1. Error Analysis
SPH4U

2. Error = Uncertainties
Experimental errors are not mistakes. They are indications of unavoidable imprecision and are better called “uncertainties.”

3. Example
What is the length to the end of the pendulum?

4. Example What is the length to the end of the pendulum?
Well, it’s certainly closer to cm than cm or cm. We’re at least within 0.05 cm.
Given a similar uncertainty at the other end. . .

5. Example The end of the mass is at what position? central value
uncertainty

6. Digital Uncertainty
For a digital readout, the error is usually half of the last digit:

7. Instrumental vs. Procedural
These read-off errors are instrumental. The dominant error is often procedural.

8. 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.

9. Standard Deviation
For a better estimate of our uncertainty, we perform multiple trials and calculate the standard deviation from the mean.

10. Standard Deviation
For a better estimate of our uncertainty, we perform multiple trials and calculation the standard deviation from the mean.
Example:

11. Sig Digs
If we don’t round the mean or standard deviation, we have:

12. 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?

13. 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:

14. 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:

15. Relative Error
This is an absolute error. Relative errors are relative to the central value.
Example:
absolute error
relative error
3/85 = 0.04

16. Error Propagation
We add absolute errors of measurements when adding or subtracting them.
Example:
Consider:
The minimum values of both are 4.9 cm and 4.8 cm: 4.9 cm cm = 9.7 cm.
The maximum values are 5.1 cm and 5.0 cm: 5.1 cm cm = 10.1 cm.

17. Error Propagation
We add relative errors of measurements when multiplying or dividing them.
Example: What is the speed of this projectile?

18. Error Propagation
We add relative errors of measurements when multiplying or dividing them.
Example: What is the speed of this projectile?

19. Error Propagation
We add relative errors of measurements when multiplying or dividing them.
Example: What is the speed of this projectile?

20. 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.
5 is 20% of 25 and
a square root is a ½ power.
Example:

21. 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.

22. 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.

23. Graphing Error
. . . draw your maximum and minimum possible lines of best fit.
Max slope = 25.0 cm/s2
Min slope = 23.8 cm/s2

24. 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

25. Graphing Error (Note that all lines pass through the origin.)
More on graphical analysis tomorrow