# 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.

# Writing backwards

My regular readers will know that I am a fan of the 5E instructional method and in particular combining it with Everyday Engineering Examples when teaching introductory engineering courses to undergraduate students. Elsewhere in this blog, there is a catalogue of lesson plans and examples originally published in a series of booklets produced during a couple of projects funded by the US National Science Foundation. Now, I have gone a step further and embedded this pedagogy in a Massive Open Online Course (MOOC) on Energy! Thermodynamics in Everyday Life. If you follow the MOOC, you’ll find some new worked examples that I explain while writing ‘backwards’ on a glass board. My film unit are very proud of the ‘backwards’ writing in these examples, which they tell me is an innovation in education filming-making. Our other major innovation is laboratory exercises that MOOC participants can perform in their kitchens. Two of these are based on everyday experiences for most participants: boiling water and waiting for a hot drink to cool down; the third is less everyday because it involves a plumber’s manometer. In each case, I am attempting to move people around Honey and Mumford’s learning cycle, which is illustrated schematically in the figure, i.e. having an experience, reviewing the experience, concluding from the experience and the planning the next steps. The intention is that students progress around the cycle in the taught component, then again in the experiments.

If you want to have a go at the one of experiments, then the instructions for the first one are available here. Alternatively you could sign up for the MOOC – its not too late!  But if you don’t want to follow the course then you can stil watch some excerpts on the University of Liverpool’s Stream website, including the backwards written examples.

Sources:

Atkin, J.M. and Karplus, R., 1962. Discovery of invention? Science Instructor, 29 (5), 45–47.

Honey P, Mumford A. The Manual of Learning Styles 3rd Ed. Peter Honey Publications Limited, Maidenhead, 1992.