Assessment Progress:

These self-assessment problems accompany the mini course Calculating precision from data.

In most of the examples of this mini course so far, you have been given a variance-covariance matrix from which to work, as if it had come out of an estimation process. The point of the current topic was to remind you of how to compute relevant values yourself, so that’s what the following questions aim to help you do.

Feel free to use Excel for doing this work.

Problem 1

Imagine that a distance is known to be 1042.1220 m. And that you measure that distance a bunch of times to yield the following, all in metres:

    \begin{equation*} \mathbf{l}_{measured}= \begin{bmatrix} 1043.817976 \\ 1043.823131 \\ 1043.818425 \\ 1043.841238 \\ 1043.832277 \\ 1043.820113 \\ 1043.82398 \\ 1043.82651 \\ 1043.831772 \\ 1043.811224 \\ 1043.822178 \\ 1043.81479 \\ 1043.818391 \\ 1043.819674 \\ 1043.820842 \\ 1043.810539 \\ 1043.820867 \\ 1043.833896 \\ 1043.827218 \\ 1043.829174 \\ 1043.817662 \\ 1043.818788 \end{bmatrix} \end{equation*}

Calculate:

a) the sample mean of the errors

b) the sample variance of the errors

c) the sample standard deviation of the errors

d) the RMS of the errors

Problem 2

For the same measurement data provided above calculate:

a) the sample mean, \bar{x}

b) the sample variance, s_x^2

c) the sample standard deviation, s_x

d) the standard error of the sample mean, SE (using SE = s_x/\sqrt(n) and n is the sample size)

Problem 3

Clearly explain:

a) what does the sample standard deviation tell us about the data?

b) what does the standard error of the mean tell us?

This may help you here.

Problem 4

a) If the standard deviation of each of a set of distance measurements like those described in Problem 1 is known to be \sigma_l = 1.2 cm, then what would be the standard error of the estimated sample mean if you measured the distance 30 times?

b) How many samples would be needed for the precision of your sample mean (measured as the standard error) to reach 1.2 mm?

c) Does this mean the accuracy of the sample mean would have reached the same level? Why or why not? Draw analogy to the data in Problem 1 to help illustrate your point.

Problem 5

For these two sets of errors in each direction:

    \begin{equation*} \mathbf{e_E}= \begin{bmatrix} 3.0 \\ 4.4 \\ 5.1 \\ 6.0 \\ 6.6 \end{bmatrix} \end{equation*}

    \begin{equation*} \mathbf{e_N}= \begin{bmatrix} 6.5 \\ 5.8 \\ 5.6 \\ 5.0 \\ 4.4 \end{bmatrix} \end{equation*}

calculate:

a) the sample standard deviation of each

b) the sample covariance

c) the sample coefficient of correlation

d) the Circular Error Probable (CEP)

e) the DRMS

f) the 2DRMS

g) the 3DRMS

Problem 6

Explain each of the outcomes in Problem 5 using words that explain the results. Be sure to put words to the intuitive meaning of the sample coefficient of correlation and sample covariance you calculated, including their signs.

This may help you here.