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? central value uncertainty

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

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

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. . . .