# 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.

# Engaging learners on-line

Filming at Quarry Bank Mill

The Everyday Engineering Examples page of this blog continue to be very popular.  More than 70 engineering schools in the USA have signed up to use this approach to teaching engineering science as part of the ENGAGE project.  The lesson plans on that page assist instructors to deliver traditional lectures that are engaging and effective.  Now, we have transferred the approach to online delivery in a MOOC that was designed to support undergraduate learning as well as to increase public engagement and understanding of engineering science.

The MOOC entitled ‘Energy: Thermodynamics in Everyday Life‘ was completed by more than 960 learners from about 35 countries who ranged in age from 13 to 78 years old with a correspondingly wide range of qualifications in terms of both subject and level.  I believe that this is the first MOOC to use Everyday Engineering Examples within a framework of the 5E lesson plans and it seems to have been effective because the completion rate was 50% higher than the average for FutureLearn MOOCs.

We also included some experiments for MOOC learners to do at home in their kitchen.  Disappointingly only a quarter of learners performed the experiments but surprisingly almost half of all learners(46%) reported that the experiments contributed to their understanding of the topics.  This might be because results and photos from the experiments were posted on a media wall by learners.  There was also a vibrant discussion throughout the five-week course with a comment posted every 8 minutes (or more than 6,500 comments in total).

More than half the undergraduates (53%) who followed the MOOC did not continue to attend the traditional lectures and roughly the same percentage agreed or agreed strongly that the MOOC could replace the traditional lecture course with only 11% disagreeing.  So maybe the answer to my question about death knell for lectures [see my post ‘Death Knell for the lecture?‘ on October 7th, 2015] is that I can hear the bell tolling.

I gave a Pecha Kucha 20×20 on these developments at an International Symposium on Inclusive Engineering Education in London last month, which is available as a short video.

# 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.