Measuring
Keeping track of precision
Two Types of numbers
• The two types are exact and measured.
• Exact numbers are arrived at through
counting. Example: How many students
are in this class?
• Measured numbers are found by using a
measuring instrument (ruler, bathroom
scale) and estimating the readout of the
instrument.
All measurements are estimates!
• We may know some parts of a
measurement with certainty but even
digital readouts are typically accurate to
within + or – 5% of the measured value.
Accuracy vs. precision
• If we measure something properly then
our measurement is accurate and reflects
what we are actually measuring.
• Precision is indicated by how many
significant digits we need to describe a
measurement. Compare 100 km/h to
102.35 km/h. Which is more precise?
• It is our goal in physics to be as accurate
and as precise as possible.
Significant Digits
• The digits 1, 2, 3, 4, 5, 6, 7, 8, 9, are
always significant (125 has 3 sig digits)
Zeros
• Zeros are not significant when only used
as placeholders (100 and 0.01 each have
1 significant digit)
• Zeros are significant when
– Sandwiched between two significant digits
• (105 has 3 sd, so does 1050)
– Found to the right of a decimal and not used
as a placeholder ( 0.050 and 5.0 each have 2
significant digits.
Check yourself
• How many Significant digits do each of the
following numbers have?
1236
2500
2500.0
0.00510
2500.00510
Important points
• The precision of a number is indicated by
the number of significant digits in the
number.
• Whether or not a digit is significant can be
determined by whether or not the digit
adds precision to the number.
Scientific Notation
• Only the digits in the first factor are
significant. (The 2nd factor is a power of
ten which is the “placeholder” for the
number).
Calculations
• When measured values become the
numbers used for a calculation the end
result of the calculation has the same
degree of precision as the least precise
number that went into the calculation.
(A chain is as strong as its weakest link)
There are many rules governing how many
sig figs an answer should have based on
the precision of the input values…
We will ignore these.
•
The rules we will use in class are as
follows:
1. All final answers should have 3 + or – 1
Significant digits.
2. All intermediate values used in a
calculation should have a minimum of 5
significant digits.
For Practice
• Significant figures practice worksheet with
answers go to:
http://misterguch.brinkster.net/PRA006.pdf
• For Rounding rules go to the lab manual
found on the grade 11 section Of Mr.
Turton’s website page 12:
http://hrsbstaff.ednet.ns.ca/max/Grade%2
011%20Documents/2008%20Fall%20Phy
sics%2011%20Lab%20Manual.pdf
Precision in the lab
• Precision of measured values is typically
indicated with absolute uncertainty .
• Precision of final results is typically
indicated with a percent uncertainty.
Absolute uncertainty
• Example: 50.5 ± 0.2 cm
• 50.5 cm is the best estimate of the
measurement.
• 0.2 cm is the uncertainty in the
measurement.
• This means that the measurement could
reasonably be as low as 50.3 cm or as
high as 50.7 cm.
Percent Uncertainty
• Example: The absolute uncertainty in 50.5
± 0.2 cm can be converted into a percent
uncertainty as follows
• 0.2/50.5 = 0.003960
• 0.003960 x 100 = 0.396% = 0.4%
• Percent uncertainty : 50.5 cm ± 0.4%
• Both types of uncertainty are represented
with one significant figure.
Uncertainty agreement
• 50.536 ± 0.2 cm is unreasonable
– Why would we be concerned with the extra
0.036 cm when using our best judgement the
measurement could be off by as much as 0.2
cm ?
– The measured or calculated value should
agree with the stated uncertainty.
• 50.5 ± 0.2 cm shows this agreement
Uncertainty from Sig Figs
• A number expressed with the appropriate
number of significant figures may be
converted into an absolute uncertainty as
± 2 of the most precise digit in the
measurement
• 105 km/h = 105 ± 2 km/h
• 1030 m = 1030 ± 20 m
• 0.105 kg = 0.105 ± 0.002 kg
A measurement made without
any knowledge of its precision is
meaningless.
That’s all folks.