For Fibonacci ratios less than 1, “bullish” and “bearish” retracements measured on log scale produced

ratios of 0.786, 0.618 and 0.382 more often than would be expected on average in a random environment

(hereafter called “positive outcome”). Two of them are significant at the 1% level and one at the 5% level.

So half of these six Fibonacci occurrence rates are significant, despite limited data, and all six have positive

outcomes. Finding positive outcomes six out of six times itself is statistically significant at a level of 1.6%.

Finding significant outcomes three out of six times is also statistically significant, at the 2% level. Projec-

tions also produced positive outcomes in five out of six tests, a result that is significant at the 11% level. So,

price trends on log scale produced positive outcomes in 11 out of 12 tests for Fibonacci ratios less than 1.

This amount is highly statistically significant (p =0.003).

For Fibonacci ratios greater than 1, price trends on

log scale produced positive outcomes in 10 out of 16 tests (p= 0.2).

In all, price trends on log scale produced positive outcomes in 21 out of 28 independent tests,

which is a highly significant result (p= 0.006).

Price trends on arithmetic scale produced positive outcomes in 18 out of 27 independent tests,

3 a result significant at the 6% level.

Time lengths and percentage price moves also produced positive outcomes in more than half

the tests: 16 times out of 28 and 16 times out of 27, respectively.

On the whole, 71 out of 110 Fibonacci occurrence rates exceed the 50th percentile,

a number greater than the 55 expected by random chance.

Similarly, 22 out of 112, more than the 11 expected, achieve the 90th

percentile; 19 out of 112, more than the 6 expected, achieve the 95th

percentile; and fully 16 out of 112, many more than the 1 expected, achieve the 99th

percentile. If these results could be assumed independent,

the observed amounts would be highly significant for each of these percentiles:

50th (p =0.001),

90th (p =0.003).