Relationship between virtual flight path and Z -score value. To obtain Z-scores (shown upper right of each graph), flight data for each butterfly tested in the flight simulator were analyzed using a Rayleigh test. Z, which is the critical value for the Rayleigh test, is calculated from the following formula: Z = nr
2, where n is the number of observations and r is the magnitude of the mean vector. Only butterflies with a Z-score 500 or above have a flight path that shows clear directionality. Virtual flight paths were constructed by starting in the center of the square and plotting each direction interval consecutively as one unit length .