Quiz 1 – Study Guide
Uncertainty, Error and Significant digits
1. Calculations with significant figures: addition or subtraction
The number that is the least precise , or has the least number of decimal places, determines
number of decimal places and subsequently the number of sig. figs.
56.25 + 7.1 = 63.35 = 63.4 m
1 decimal place because of the 7.1 and total of 3 sig. digits
Multiplication and Division
The limiting term is the number with the smallest number of significant digits.
For example:
4.56 x 1.4 = 6.384
since 1.4 has only 2 significant digits the answer must have 2
significant digits
answer = 6.4
2. Error and Uncertainty : Random error and Systematic Error
Types of Errors
Error can be divided into two main classes: random errors; and systematic errors.
Random errors are due to unpredictable variations in the performance of the instrument, the
operator and/or environmental conditions. If readings of a measurement are above and below
the “correct” value with equal probability, then the errors are random. In other words random
errors fluctuate about some value. Random errors can be caused by such things as:
- vibrations and air currents in mass readings;
- temperature variations;
- variation in the thickness of a surface being measured e.g. thickness of a wire;
- using a less sensitive instrument when a more sensitive instrument is available;
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- misreading scales e.g. human parallax error; ( see fig. 2.2 p. 11 Tsokos)
- poorly timed actions e.g. reactions time when you click a stopwatch;
Averaging repeated experiments minimizes the effects of random errors. Consistent
experimental procedures can also minimize random errors.
Example : say the following data were taken form a person trying to measure the length of a
stick. Th true value of the stick is 5.0 cm labeled T
I I
I I
T
I
I I
The other lines represent the measurements: 4.5, 4.6 , 4.7 , 4.8 , 5.2, 5.3 , and 5.4.
by taking the average x = 4.5 + 4.6 + 4.7 + 4.8 + 5.2 + 5.3 + 5.4
= 4.9
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and so the random error has been reduced by averaging out.
Systematic errors are due to the system or apparatus being used. A systematic error is when
every data point deviates from the “correct” value by the same fixed amount, resulting in a
random set of measurements to be spread about a value that is not the accepted value. They
are reproducible inaccuracies that are consistently in the same direction. Hence, systematic
errors are predictable. They can often be detected by repeating the measurement using a
different method or different apparatus and comparing the results. Systematic errors can result
from:
- badly made equipment/instruments (e.g. a stopwatch running too fast or slow; - - movement in the support for a pendulum);
-slight
- poorly calibrated instruments;
- an instrument having a zero error, a form of calibration;
-incorrect method or procedure used to take the measurement (e.g. error in
-temperature measurement due to poor thermal contact between the thermometer
and the substance being measured). when the act of measuring changes the measurement
(e.g. temperature of a small sample, voltmeter, ammeter, etc.)
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Systematic errors can be eliminated or corrected before the investigation is carried out on
most occasions. Instruments can deteriorate or be damaged by use so it is important to
calibrate instruments regularly by having them measure an accepted standard. Systematic
errors are not reduced by repeating readings. For example; if a measuring instrument is
incorrectly calibrated, a systematic error will appear which cannot be corrected by making
repeated measurements. Zero Error on an instrument = systematic error
Example : say the following data were taken from a person trying to measure the length of a
stick. The true value of the stick is 5.0 cm labeled T
I I I II I I
T
The other lines represent the measurements: 3.9 , 4.2, 4.3, 4.5, 4.55, 4.6, 4.7
x = 3.9 + 4.2+ 4.3 + 4.5 + 4.55 + 4.6 + 4.7
= 4.4
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So no matter how many times you repeat the measurement you will not be able to “ average
out “ the error. There is something wrong with the instrument.
Tsoko text p. 13: 1 and 2
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Mistakes on the part of individuals such as the following are definite sources of error but they
are not considered as experimental error.:




writing the wrong value into a notebook;
wrongly transferring raw data to the final report.
poor arithmetic and computational skills.
using the wrong theory or equations;
In general, all errors are usually a combination of random and systematic error.
3. Error and Uncertainty : Accuracy and Precision
Accuracy and Precision
When obtaining experimental data, we need to be aware of the errors that are associated with
accuracy and precision. These two terms have distinct meanings in science.

Accuracy is an indication of how close a measurement is to the accepted value found in
a handbook. It refers to how well we measure a value. An accurate experiment has a
low systematic error.

Precision is the degree of exactness to which the measurement of a quantity can be
reproduced (an indication of how close repeated measurements are to each other). A
measurement of 1.257 m is more precise than a measurement of 1.25 m. The
achievable precision depends on the instrument. On a meter ruler, the smallest division
may be 1cm. If we specify a length more precisely than to the nearest 1cm, we are
making an estimate. Every instrument has a limited precision that cannot be improved.
A precise experiment has a low random error.
Accuracy and precision are different as shown in Figure 1.1 (Kerr, Kerr and Ruth 2002; 15).
Consider the following example: a technician is fine tuning a computer monitor by aiming an
electron gun at a pixel in the screen
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In the first example there is low accuracy and precision. The technician adjusts the collimator to
reduce the scattering of electrons and changes the magnetic field so the electrons hit the pixel
target. In the second example the collimator has been adjusted to increase the precision but
the magnetic field still needs adjustment. In the third example both adjustments have been
made. What should the technician do to correct figure 4?

Examples: T = true value
1)
IIII
III = measurements
T
precise but not accurate
2) I I T I I
accurate but not precise
3) IITIII
precise and accurate
4) II
I
T I
[Brainpop – Precision and Accuracy]
III
all over the place, neither accurate nor precise
Systematic error, random error, precise, and accurate:
1) An instrument that has systematic error ( calibration problem or zero error) will not have
accurate measurements. But it can be precise:
IIII
T
precise but not accurate
2) A instrument may have no systematic error but there may be random error when a
person takes measurements. In this case the instrument can be accurate but not precise:
I IT I I
accurate but not precise
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4. Error and Uncertainty : Absolute Uncertainty / Relative or %
Uncertainty
A) Absolute Uncertainty
Absolute uncertainty simply refers to the size of an uncertainty and its units. Absolute
uncertainty can be calculated by a number of different methods:

Equipment uncertainty (also called reading error) is  the smallest unit on
the scale. When making any measurement we read down to the smallest unit on
the scale and then we estimate the next figure. This last figure is uncertain; so
the actual the degree of uncertainty is half the limit of reading. However, most
measurements (exception: volume in a measuring cylinder and thermometer)
require two readings: one to zero the scale and one to make the measurement;
the uncertainty is doubled to  the smallest unit on the scale. For example,
when using a ruler the smallest graduation is 1mm and the uncertainty is
 1mm or 0.001m.
For digital instruments we take the uncertainty to be the smallest division. For example, a
stopwatch that reads to 25.38 s will have an uncertainty of + 0.01 s.
Thermometer, dial instruments and graduated cylinder = half the smallest scale
Ruler = smallest division
Note: uncertainty in raw data is usually larger than the equipment uncertainty when other
considerations are taken into account e.g. reaction time when using a stopwatch,
parallax error when taken from afar, etc.
Maximum uncertainty: it is sufficient for IB Physics Labs to approximate
measurement uncertainty by the following:
The average of the measurements + maximum uncertainty
maximum uncertainty = (maximum value – minimum value)
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For example, if the length of a desk was measured five times, and the values obtained were
1.01 m, 1.02 m, 0.98 m, 0.99 m and 1.00 m.
Step 1 : Calculate the avg.: 1.01 + 1.02 + 0.98 + 0.99 + 1.00
= 1.00 m
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Step 2 : Calculate the maximum uncertainty:
maximum uncertainty = (maximum value – minimum value)
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2
maximum value is 1.02
and minimum value is 0.98 so
maximum uncertainty = ( 1.02 – 0.98 ) = 0.02
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So the length would be quoted as 1.00  0.02 m. In other words, we are confident that the true
length of the desk is between 0.98 m and 1.02 m. The  0.02 m is called the absolute
uncertainty in the measurement.
5. Graphical Analysis – Error Bars, Line of Best Fit, Slope
Read Text Tsoko p. 12 and see example fig. 2.4 about error bars and line of best fit
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Line of best fit must intersect one part of all error bars and try to pass through origin
May Exam 2009 Paper 1 # 1 : error bars and error
A. Measurements show significant systematic error but small random error.
B. Measurements show significant random error but small systematic error.
C. The measurements are accurate but not precise.
D. The weight of an object is proportional to its mass.
Answer : A
NOTE small error bars = small random error
systematic error = when the best fit line can not be made to pass through zero.
Significant systematic error when line not really close to zero.
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Data Analysis Question
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(ii) At T = 1.0 N the speed v = 27 + 5ms-1 . Calculate the uncertainty in v2.
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Data Analysis Question - Answers
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Some More Uncertainty Examples check answers at end of packet
Example 1 - Find the overall % uncertainty and absolute uncertainty for the following
equation and the measurements:
X = 6a + 4b
a = 40 + 2
b = 20 + 2
Show work:
Example 2
Formula X = a + b : Two forces have magnitude of a = 50 N + 3N and b = 25 N + 2N .
For X = a + b ( sum of the forces) :
a) state the measurement with uncertainty
b) calculate the % uncertainty of the final measurement
c) calculate the relative uncertainty of the final measurement
SHOW WORK:
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Example 3
The force exerted by a gas on a piston of radius 8.0 + 0.4 mm is measured to be 30 + 2 N.
Calculate the uncertainty in the area of the piston ( area = π r2 )
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ANSWERS
Example 1
X = 6a + 4b
a = 40 + 2
b = 20 + 2
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Example 2
Formula X = a + b : Two forces have magnitude of a = 50 N + 3N and b = 25 N + 2N .
For X = a + b ( sum of the forces) :
a) state the measurement with uncertainty
b) calculate the % uncertainty of the final measurement
c) calculate the relative uncertainty of the final measurement
x=a +b
a = 50 N + 3N
b = 25 N + 2N
a) final measurement and uncertainty
75 N +
just add:
5N
b) % uncertainty = 5 = 0.066 = 0.07 = 7 % ( 1 sig. dig)
75
c) relative uncertainty = 5 = 0.066 = 0.07 ( 1 sig. dig)
75
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Example 3
The force exerted by a gas on a piston of radius 8.0 + 0.4 mm is measured to be 30 + 2 N.
Calculate the uncertainty in the area of the piston ( area = π r2 )
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