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Quantity Street, or why there aren’t any Green Triangles left

It’s Boxing Day, and in between servings of leftovers, you notice the distinctive purple octagon of Quality Street on the table. But when you take the lid off, hoping for a Green Triangle or a Purple One, you find there’s nothing left but Strawberry Delights. Is it just bad luck? Or skewed odds?

That’s the question that one ITV News journalist has asked for the last two years on Twitter. Stephen Hull has photographed the arrayed contents of his tin, and noted that the distribution isn’t very even. So, is it rigged? Nestle, who have made Quality Street since buying Rowntree Mackintosh in 1988, have never publicly announced the intended distribution of sweets in their tins. But we can use a simple statistical test to try to find out: the chi-squared test. This test can be used with data that fit into distinct categories, to check whether the distribution is as expected or not.

Suppose, for the purposes of the test, that on average, each of the 11 types of sweet is just as common as the next. Since Stephen’s tin had 91 sweets overall, this means that the expected number of each was 8.27. The chi-squared test tries to decide whether this hypothesis is reasonable, by constructing a test statistic that should follow the chi-squared distribution, and comparing it to that distribution.

The table below shows how many of each sweet were in Stephen’s tin, and how many we would have expected if the types were all equally common, on average. The final column calculates the test statistic, which is 8.24.

This statistic measures how different the actual tin of sweets was from the expected one, with a bigger statistic meaning more differences. If we took a random number from the appropriate chi-squared distribution, it would be this big or bigger about 60% of the time. In other words, assuming that each type of sweet is as common as the next, this distribution for a particular tin isn’t too unusual.

Now, as I mentioned, journalists have started to take an interest in this question, and naturally they’ve asked Nestle how the makeup of each tin is determined. A Nestle spokesperson told the i newspaper in 2018 that each tub is split roughly into thirds according to three categories of sweet, with five different types of chocolates, four different toffees and fudges, and only two types of fruit creme. This gives us an alternative hypothesis to test: that there really are more Strawberry Delights and Orange Cremes than the others, since they are the only ones in their category.

We can re-run the same test as before, but calculating our expected numbers of sweets based on this new theory:

Here, we’ve calculated the expected number of each sweet by allocating 91 / 3 = 30.33 sweets to each of the three categories, and then dividing these equally among the types in that category. The test statistic this time is even lower, at 6.15, suggesting this hypothesis could be a better fit. A random tin would have a test statistic as big as this about 80% of the time, assuming the hypothesis about equal numbers of each category is correct.

Purely in the interests of scholarly research, I popped round to the local supermarket to pick up a tub of Quality Street for myself. My slightly smaller tub had a total of 66 sweets, and I re-ran the previous tests using a combined sample from Stephen Hull’s tin and my tub.

If the sweets were equally distributed, we would get a test statistic of 9.54 or bigger about half of the time, and if they were distributed by category we would get a statistic of 12.55 or bigger about one time in four. These are both definitely plausible.

So, based on this sample size, we can’t conclude much: the distribution might be biased to include equal numbers of each category, or it might not be biased at all. This analysis could be taken further by considering factors like cost, weight or nutritional information of each sweet, which Nestle have also suggested are taken into account in their plans. Alternatively, the same chi-squared tests could tell us more if we bought more tins of sweets to give us a bigger sample size, but since I’ve got a dentist’s appointment in a few weeks, I think I’d better leave it there. I hope you have a happy and peaceful holiday period, and find all your favourite sweets in the tin.

Michael Scanlon

December 2021