed between the set-squares, and the
set-squares are aligned with the ruler, in the manner of
the jaws of calipers. The readings on the ruler at opposite
ends of a diameter are 4.15 cm and 2.95 cm. Each
reading has an uncertainty of ±0.05 cm.
a What is the diameter of the cylinder?
b What is the uncertainty in the diameter?
Physics Worksheet
3. Read and answer Question number 21 from page 17.
A value of the volume V of a cylinder is determined by
measuring the radius r and the length L. The relation
between V, r and L is V = πr2L
In an experiment, r was measured as 3.30 ± 0.05 cm, and
L was measured as 25.4 ± 0.4 cm. Find the value of V,
and the absolute uncertainty in this value.
Physics Worksheet
4. Read and answer Question number 22 from page 17.
The mass and dimensions of a metal rectangular block
are measured. The values obtained are: mass = 1.50 ±
0.01 kg, length = 70 ± 1 mm, breadth 60 ± 1 mm and
depth 40 ± 1 mm. Determine the density of the metal and
its absolute uncertainty in kg m−3.
Physics Worksheet
5. The figure below shows part
of a thermometer. Determine the
correct reading on the
thermometer with its absolute
uncertainty.
Reading:
____________________
Physics Worksheet
6. A student records 5 repeat readings using a micrometer
screw gauge in mm shown in the table below.
(a) Calculate the average of the readings in the Table. Give your
answer to an appropriate number of significant figures.
(b) Give two reasons why taking repeat readings provides more
accurate data.
(c) Another student repeats the same experiment using a micrometer
screw gauge and obtains an average value of 1.53 ± 0.03 mm.
Calculate the percentage uncertainty in the student’s average.