measurements and uncertainities for IBDP 11th graders

1. Measurements and
Uncertainities
IB Grade 11 Physics Lesson 1

2. The realm of physics
• 1.1.1 State and compare quantities to the
nearest order of magnitude.
Throughout the study of physics we deal with a wide range of
magnitudes. We will use minuscule values such as the mass of an
electron and huge ones such as the mass of the (observable)
universe. In order to easily understand the magnitude of these
quantities we need a way to express them in a simple form, to
do this, we simply write them to the nearest power of ten
(rounding up or down as appropriate).
That is, instead of writing a number such as 1000, we write 103 .

3. • The use of orders of magnitude is generally
just to get an idea of the scale and differences
in scale of values. It is not an accurate
representation of a value. For example, if we
take 400, it’s order of magnitude is 102 , which
when we calculate it gives 10 x 10 = 100. This
is four times less than the actual value, but
that does not matter. The point of orders of
magnitude is to get a sense of the scale of the
number, in this case we know the number is
within the 100s.

4. 1.1.2 State the ranges of magnitude of distances, masses and
times that occur in the universe, from smallest to greatest.
• Distances:
• sub-nuclear particles: 10-15 m
• extent of the visible universe: 10+25m
• Masses:
• mass of electron: 10-30 kg
• mass of universe: 10+50 kg
• Times:
• passage of light across a nucleus: 10-23 s
• age of the universe : 10+18 s

5. 1.1.3 State ratios of quantities as
differences of orders of magnitude.
• Using orders of magnitude makes it easy to
compare quantities, for example, if we want
to compare the size of an an atom (10-10 m) to
the size of a single proton (10-15 m) we would
take the difference between them to obtain
the ratio. Here, the difference is of magnitude
105meaning that an atom is 105 or 100000
times bigger than a proton.

6. 1.1.4 Estimate approximate values of everyday
quantities to one or two significant figures and/or to
the nearest order of magnitude.
• Significant figures
To express a value to a certain amount of significant
figures means to arrange the value in a way that it
contains only a certain amount of digits which
contribute to its precision.
• For example, if we were asked to state the value of an
equation to three significant figures and we found the
result of that value to be 2.5423, we would state it as
2.54.
• Note that 2.54 is accurate to three significant figures as
we count both the digits before and after the point.

7. • The amount of significant figures includes all
digits except:
• leading and trailing zeros (such as 0.0024 (2
sig. figures) and 24000 (2 sig. figures)) which
serve only as placeholders to indicate the
scale of the number.
• extra “artificial” digits produced when
calculating to a greater accuracy than that of
the original data, or measurements reported
to a greater precision than the equipment
used to obtain them supports.

8. Rules for identifying significant figures:
All non-zero digits are considered significant (such as
14 (2 sig. figures) and 12.34 (4 sig. figures)).
Zeros placed in between two non-zero digits (such as
104 (3 sig. figures) and 1004 (4 sig. figures))
Trailing zeros in a number containing a decimal point
are significant (such as 2.3400 (5 sig. figures) note
that a number 0.00023400 also has 5 sig. figures as
the leading zeros are not significant).

9. Note that a number such as 0.230 and 0.23 are
technically the same number, but, the former (0.230)
contains three significant figures, which states that it
is accurate to three significant figures. On the other
hand, the latter (0.23) could represent a number
such as 2.31 accurate to only two significant figures.
The use of trailing zeros after a decimal point as
significant figures is is simply to state that the
number is accurate to that degree.

10. Another thing to note is that some numbers with no decimal
point but ending in trailing zeros can cause some confusion.
For example, the number 200, this number contains one
significant figure (the digit 2). However, this could be a
number that is represented to three significant figures which
just happens to end with trailing zeros.
Typically these confusions can be resolved by taking the
number in context and if that does not help, one can simply
state the degree of significance (for example “200 (2 s.f.)” ,
means that the two first digits are accurate and the second
trailing zero is just a place holder.

11. Expressing significant figures as orders of magnitude:
To represent a number using only the significant digits can
easily be done by expressing it’s order of magnitude. This
removes all leading and trailing zeros which are not
significant.
For example:
0.00034 contains two significant figures (34) and fours leading
zeros in order to show the magnitude. This can be
represented so that it is easier to read as such: 3.4 x 10-4 .

12. Note that we simply removed the leading zeros and
multiplied the number we got by 10 to the power of
negative the amount of leading zeros (in this case 4). The
negative sign in the power shows that the zeros are leading.
A number such as 34000 (2 s.f.) would be represented as 34
x 103 .

13. Again, we simply take out the trailing zeros, and multiply the
number by 10 to the power of the number of zeros (3 in this
case).
There are a couple of cases in which you need to be careful:
A number such as 0.003400 would be represented as 3.400 x
10-3 . Remember, trailing zeros after a decimal
point are significant.
56000 (3 s.f.) would be represented as 560 x 102 . This is
because it is stated that the number is accurate the 3
significant zeros, therefor the first trailing zero is significant
and must be included. Note that this can be used as another
way to express a value such as 56000 to three significant
figures (as opposed to writing “56000 (3 s.f.)”).

14. Rounding
When working with significant figures you will often have
to round numbers in order to express them to the
appropriate amount of significant figures.
For example:
State 2.342 to three significant figures: would be written
2.34.
When representing the number 2.342 to three significant
figures we rounded it down to 2.34. This means that when
we removed the excess digit, it was not high enough to
affect the last digit that we kept.

15. Whether to round up or down is a simple decision:
If the first digit in the excess being cut off is lower
than 5 we do not change the last digit which we
are keeping. If the first digit in the excess which is
being cut off is 5 or higher we increment the last
digit that we are keeping (and the rest of the
number if required).

16. For example:
State 5.396 to three significant figures: would be
written 5.40. This is because we remove the last
digit (6) in order to have three significant digits.
However, 6 is large enough for it to affect the digit
before it, therefor we increment the last digit of
the number we are keeping by 1. Note that the
last digit of the number that we are keeping is 9,
therefor incrementing it by 1 gives 10. Thus we
also need to increment the second to last digit by
1. This gives 5.40 which is accurate to three
significant figures.

17. You might think that if a number such as 5.4349 were to be
rounded to 3 s.f. it would give 5.44 as the last digit is 9
which is large enough to affect the previous digit which
would then become 5. Now that 5 would be large enough
to affect the last digit of the number we are keeping which
would become 4 (thus 5.44). However, this is not the case,
when rounding, we only look at the digit immediately after
the one we are rounding to, whether or not that digit
would be affected by the one after it is not taken into
account. Therefor, the correct result of this question would
be 5.43.

18. 1.2.1 State the fundamental units in the SI system.
Many different types of measurements are made in physics.
In order to provide a clear and concise set of data, a specific system of units is
used across all sciences.
This system is called the International System of Units
(SI from the French "Système International d'unités").
The SI system is composed of seven fundamental units:

19. Figure 1.2.1 - The fundamental SI units
Quantity Unit name Unit symbol
mass kilogram kg
time second s
length meter m
temperature kelvin K
Electric current ampere A
Amount of substance mole mol
Luminous intensity candela cd
Note that the last unit, candela, is not used in the IB diploma program.

20. 1.2.2 Distinguish between fundamental and derived
units and give examples of derived units.
• In order to express certain quantities we combine the SI
base units to form new ones. For example, if we wanted to
express a quantity of speed which is distance/time we write
m/s (or, more correctly m s-1). For some quantities, we
combine the same unit twice or more, for example, to
measure area which is length x width we write m2.
• Certain combinations or SI units can be rather long and
hard to read, for this reason, some of these combinations
have been given a new unit and symbol in order to simplify
the reading of data.
For example: power, which is the rate of using energy, is
written as kg m2 s-3. This combination is used so often that
a new unit has been derived from it called the watt
(symbol: W).

21. Below is a table containing some of the SI derived units you will often encounter:
Table 1.2.2 - SI derived units
SI derived unit Symbol SI base unit Alternative unit
newton N kg m s-2 -
joule J kg m2 s-2 N m
hertz Hz s-1 -
watt W kg m2 s-3 J s-1
volt V kg m2 s-3 A-1 W A-1
ohm Ω kg m2 s-3 A-2 V A-1
pascal Pa kg m-1 s-2
N m-2

22. 1.2.3 Convert between different units of quantities.
• Often, we need to convert between different units. For
example, if we were trying to calculate the cost of
heating a litre of water we would need to convert
between joules (J) and kilowatt hours (kW h), as the
energy required to heat water is given in joules and the
cost of the electricity used to heat the water is a
certain price per kW h.
• If we look at table 1.2.2, we can see that one watt is
equal to a joule per second. This makes it easy to
convert from joules to watt hours: there are 60 second
in a minutes and 60 minutes in an hour, therefor, 1 W h
= 60 x 60 J, and one kW h = 1 W h / 1000 (the k in kW h
being a prefix standing for kilo which is 1000).

23. 1.2.4 State units in the accepted SI format.
• There are several ways to write most derived
units. For example: meters per second can be
written as m/s or m s-1. It is important to note
that only the latter, m s-1, is accepted as a valid
format. Therefor, you should always write
meters per second (speed) as m s-1 and meters
per second per second (acceleration) as m s-2.
Note that this applies to all units, not just the
two stated above.

24. 1.2.5 State values in scientific notation and in
multiples of units with appropriate prefixes.
• When expressing large or small quantities we
often use prefixes in front of the unit. For
example, instead of writing 10000 V we write 10
kV, where k stands for kilo, which is 1000. We do
the same for small quantities such as 1 mV which
is equal to 0,001 V, m standing for milli meaning
one thousandth (1/1000).
• When expressing the units in words rather than
symbols we say 10 kilowatts and 1 milliwatt.

25. 1.2.6 Describe and give examples of random and
systematic errors.
• Random errors
A random error, is an error which affects a reading at random.
Sources of random errors include:
• The observer being less than perfect
• The readability of the equipment
• External effects on the observed item
• Systematic errors
• 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

26. • 1.2.7 Distinguish between precision and accuracy.
• Precision
A measurement is said to be accurate if it has
little systematic errors.
• Accuracy
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).

27. • 1.2.8 Explain how the effects of random errors
may be reduced.
• The effect of random errors on a set of data
can be reduced by repeating readings. On the
other hand, because systematic errors occur
at each reading, repeating readings does not
reduce their affect on the data.

28. 1.2.9 Calculate quantities and results of calculations to
the appropriate number of significant figures.
• The number of significant figures in a result should mirror
the precision of the input data. That is to say, when dividing
and multiplying, the number of significant figures must not
exceed that of the least precise value.
• Example:
Find the speed of a car that travels 11.21 meters in 1.23
seconds.
• 11.21 x 1.13 = 13.7883
• The answer contains 6 significant figures. However, since
the value for time (1.23 s) is only 3 s.f. we write the answer
as 13.7 m s-1.
• The number of significant figures in any answer should
reflect the number of significant figures in the given data.

29. 1.2.10 State uncertainties as absolute, fractional and
percentage uncertainties.
• Absolute uncertainties
When marking the absolute uncertainty in a piece of
data, we simply add ± 1 of the smallest significant
figure.
• Example:
• 13.21 m ± 0.01
0.002 g ± 0.001
1.2 s ± 0.1
12 V ± 1

30. • Fractional uncertainties
To calculate the fractional uncertainty of a piece
of data we simply divide the uncertainty by the
value of the data.
• Example:
• 1.2 s ± 0.1
Fractional uncertainty:
0.1 / 1.2 = 0.0625

31. • Percentage uncertainties
To calculate the percentage uncertainty of a
piece of data we simply multiply the fractional
uncertainty by 100.
• Example:
• 1.2 s ± 0.1
• Percentage uncertainty:
• 0.1 / 1.2 x 100 = 6.25 %

32. 1.2.11 Determine the uncertainties in results.
Simply displaying the uncertainty in data is not
enough, we need to include it in any calculations we
do with the data.
Addition and subtraction
When performing additions and subtractions we
simply need to add together the absolute
uncertainties.
Example:
Add the values 1.2 ± 0.1, 12.01 ± 0.01, 7.21 ± 0.01
1.2 + 12.01 + 7.21 = 20.42
0.1 + 0.01 + 0.01 = 0.12
20.42 ± 0.12

33. Multiplication, division and powers
When performing multiplications and divisions, or,
dealing with powers, we simply add together the
percentage uncertainties.
Example:
Multiply the values 1.2 ± 0.1, 12.01 ± 0.01
1.2 x 12.01 = 14
0.1 / 1.2 x 100 = 8.33 %
0.01 / 12.01 X 100 = 0.083%
8.33 + 0.083 = 8.413 %
14 ± 8.413 %

34. Other functions
For other functions, such as trigonometric ones, we calculate the mean,
highest and lowest value to determine the uncertainty range. To do this, we
calculate a result using the given values as normal, with added error margin
and subtracted error margin. We then check the difference between the best
value and the ones with added and subtracted error margin and use the
largest difference as the error margin in the result.
Example:
Calculate the area of a field if it's length is 12 ± 1 m and width is 7 ± 0.2 m.
Best value for area:
12 x 7 = 84 m2
Highest value for area:
13 x 7.2 = 93.6 m2
Lowest value for area:
11 x 6.8 = 74.8 m2
If we round the values we get an area of:
84 ± 10 m2

35. 1.2.12 Identify uncertainties as error bars in graphs.
When representing data as a graph, we represent uncertainty in the data points by
adding error bars. We can see the uncertainty range by checking the length of the
error bars in each direction. Error bars can be seen in figure 1.2.1 below:
Figure 1.2.1 - A graph with error bars

36. 1.2.13 State random uncertainty as an uncertainty range (±) and represent it
graphically as an "error bar".
In IB physics, error bars only need to be used when the
uncertainty in one or both of the plotted quantities are
significant. Error bars are not required for trigonometric
and logarithmic functions.
To add error bars to a point on a graph, we simply take the
uncertainty range (expressed as "± value" in the data) and
draw lines of a corresponding size above and below or on
each side of the point depending on the axis the value
corresponds to.

37. Example:
Plot the following data onto a graph taking into
account the uncertainty.
Time ± 0.2 s Distance ± 2 m
3.4 13
5.1 36
7 64

38. Figure 1.2.2 - Distance vs. time graph with
error bars

39. In practice, plotting each point with its specific error bars
can be time consuming as we would need to calculate
the uncertainty range for each point. Therefor, we often
skip certain points and only add error bars to specific
ones. We can use the list of rules below to save time:
Add error bars only to the first and last points
Only add error bars to the point with the worst
uncertainty
Add error bars to all points but use the uncertainty of the
worst point
Only add error bars to the axis with the worst uncertainty

40. Gradient
To calculate the uncertainty in the gradient, we simply add error bars to the first and
last point, and then draw a straight line passing through the lowest error bar of the one
points and the highest in the other and vice versa. This gives two lines, one with the
steepest possible gradient and one with the shallowest, we then calculate the gradient
of each line and compare it to the best value. This is demonstrated in figure 1.2.3
below:

41. Intercept
To calculate the uncertainty in the intercept, we do the same thing as when calculating the
uncertainty in gradient. This time however, we check the lowest, highest and best value for
the intercept. This is demonstrated in figure 1.2.4 below:
Figure 1.2.4 - Intercept uncertainty in a graph
Note that in the two figures above the error bars have been exaggerated to improve
readability.

42. Prepared by Mesut Mızrak
IBDP Physics Teacher
2014