Self-assessment for: Calculating precision from data

These self-assessment problems accompany the section of our notes on 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 a spreadsheet for doing this work. If you’re in my class, you’re asked to use Google Sheets and make the link available when you hand in the self-assessments as has been our practice so far.

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*}

The same numbers are listed here so you can copy and paste them:

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

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?

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.


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