[NOTE this primarily about Weight Watchers prior Points system that was in use for 1997-2010 in the US there is a note at the end about the application to PointsPlus]
From US Patent 6,436,036:
Advantageously, the activity points are determined in accordance with the equation:
(k4 x kg body weight x minutes of activity)/100 …
Low Intensity = walking at 2.5 mph on a flat surface, k4 = .051
Moderate = fast walking at 4.0 mph on a flat surface, k4 = .0711
High = running at 6.0 mph on a flat surface, k4 = .1783 ….
The O2 cost per minute for each of these activities can be calculated.. and caloric consumption values. The foregoing calculation are tempered by experience tending to show that program participants tend to over-estimate physical activities and under estimate food intake…
Cutting to the punch line, it doesn’t directly match up to calories. That said the division by 100 probably explains where one gets the intuitive sense that 1 AP = approx 100 calories especially given that the primary Points formula is denominated by 50 kCalories.
And a bit of experimentation gets you to the following result for walking:
Or about 100 calories for an activity point.
The newer patents discussing PointsPlus do not provide a separate formula for activity points. Assuming the same 2-for-1 ratio is in effect and given that 1 PointsPlus is denominated by 35 kCalories, 1 AP =~ 70 kCal under PointsPlus.