It's interesting to compare these angles with those that come up in engineering.

• One second of arc (the Ramsden theodolite): r = 0.99999999925

• 1/5000 to 1/1000 inch per inch (accuracy of a try-square): r = 0.99999950 to 0.999999980

• 1 minute of angle (the accuracy that a rifle should hold to) = 1 inch at 100 yards: r = 0.999999970

• 1 degree: r = 0.99985

• 1 compass point (1/32 of a revolution, limit of navigational accuracy in the days of sail): r = 0.981

And in contrast:

• Correlation that implies 1 bit of mutual information (for a bivariate Gaussian): r = 0.866, angle = 30 degrees (exactly).

• Correlation considered high in the softer disciplines: r = 0.8, angle = 36.9 degrees.

• Correlation considered publishable: r = 0.2, angle = 78.5 degrees. This means that if the truth is due North, you're proceeding East by North (one compass point away from East).