Experimental Design & Practice 1 (Propagation of Error)

Question 1

Error is written following the convention:

Answer 1

𝑥 = mean 𝑥± ∆𝑥, (error)
or
± 𝛿𝑥, (error)
or
± 𝜎𝑥, (standard deviation)

Question 2

Purely random errors have what kind of distribution?

Answer 2

Purely random errors have a Gaussian (Normal) distribution:

Question 3

Gaussian (Normal) Distribution:

Answer 3

𝑓 𝑥 =
1/(𝜎 x square root 2𝜋 ) 𝑒𝑥𝑝 − 1/2 (𝑥− 𝑥ҧ /𝜎)^2

𝑥ҧ= mean value of distribution
𝜎 = standard deviation of distribution
𝑓 𝑥 = The probability distribution function

Question 4

what percentage of measurements lie within one standard deviation of the mean for The Gaussian distribution

Answer 4

~68% of measurements should lie within 1σ of mean of x.

Question 5

The standard deviation of a set of measurements is given by:

Answer 5

square root ( sum (xi - mean x)^2 all divided by n)

Question 6

Theory suggest the standard uncertainty, from the standard deviation should be:

Answer 6

𝛿𝑥 = 𝜎𝑥 / square root n

Therefore the standard uncertainty decreases with
the number of experiments!!

Question 7

Significant figures

Answer 7

Significant figures (or digits) are the number
of reliable digits of a measurement or a
calculation.

Significant figures are determined by the
error.

Question 8

How many signifacnt figures should you give for the standard error

Answer 8

• The error, by convention, should be given to
  one significant figure, unless the value
  starts with a “1”, then you need to use two
  significant figures.
• Mean and error then are given consistency to
  the same number of decimal places.

Question 9

is standard deviation only used to calculate all types of error

Answer 9

The standard deviation is
only used to determine random
errors. You still have the error
given by the inability/limitation
of the caliper to measure more
decimal places.

Question 10

Precision

Answer 10

The precision of measurements with no distribution,
e.g. successive measurements with the same
instrument yielding the same value regardless of the
number of measurements, is determined by the
instrument or device.

Question 11

How to calculate precision

Answer 11

The precision is given by half of a division or half of
the last digit.

e.g. a conventional 30 cm ruler has a precision of 0.5
mm.

If you read 5.00 V from a voltmeter, then its precision
would be 0.005 V, and the measurement would be
5.000 ± 0.005 V (remember the rule about decimals)

Question 12

If only the maximum and minimum values are
known (range), or it is expected that the values
are evenly distributed:

The uncertainty (rather than the error) can be
taken as:

Answer 12

half width (a) or resolution / square root 3

Question 13

What is a “method” section in an experimental report? (4)

Answer 13

A methods section:
* Is an account of what you did.
* Is written in the past tense.
* Is usually written in the passive voice.
* Should enable the reader to repeat your experiment in exactly the way you did it,
even if your procedure wasn’t perfect

Question 14

What is a standard operating
procedure (SOP)?

Answer 14

• Is a set of instructions.
• Is written in the imperative tense.
• Can be written as narrative or as a numbered list
  of steps.
• Describes a procedure which represents best
  practice.
• If followed accurately enables different
  technicians/labs/factories to get results to the
  same (high) precision.

Question 15

Let’s consider a general function, u, of several independent
variables, x, y, z…:
𝑢 = 𝑓(𝑥, 𝑦, 𝑧, … )
𝛿𝑢 is the uncertainty on u
𝛿𝑢 is given by:

Answer 15

For a function 𝑢 = 𝑓(𝑥𝑖
, 𝑥𝑖+1 … 𝑥𝑛)
𝜕𝑓 / 𝜕𝑥𝑖
is the partial derivative of f with respect to variable 𝑥𝑖
whilst assuming all other variables are constant.