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10 January 2019
Christmas may be a fading memory but no doubt the festive season brought with it a cornucopia of confectionary delights, the remains of which may tell a statistical story about your local favoured and not-so-favoured treats. The arrival of the first of many selection boxes to Cambridge Maths Towers this holiday period inevitably prompted the questions “Which one in this box is best?” and “Do others agree with my answer or am I out on a limb?” (rationally, not a bad situation to be in when in possession of a team-owned box of comestibles; emotionally, often the cause of much frustration). In short: how does this selection fare in terms of “which is the most popular”?
After a short-lived argument about whether Whispas and Twirls are genuinely different chocolates (for the record, they are significantly different enough that two of our team could tell them apart in an impromptu blind taste test), we sent out a tweet to collect some data:
We knew that ranking was a good idea for our purposes (thinking about what was the most and least popular). But what does popular even look like when it comes to the distribution of rankings for each sweet? Instinctively it feels like this should be a straightforward question to answer – but it is surprisingly difficult when scrutinised. After collecting the responses, we considered the basic tools that we had at our disposal.
Average rank is a tempting starting point. With each sweet scoring from 1 to 7, perhaps we can determine the best sweet by reducing the data to a numerical summary? See above for this data, organised in this manner.
But: the recurring question that the research on measures of centrality often prompts applies here – what does average mean in this context? Will any type of average tell us which sweet is most “popular”?
Averaging ordinal data is beset with problems. Despite being relatively simple to calculate (and thus ubiquitous in the statistics classroom), the values themselves don’t have a true numerical meaning, referring as they do to a subjective scale unique to each individual’s confectionary preferences. Mode is the most likely appropriate value to consider, but unfortunately the likely candidates for most popular, Twirl and Caramel, do not have a unique mode. Worse still, we have no idea whether the mode even represents a rank that is substantially more common.
It seems that a graphical representation might be useful to make sense of the data. What might “popular” look like in graphical form? And what do the different shapes we might see in a graph of the rankings for each sweet allow us to infer about how people feel about them?
Before looking at any graphs, what does your instinct tell you will be the signature of a popular sweet? A symmetrical distribution? A skew towards one end? All the bars the same height? What might each of these shapes tell you?
Furthermore, how can we best represent the data in a way that makes these shapes visible? It’s tempting to draw charts that meet strict criteria, but there is a potential trade-off between following convention and bending the rules to highlight or hide information. If we want to perceive the shape of the entire distribution, rather than focus on the frequency of individual ranks there is perhaps some benefit to presenting the ‘bars’ as a single object, and removing the numbers from the scales
It was at this point that a disagreement arose. As is so often the case when considering the construction of graphs, a balance point between function and form was up for discussion. Could we produce a graph that didn’t need a title, for example? Could we make something beautiful that also showed the data points and shape of the distributions clearly? Lucy, confident that she had a vision that would work in the service of these multiple goals, started work. Two hours later, this appeared:
Looking at the graphs, how easy is it to perceive shapes in the data? And what non-standard methods have been used to communicate information? Do these graphs, as was our goal, help to answer the question about which of the chocolates are most and least popular?
This kind of highly contextual statistical problem gives rise to all sorts of opportunities for using data to make purposive decisions. The question “Which sweet is most popular?” could be reframed as “Which of the individual chocolate snacks should you give as a gift to someone if you don’t know what they like?” or alternatively, “If you were asked to choose a total of 100 chocolate snacks to stock a tuck shop, how many of each should you choose?”. The second of these questions begins to link data and proportional reasoning, an essential facet of statistical reasoning.
How could you use these graphs to promote decision making with data?
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