Transcript Error Analysis - msamandakeller

Error Analysis
SPH4U
Error = Uncertainties
Experimental errors are not mistakes.
They are indications of unavoidable imprecision
and are better called “uncertainties.”
Example
What is the length to the end of the pendulum?
Example
What is the length to the end of the pendulum?
Well, it’s certainly closer to 128.9 cm than 128.8 cm
or 129.0 cm. We’re at least within 0.05 cm.
Given a similar uncertainty at the other end. . .
Example
The end of the mass is at what position?
128.9  0.1cm
central value
uncertainty
Digital Uncertainty
For a digital readout, the error is usually half of
the last digit:
138.2  0.05 lb
Instrumental vs. Procedural
These read-off errors are instrumental.
The dominant error is often procedural.
Instrumental vs. Procedural
These read-off errors are instrumental.
The dominant error is often procedural.
Example:
The uncertainty in the time
for five oscillations of the
pendulum is dominated by
human reaction time, not the
precision of the stopwatch.
Standard Deviation
For a better estimate of our uncertainty, we
perform multiple trials and calculate the
standard deviation from the mean.
Standard Deviation
For a better estimate of our uncertainty, we
perform multiple trials and calculation the
standard deviation from the mean.
Example:
Sig Digs
If we don’t round the mean or standard
deviation, we have:
Sig Digs
If we don’t round the mean or standard
deviation, we have:
This is obviously absurd. We don’t have
nanosecond precision in our error.
And if we’re uncertain about the tenths place,
why would we bother writing decimals out to
the millionth place in our central value?
Sig Digs
If we don’t round the mean or standard
deviation, we have:
General rule:
Round the central value to the rightmost
decimal place at which the error applies.
Here, to the tenths:
Sig Digs
If we don’t round the mean or standard
deviation, we have:
General rule:
You may wish to keep an extra digit for each,
especially when performing calculations:
Relative Error
This is an absolute error.
Relative errors are relative to the central value.
Example:
absolute error
relative error
3/85 = 0.04
Error Propagation
We add absolute errors of measurements when
adding or subtracting them.
Example:
d  5.0  0.1 cm  4.9  0.1 cm
d  9.9  0.2 cm
Consider:
The minimum values of both are 4.9 cm and 4.8 cm: 4.9 cm + 4.8 cm = 9.7 cm.
The maximum values are 5.1 cm and 5.0 cm: 5.1 cm + 5.0 cm = 10.1 cm.
Error Propagation
We add relative errors of measurements when
multiplying or dividing them.
Example: What is the speed of this projectile?
Error Propagation
We add relative errors of measurements when
multiplying or dividing them.
Example: What is the speed of this projectile?
Error Propagation
We add relative errors of measurements when
multiplying or dividing them.
Example: What is the speed of this projectile?
Error Propagation
Similarly, when taking the root or power of a
measurement, we work with the relative error
and multiply the relative error by the power.
Example:
v  25  5
m2
s2
v  5.0 ms  12 20% 
v  5.0 ms  10%
v  5.0  0.5 ms
5 is 20% of 25 and
a square root is a
½ power.
Conversions
But the error in any conversion factor is 0%.
Just convert the error too.
Example:
An Italian police Lamborghini
has a top speed of 309 ± 5 km/h,
which converts to 85.8 ± 1.4 m/s.
Graphing Error
If you are using measurements as data points on
a graph and calculating the slope. . . .
The height and width of
your error bars for each
data point may be
determined by your
standard deviation from
multiple trials.
Graphing Error
. . . draw your maximum and minimum possible
lines of best fit.
Max slope = 25.0 cm/s2
Min slope = 23.8 cm/s2
Graphing Error
. . . and determine your central value and
deviation.
Max slope = 25.0 cm/s2
Min slope = 23.8 cm/s2
Slope = 24.4 ± 0.6 cm/2
Graphing Error
(Note that all lines pass through the origin.)
More on graphical
analysis tomorrow. . . .