Tag Archives: measurements

How many repeats do we need?

This is a question that both my undergraduate students and a group of taught post-graduates have struggled with this month.  In thermodynamics, my undergraduate students were estimating absolute zero in degrees Celsius using a simple manometer and a digital thermometer (this is an experiment from my MOOC: Energy – Thermodynamics in Everyday Life).  They needed to know how many times to repeat the experiment in order to determine whether their result was significantly different to the theoretical value: -273 degrees Celsius [see my post entitled ‘Arbitrary zero‘ on February 13th, 2013 and ‘Beyond  zero‘ the following week]. Meanwhile, the post-graduate students were measuring the strain distribution in a metal plate with a central hole that was loaded in tension. They needed to know how many times to repeat the experiment to obtain meaningful results that would allow a decision to be made about the validity of their computer simulation of the experiment [see my post entitled ‘Getting smarter‘ on June 21st, 2017].

The simple answer is six repeats are needed if you want 98% confidence in the conclusion and you are happy to accept that the margin of error and the standard deviation of your sample are equal.  The latter implies that error bars of the mean plus and minus one standard deviation are also 98% confidence limits, which is often convenient.  Not surprisingly, only a few undergraduate students figured that out and repeated their experiment six times; and the post-graduates pooled their data to give them a large enough sample size.

The justification for this answer lies in an equation that relates the number in a sample, n to the margin of error, MOE, the standard deviation of the sample, σ, and the shape of the normal distribution described by the z-score or z-statistic, z*: The margin of error, MOE, is the maximum expected difference between the true value of a parameter and the sample estimate of the parameter which is usually the mean of the sample.  While the standard deviation, σ,  describes the difference between the data values in the sample and the mean value of the sample, μ.  If we don’t know one of these quantities then we can simplify the equation by assuming that they are equal; and then n ≥ (z*)².

The z-statistic is the number of standard deviations from the mean that a data value lies, i.e, the distance from the mean in a Normal distribution, as shown in the graphic [for more on the Normal distribution, see my post entitled ‘Uncertainty about Bayesian methods‘ on June 7th, 2017].  We can specify its value so that the interval defined by its positive and negative value contains 98% of the distribution.  The values of z for 90%, 95%, 98% and 99% are shown in the table in the graphic with corresponding values of (z*)², which are equivalent to minimum values of the sample size, n (the number of repeats).

Confidence limits are defined as: but when n = , this simplifies to μ ± σ.  So, with a sample size of six (6 = n   for 98% confidence) we can state with 98% confidence that there is no significant difference between our mean estimate and the theoretical value of absolute zero when that difference is less than the standard deviation of our six estimates.

BTW –  the apparatus for the thermodynamics experiments costs less than £10.  The instruction sheet is available here – it is not quite an Everyday Engineering Example but the experiment is designed to be performed in your kitchen rather than a laboratory.

Red to blue

Some research has a very long incubation time.  Last month, we published a short paper that describes the initial results of research that started just after I arrived in Liverpool in 2011.  There are various reasons for our slow progress, including our caution about the validity of the original idea and the challenges of working across discipline boundaries.  However, we were induced to rush to publication by the realization that others were catching up with us [see blog post and conference paper].  Our title does not give much away: ‘Characterisation of metal fatigue by optical second harmonic generation‘.

Second harmonic generation or frequency doubling occurs when photons interact with a non-linear material and are combined to produce new photons with twice the energy, and hence, twice the frequency and half the wavelength of the original photons.  Photons are discrete packets of energy that, in our case, are supplied in pulses of 2 picoseconds from a laser operating at a wavelength of 800 nanometres (nm).  The photons strike the surface, are reflected, and then collected in a spectrograph to allow us to evaluate the wavelength of the reflected photons.  We look for ones at 400 nm, i.e. a shift from red to blue.

The key finding of our research is that the second harmonic generation from material in the plastic zone ahead of a propagating fatigue crack is different to virgin material that has experienced no plastic deformation.  This is significant because the shape and size of the crack tip plastic zone determines the rate and direction of crack propagation; so, information about the plastic zone can be used to predict the life of a component.  At first sight, this capability appears similar to thermoelastic stress analysis that I have described in Instructive Update on October 4th, 2017; however, the significant potential advantage of second harmonic generation is that the component does not have to be subject to a cyclic load during the measurement, which implies we could study behaviour during a load cycle as well as conduct forensic investigations.  We have some work to do to realise this potential including developing an instrument for routine measurements in an engineering laboratory, rather than an optics lab.

Last week, I promised weekly links to posts on relevant Thermodynamics topics for students following my undergraduate module; so here are three: ‘Emergent properties‘, ‘Problem-solving in Thermodynamics‘, and ‘Running away from tigers‘.