Theory of Errors
Exercise 1 : Types of Errors
In the following examples, state whether the error is systematic (constant $S$) or random $R$.
- A 100-metre race is held in a school during a sports day. The judges start their stopwatches when the sound of the starting pistol reaches them. What type of error arises in this case?
- The timing of the start and end of the 100-metre race is subject to individual fluctuations, e.g. reaction time. What kind of error is it?
- The zero point of a voltmeter has been wrongly set. The measurements are therefore subject to an error. What kind of error is it?
- The resistance of a copper coil is obtained by measuring the current flowing through when a voltage is applied to it. As the coil warms up the resistance increases. What kind of error will ensue?
Exercise 2 : Sample Statistics
- Nine different rock samples are taken from a crater on the moon whose densities are determined with the following results: $3.6, 3.3, 3.2, 3.0, 3.2, 3.1, 3.0, 3.1, 3.3 \frac{g}{cm^3}$. Calculate the mean value and standard deviation of the parent population.
- The velocity of a body travelling along a straight line is measured 10 times. The results are $1.30, 1.27, 1.25, 1.26, 1.29, 1.31, 1.23, 1.33, 1.24 \frac{m}{s}$. Calculate the mean value and the standard deviation.
Exercise 3 : Uniform Distribution Moments
A continuous random variable has the following density function:
$$ f(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} $$Calculate the mean value and the variance.
Exercise 4 : Normal Probability
A measured variable is normally distributed with a mean value $\mu = 8$ and a standard deviation $\sigma = 1$. What percentage of all test data are smaller than 7?
Exercise 5 : Error Propagation — Area and Density
- The sides of a rectangle are $x = 120 \pm 0.2 cm$ and $y = 90 \pm 0.1 cm$. Calculate the area and the standard deviation.
- Calculate the density and the standard deviation of a sphere of diameter $6.2 \pm 0.1 mm$ and mass $1000 \pm 0.1 g$.
Exercise 6 : Linear Fit (Spring Balance)
The sensitivity of a spring balance is to be determined. To achieve this we place different masses $m$ on the balance and record the deflection $S$. The results are:
| mass (mg): | 2000 | 3000 | 4000 | 5000 | 6000 |
|---|---|---|---|---|---|
| deflection (mm): | 16 | 27 | 32 | 35 | 40 |
If a straight line is to be fitted through these data points, i.e. $S = am + b$, calculate the values of $a$ and $b$.
Exercise 7 : Linear Regression with Uncertainties
In the spring-balance experiment you may instead have measured additional points with small known uncertainties. Suppose the measured deflections (mm) and their standard uncertainties are: $(16\pm0.5),(27\pm0.5),(32\pm0.5),(35\pm0.5),(40\pm0.5)$. Perform a weighted least-squares fit to $S=am+b$ and compute $a,b$ and their standard uncertainties.
Exercise 8 : Weighted Angle Average
An angle has been measured several times with two theodolites and the following values were obtained: $73^\circ 2'7’’ \pm 10’$ and $73^\circ 2'12’’ \pm 20’’$. Calculate the weighted average.
Exercise 9 : Chi-square Test for Calibration
A calibration experiment yields counts in 5 bins: [102, 98, 95, 105, 100]. The expected (ideal) distribution is uniform. Perform a chi-square goodness-of-fit test at the 5% level and state whether the instrument agrees with the expected uniform response.
Exercise 10 : Monte Carlo Error Propagation
Let $z=\dfrac{xy}{x+y}$ where $x=1.23\pm0.02$ and $y=2.34\pm0.03$ (independent). Estimate the value and standard uncertainty of $z$ using (a) first-order error propagation and (b) a Monte Carlo simulation (draw 10,000 samples from Gaussian distributions and compute the empirical mean and standard deviation).
Exercise 11 : Significant Figures and Rounding Effects
Discuss how rounding to different numbers of significant figures influences propagated results. As a concrete example, compute the area of a rectangle with sides $x=12.34,cm$ and $y=5.678,cm$, (a) using full precision then rounding only the final result to 3 s.f., and (b) rounding each side to 3 s.f. before computing the area. Compare differences.
Exercise 12 : Maximum Likelihood Estimation for Normal Errors
You measure a quantity several times: $8.1,8.0,8.3,7.9,8.2$. Assuming independent Gaussian measurement errors with unknown mean $\mu$ and known standard deviation $\sigma=0.1$, derive the maximum likelihood estimator for $\mu$ and compute its value and standard error.