The document explains two types of errors in experimental physics: random errors, caused by chance and variability in measurements, and systematic errors, which consistently deviate from the true value, often due to inaccurate instruments or procedures. It outlines how to determine measurement uncertainty, emphasizing the importance of writing measurements in the format 'value ± uncertainty,' and describes how to calculate absolute and percentage uncertainties for addition, subtraction, multiplication, and division. Furthermore, the text defines accuracy and precision in measurements and provides examples to illustrate different relationships between measured values and true values.

1. Experimental Physics
Unit - 3

2. Errors
• There are two types of errors: random and systematic.
• Random error occurs due to chance. There is always some variability
some variability when a measurement is made.
• Random error may be caused by slight fluctuations in an instrument,
the environment, or the way a measurement is read, that do not cause
the same error every time.
• In order to address random error, scientists utilized replication.
• Replication is repeating a measurement many times and taking the
and taking the average.

3. • Systematic error gives measurements that are consistently
different from the true value in nature, often due to
instruments or the procedure.
• Instrumental error happens when the instruments being
used are inaccurate, such as a balance that does not work.
• Procedural error occurs when different procedures are
used to answer the same question and provide slightly
two people are rounding, and one rounds down and the
is procedural error.
• Human error is due to carelessness or to the limitations of
human ability. Two types of human error are
estimation error.

7. Uncertainty — Overview &
Importance
• Measuring instruments and techniques are never
perfect which leads to some uncertainty.
Measurement uncertainty is the degree of error in a
measured value.
• All instruments have uncertainty in their measurements.
• Often, the instrument's owner's manual will tell you its
uncertainty.
• However, if it does not tell you, a general rule of thumb
is to use
1
2
unit of the smallest significant digit.
• For example, if you measure something to be 52.5 grams
your uncertainty would be 0.05 grams.
• Should be written as : 52.5𝑔 ± 0.05𝑔

8. Writing Uncertainties
• When we write our uncertainty we use the
format: measurement ± uncertainty.
• For example, let's say we measure the mass of a
substance to be 5.05 grams and our uncertainty is 0.04
grams.
• We would write our value as 5.05g ±0.04g.
• Our uncertainty tells us that we are confident that our
measurement lies between 5.01g and 5.09g.

9. Uncertainties and Significant
Figures
• Uncertainties follow the rules of significant figures.
• It is always the uncertain digit that sets the number of
significant figures. For example, you could not have a mass
of 5.05g ±0.1g.
• In this case, we would be saying that the tenths place is
uncertain.
• However, we were able to measure to the hundredths place.
• That does not make sense!

10. Calculating Uncertainties
• Eventually, you will use statistics to calculate your
uncertainty.
• You calculate uncertainty by taking multiple
measurements and then using the standard deviation.
• The number of measurements you should take will
depend on your level.
• 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑖𝑡𝑦 =
1
2
∗ (𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒)

11. Absolute
uncertaintie
s &
Percentage
uncertaintie
s
• 52.5𝑔 ± 0.05𝑔
• 0.05g is the absolute uncertainty
•
0.05
52.5
× 100 = 0.095%
• 0.095% is the percentage uncertainty
• % 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑖𝑡𝑦 =
𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑖𝑡𝑦
𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
× 100 %

12. Uncertainties
in
Calculations
: Addition
and
Subtraction
• In some experiments, you will measure
multiple variables to use in a calculation.
• However, each measurement will have its own
uncertainty.
• When you must add or subtract measurements,
the units stay the same.
• When you have the same units, you add the
absolute uncertainties.
• Absolute uncertainties have the same units as
the measurement.
• Addition: 5.05g ±0.04g + 1.01g ±0.03g =
6.01g ±0.06g
• Subtraction: 5.05g ±0.04g - 1.01g ±0.03g =
4.04g ±0.06g

13. Uncertainties in Calculations : Multiplication &
Division
• When we do multiplication or
division, the units in our final answer
are different than our
measurements.
• When we multiply or divide, we
must convert our absolute
uncertainties to percentage
uncertainties.
• Once we have our percentage
uncertainties, we add them
together.
• Finally, we convert our relative
uncertainty back to an absolute
uncertainty:
• Final Answer:
9.29m3 ±4.65% → 9.29m3 ±0.44
m3
𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑖𝑡𝑦 = 9.29 ×
465
100
= 0.44𝑚3

14. Uncertainties in Calculations : Raising
to the power
• If a measurement is raised to a power, for example squared or cubed, then
the percentage uncertainty is multiplied by that power to give the total
percentage uncertainty.
• Example:A builder wants to calculate the area of a square tile. He uses a
rule to measure the two adjacent sides of a square tile and obtains the
following results:
• Length of one side = 84 mm ± 0.5mm Length of perpendicular side = 84
mm ± 0.5mm.
• The percentage uncertainty in the length of each side of this square tile is
given by: Percentage uncertainty = (0.5/84) × 100% = 0.59 % = 0.6 %
• The area of the tile A is given by A = 84 × 84 = 7100 mm2
• Note that this is to 2 SF since the measurements are to 2 SF.

15. • Percentage uncertainty in A = 2 × 0.6% = 1.2%
• Therefore, the uncertainty in A = 7100 × 1.2% = 85 mm2
• So
• A = 7100 mm2 ± 1.2% or A = 7100 mm2 ± 85 mm

16. Accuracy and Precision
• Accuracy is how close the measured value is to the
correct value. Precision is how close repeated
measurements of the same object are to each other.
• When taking measurements of anything there can be
some measurement uncertainty due to limitations of
the measuring instrument or human error.
• There are 2 terms used to describe the amount of
uncertainty in a measurement.
• Accuracy: the closeness of a measured value to
the true value
• Precision: the closeness of different measured
values to each other

17. Comparison Example
Imagine you measure the mass of a metal sphere with a true mass of 15.000g.
If three measurements are taken the possible options are:
•Accurate and precise: average of measurements is close to true mass and the measurements are
all approximately the same
•Average: 15.001g, measurements: 15.000g, 15.001g, 15.001g
•Accurate and not precise: average of measurements is close to true mass, but measurements
are not close together
•Average: 15.001g, measurements: 15.811g, 14.301g, 14.891g
•Not accurate and precise: average of measurements is not close to true mass, but measurements
are close to each other
•Average: 15.827g, measurements: 15.826g, 15.8297g, 15.825g
•Not accurate and not precise: average of measurements is not close to true mass and
measurements are not close to each other
•Average: 15.289g, measurements: 16.020g, 15.324g, 14.523g