Being a data detective

The last skill in this chapter is not calculation but reasoning: knowing what the numbers CAN and CANNOT say, often before working anything out.

The daily water usage from a tap, in litres, was 5.6, 8, 3.09, 12.9, 6.5, 12.1, 11.3, 20.5, 7.4. Could the mean or median lie between 25 and 30?

No — and you can see it without calculating. The mean is a fair share of the total, and every single value is below 21, so the share cannot possibly exceed the largest value. The median is one of the values, or the average of two, so it cannot exceed the largest either.

That gives a general rule worth keeping: A Mean or median can never be less than the minimum value or greater than the maximum value in the data. If your answer falls outside that range, you have made a mistake.

Next, comparing two groups. The weights of some newborn babies, in kilograms: boys 3.5, 4.1, 2.6, 3.2, 3.4, 3.8; girls 4.0, 3.1, 3.4, 3.7, 2.5, 3.4. Fill a dot plot for each and compare centre AND spread, as you did with the Grade 5 heights.

Compare groups that are wildly different and the picture is easier. Sumo wrestlers weighing 295.2, 250.7, 234.1, 221.0 and 200.9 kg against ballet dancers: the two groups do not overlap at all, and any measure of centre will say so clearly.

Finally, inference from context. Two cities' tables give monthly figures in HOURS. City 1 runs low in January and December and peaks around June; City 2 is the reverse, highest in January. What might the data be? Monthly hours of sunshine — and the reversal says the two cities are in opposite hemispheres, so it is summer in one while it is winter in the other. Reading the pattern, not just the numbers, is what makes a data detective.