MEASUREMENT UNCERTAINTIES

1. Table Labeling
Conventions

2. • To label a table, it is
conventional to have the
independent variable (the
variable being controlled in
an experiment ) in the first
column from the left and the
dependent variable (the
variable that is being
observed or calculated
depending on the
independent variable) in the
column on the right.
Independent
Variable
Dependent
Variable

3. Example
• A student passes different values of electrical current through a
resistor. For each value of current, he records the voltage across the
resistor.
Electric Current
(Amperes)
Voltage (Volts)
1 0.1
2 0.2
3 0.3
4 0.4
The headings of the table are ideally
represented as:

4. Graph Labeling Convention

5. • In creating a graph labeling convention, it is customary to place the
independent variable on the horizontal axis while the dependent
variable in the vertical axis.
10 2.12
20 4.23
30 6.55
40 8.72

6. 10 20 30 40
0
1
2
3
4
5
6
7
8
9
10
Time, t (seconds)
Speed
,v
(m/s)
In the figure, the graph labeling convention of the independent variable time,
t, is placed on the x-axis while the dependent variable speed, v, is placed on
the y-axis.

7. Units, Measurements, and
Quantities: The
Measurements of Uncertainty

8. Essential idea:
Scientists aim towards designing experiments that can give a
“true value” from their measurements, but due to the limited precision
in measuring devices, they often quote their results with some form of
uncertainty.
Nature of science:
Uncertainties:“All scientific knowledge is uncertain.When the scientist
tells you he does not know the answer, he is an ignorant man.When he
tells you he has a hunch about how it is going to work, he is uncertain
aparamount importance, in order to make progress, that we recognize this
ignorance and this doubt. Because we have the doubt, we then propose
looking in new directions for new ideas.”
– Feynman, Richard P. 1998.The Meaning of It All:
Thoughts of a Citizen-Scientist.
Reading, Massachusetts, USA. Perseus. P 13.

9. • Measurements have a degree of uncertainty that comes from different
sources.
• Uncertainty analysis or Error analysis is used to calculate
uncertainty
• A value that is measured should have an estimate together with the
uncertainty value.

10. Measurement is not one
particular value, rather it is a
range of values.
For example, the result
(20.1 ± 0.1) cm basically
communicates that the
person making the
measurement believes the
value to be closest to 20.1
cm but it could have been
anywhere between 20.0 cm
and 20.2 cm.

12. Accuracy and Precision

13. Accuracy vs Precision
• Accuracy is defined as how close a measured value to a true
or accepted value is.
• Precision is defined as how good a measurement can be
determined. It is the amount of consistency of independent
measurements.

14. precise, not
accurate
accurate, not
precise
neither precise,
nor accurate
both accurate
and precise

15. Measurements that are close to the known value are said to be
accurate, whereas measurements that are close to each other are
said to be precise.
Precision determined the quality of the measurement while accuracy
shows the closeness of your answer to the “exact” answer.

16. • Precision is expressed as a relative or fractional uncertainty:
Example:
If mass (m) = 75.5 ± 0.5 g, the relative or fractional uncertainty is
obtained as:

17. A student measures the mass of a rock and
records it as 120.0 g ± 1.5 g. What is the
relative uncertainty of the measurement?
Tr y This!

18. A thermometer reads the temperature of a
liquid as 78.6°C with an uncertainty of ±0.3 °C.
What is the relative uncertainty of the
temperature reading?
Tr y This!

19. • Accuracy is expressed using relative error.
For example, if the expected value for mass = 80.0 g, the
relative error is:

20. A student uses a digital scale to measure sugar
for an experiment.The scale reads 120.0 g with
an uncertainty of ±1.5 g.The expected amount
of sugar based on the recipe is 125.0 g.
a.What is the relative uncertainty of the
measurement?
b.What is the relative error compared to the
expected value?
PRACTICE PROBLEM1

21. A thermometer shows a patient's temperature
as 37.2°C with an uncertainty of ±0.2 °C.The
standard healthy body temperature is 37.0 °C.
a.What is the relative uncertainty of the
thermometer reading?
b.What is the relative error from the expected
value?
PRACTICE PROBLEM2

22. A student measures the volume of water in a
beaker and gets 250.0 mL with an uncertainty
of ±1.0 mL.The volume was supposed to be
exactly 255.0 mL.
a. Calculate the relative uncertainty of the
measurement.
b. Find the relative error compared to the
expected volume.
PRACTICE PROBLEM3

23. A physics student uses a ruler to measure a
metal rod and gets 40.0 cm, with an uncertainty
of ±0.4 cm.According to the manual, the actual
length should be 39.5 cm.
a.What is the relative uncertainty of the
measurement?
b.What is the relative error from the expected
length?
PRACTICE PROBLEM4

24. Forms of Error

25. Forms of Error
•In performing measurements, errors are sometimes
committed.
•They are classified as either random or systematic
depending on the situation on how the measurement
was obtained

26. Random Errors
• Variations in the measured data brought by the limitations of the
measuring device.
• Refers to random fluctuations in the measured data due to:
the readability of the instrument
the effects of something changing in the surroundings between measurements
the observer being less than perfect
▪ Random errors can be reduced by averaging.
A precise experiment has small random error.

27. Systemic Error
•Reproducible inaccurate data that are constantly in
the same direction.
• Example:
if a systematic error is identified due to calibration of
a measuring instrument based on standard, applying a
correction factor to compensate the effect can reduce the
favored measurement.

28. Systemic Error
• error due to the instrument being “out of adjustment.”
• An instrument with a zero offset error.
• A meter stick might be worn off or rounded at one end
• An instrument that is improperly calibrated
▪ Systematic errors are usually difficult to detect.
▪ Systematic errors can be detected using
different methods of measurement

29. • A measurement is said to be accurate if it has little systematic errors.
• A measurement is said to be precise if it has little random errors.
• A measurement can be of great precision but be inaccurate (for
example, if the instrument used had a zero offset error).
precise, not
accurate
accurate, not
precise
neither precise,
nor accurate
both accurate
and precise
RE
S
E
RE
SE
RE
SE

RE
S
E

30. ·This is like the rounded-end ruler. It will produce a systematic error.
·Thus its error will be in accuracy, not precision.

31. Causes of Error in Doing Physics
Laboratory Experiments

32. Inadequate Definition
•Either Systematic or Random
•For example, if two students measure the length of a
rope, they possible get different results because either
one may stretch the rope with a different force

33. Unable to Include a Factor
•Systematic
•For example, when measuring free fall, air resistance
was not considered.
•A good way to analyze this source of error is to
discuss all aspects that could probably affect the result
before doing the experiment so that considerations
can be made before doing the measurements.

34. Factors Due to the Environment
•Either systematic or random
•Errors brought by the environment such as vibration,
temperature, noise, or other conditions that may
affect the measuring instrument.

35. Limited Scale of the Instrument
•Random
•For example a meter cannot measure exactly in the
smallest scale division.

36. Unable to Calibrate or Check
Zero Scale of the Instrument
•Systematic
•If possible, always check the calibration of the
instrument before taking measurements

37. Variation in the Physical
Environment
•Take several measurements over a whole range that is
being explored.This will reveal variations in the
experiment that might not be noticed.

38. Parallax
• Either systematic or random
• Whenever an experimenter’s eye is not aligned with a pointer in the
scale, the reading may differ either too high or low

39. Personal Errors
•Errors that occur from carelessness, poor method, or
bias measurement from the experimenter

40. Approximating Uncertainty
in Repetitive
Measurements

41. • To show the differences in the measurements, use the
average deviation. It illustrates the average of individual
measurements that varies from the mean with 50%
confidence.
• Standard Deviation is a mathematical way to characterize
the spread of a set of data. It is slightly greater than average
deviation and is used because of its link with normal
distribution that is often encountered in statistics.

42. • The importance of standard deviation is if one can have
more measurements using the same meter stick, one can
expect that the new measurement is within the range of 0.12
cm with an average 31.19 cm with 68% confidence level.

43. • Suppose you measure the length of a rod 5 times and get the
following results:
• Measurements: 10.2 cm, 10.4cm, 10.3cm, 10.5cm, 10.1cm
• STEP 1: Compute the mean (Average)

44. • STEP 2: Find the Deviation from the Mean and Square It
Measurement Deviation (x - x )
̄ Squared Deviation
(x - )²
x
̄
10.2 10.2-10.3 (-0.1) 0.01
10.4
10.3
10.5
10.1

45. • STEP 4: Calculate the Standard Deviation
• STEP 3: Calculate theVariance
Square root of the computed variance

46. Measurem
ents
Width of
Paper (cm)
Deviation (cm)
1 31.33
2 31.15
3 31.26
4 31.02
5 31.20

47. • Temperatures (°C): 27, 32, 29, 35, 30, 31, 37
Measurement Deviation (x - x )
̄ Squared Deviation
(x - )²
x
̄
27
32
29
35
30
31
37

48. A balance is used to measure the mass of an
object five times.The results (in grams) are:
10.02, 10.05, 9.98, 10.01, 10.03
Question:
Determine the mean, variance, and standard
deviation of the mass measurements.
PROBLEM 1:

49. During a study of human error in length
measurements, a group of students measured the
same object and recorded the following lengths
(in cm):
30.2, 30.5, 30.3, 30.6, 30.4
Question:
Compute the mean, then find the variance and
standard deviation of these measurements.
PROBLEM 2:

50. Volumes (in mL):
100.1, 100.0, 99.9, 100.2, 100.1, 100.0,
100.2
Question:
What is the mean, variance, and standard
deviation of these volume readings?
PROBLEM 3: