$ ./numbers

85 58 6 47 7 66 55 64 61 42

Average = 49.10
Min = 6, Max = 85, Total = 91
Average of Min and Max = 45.50

$ ./numbers

75 21 85 19 41 73 66 88 53 81

Average = 60.20
Min = 19, Max = 88, Total = 107
Average of Min and Max = 53.50

$ ./numbers

26 1 26 12 89 35 70 79 81 54

Average = 47.30
Min = 1, Max = 89, Total = 90
Average of Min and Max = 45.00

—————————–

As you can see, they are close, but they ain’t the same thing.
In these examples it looks like they can be up to 12% off. There
can be a significant difference.

w_1 c(x_1) + w_2 c(x_2)

where the w_i are 1 and the x_i are ±1/sqrt(3). To convert the integral into the mean value, divide by the size of the range (2, in the above example), thus making the weights 1/2. In Jan’s example, the range is [0, 24], so scale by adding 1 and multiplying by 12. Now the x_i are 12(1±1/sqrt(3)) ~~ 5.0718 and 18.9282. Jan “added f(5.071667) and f(18.928333) and divided by two” because, among the x-values for which f had been recorded, those were nearest to the values demanded by 2-point Gaussian quadrature.

Gaussian quadrature played no part in the choice of “4, 8, 16, and 22 hours” as the times for which the values of f were requested for the purpose of approximating f by a cubic. Perhaps these times were chosen because they are a useful set of 4 for approximating the change of temperature throughout a winter day: for some latitudes, 8 is just after sunrise, so f(4) and f(8) show how the rising sun makes the temperature rise; 4pm is about sunset, so f(16) shows how the temperature has risen as a result of the sunshine in the daylight hours.

What is interesting about the (high + low)/2 method (the “traditional” way of finding daily average temperature) is that in effect, it is saying that the integral of a function over a given range can be approximated by taking the average of the maximum and minimum values of the function on that range. I learned the trapezoid rule, and Simpson’s rule, but this is an approximation technique that no one ever mentioned in Intro to Calculus. Why does it work? Is it related to Gaussian quadrature? That is, does it work because the minimum tends to occur at or near 5.071667 hours from midnight, and the maximum at 18.928333 hours from midnight? I would think the maximum occurs earlier in the day.