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A Student's Guide to Data and Error Analysis

Herman J. C. Berendsen

Chapter 5

Processing of experimental data - all with Video Answers

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Chapter Questions

01:25

Problem 1

Could the data given in Table 2.1 on page 6 be sampled from a normal distribution? If so, estimate $\hat{\mu}$ and $\hat{\sigma}$ by drawing a straight line through the cumulative distribution function of Fig. Figure 2.1.

R M
R M
Numerade Educator

Problem 2

Prove (5.6).

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Problem 3

If you subtract a constant from all values of $x$ and then compute the msd using (5.6), is a further correction still required?

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10:07

Problem 4

Generate 1000 normally distributed variables with mean $c$ and s.d. 1 . Compare the rmsd computed by both (5.4) and (5.6). Vary the constant $c$ (e.g. 1.e6, 1.e7, 1.e8, 1.e9).

Michael Twiton
Michael Twiton
Numerade Educator
08:45

Problem 5

(refer to Table 5.1 on page 61 )

A series of $n$ independent measurements of a physical quantity yields an average of 75.32578 and a mean squared deviation of 25.64306 . Report, with the correct number of digits, your best estimates of the mean and standard deviation of the underlying probability distribution, for two cases: (a) $n=15$, (b) $n=200$.

Jameson Kuper
Jameson Kuper
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02:32

Problem 6

You live in Germany and want to calibrate the speedometer of your car. On a quiet, mostly straight and level Autobahn section you keep your speed as accurate as possible at $130 \mathrm{~km} / \mathrm{hr}$ on your speedometer. Your companion measures with a stopwatch the time between passing two kilometer marks that are exactly 1 km apart. She finds the following nine intervals (in s): ${ }^7$ $29.04,29.02,29.24,28.89,29.33,29.35,29.00,29.25,29.43$
1. Compute the following properties of the measured set of time intervals:
(a) the average,
(b) the average squared deviation from the average,
(c) the root-mean-squared average deviation from the average,
(d) the range, median and the first and third quartiles.
2. Compute the best estimates for the following properties of the underlying distribution function:
(a) the mean $\hat{\mu}$,
(b) the variance $\hat{\sigma}^2$,
(c) the standard deviation $\hat{\sigma}$,
(d) the standard uncertainty of the estimated mean,
(e) the uncertainty of the last three values.
3. What is (the best estimate for) your car's real velocity? What is the standard uncertainty of this value? How large is the speedometer's deviation and what is the relative accuracy of that deviation? Give all values with the correct number of significant digits.
4. If you as driver assert that you have kept the speed within a deviation of $\pm 0.5 \mathrm{~km} / \mathrm{hr}$, does this knowledge influence your conclusions in any way?
5. Assuming that the (biased) bootstrap yields a reliable sampling distribution of the mean, generate a bootstrap distribution of 2000 samples and compute the $80 \%, 90 \%$ and $95 \%$ confidence limits for the time interval.
6. Using this bootstrap distribution, compute the $80 \%, 90 \%$ and $95 \%$ confidence limits for the velocity.
7. Assuming the underlying distribution to be normal $N(\hat{\mu}, \hat{\sigma})$, compute the $80 \%, 90 \%$ and $95 \%$ confidence limits for the velocity.
8. Assuming the underlying distribution to be normal with unknown s.d., compute the $80 \%, 90 \%$ and $95 \%$ confidence limits for the velocity according to Student's t-distribution.

Lucas Finney
Lucas Finney
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06:15

Problem 7

You are a member of a CODATA committee with the task to update Avogadro's number. The following reliable data are at your disposal:
- the already known number (see data sheet PHYSICAL CONSTANTS on page 209)
- a series of measurements by scientist A with result: $6.02214148(75) \times 10^{23}$
- a series of measurements by scientist $B$ with result: $6.02214205(30) \times 10^{23}$
- a series of measurements by scientist C with result: $6.0221420(12) \times 10^{23}$
Give the weighted mean and its standard uncertainty.

Nicole Powell
Nicole Powell
Numerade Educator
02:10

Problem 8

Plot the bootstrap distribution, the histogram of which is given in Fig. 5.4, on a probability scale. Is this distribution compatible with a normal distribution? Estimate graphically the mean and s.d. and compare to the values given in the text.

Neel Faucher
Neel Faucher
Numerade Educator
03:10

Problem 9

(This advanced exercise requires reading of Appendix $A 3$ and Appendix A5.)
Determine - using the characteristic function - the distribution function of the sum of three samples, each randomly chosen with equal probability from the three values $-1,0$ and 1 . Note that the distribution function for the sum of three values equals the convolution of the distribution functions of each value. Determine its variance. Compare your result with Fig. 5.4.

Maxime Rossetti
Maxime Rossetti
Numerade Educator