Weighted hits move the centroid through a fiducial center line. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Left-heavy weights stay before the center line
Weights 3 and 1 place the centroid at 5/4 meters.
m = x max − x c = 3 / 2 − 5 / 4 = 1 / 4 , pass m=x_{\max}\mathbin{-}x_c=3/2\mathbin{-}5/4=1/4,\quad\text{pass} m = x m a x − x c = 3/2 − 5/4 = 1/4 , pass Cluster centroid Weighted hits place a centroid inside the span. w=3 w=1 centroid span=1..2 m numerator=5 m totalWeight=4 count centroid=5/4 m span=1..2 m
Equal weights land on the center line
Weights 1 and 1 place the centroid at 3/2 meters.
m = x max − x c = 3 / 2 − 3 / 2 = 0 , pass m=x_{\max}\mathbin{-}x_c=3/2\mathbin{-}3/2=0,\quad\text{pass} m = x m a x − x c = 3/2 − 3/2 = 0 , pass Cluster centroid Weighted hits place a centroid inside the span. w=1 w=1 centroid span=1..2 m numerator=3 m totalWeight=2 count centroid=3/2 m span=1..2 m
Right-heavy weights cross the center line
Weights 1 and 3 place the centroid at 7/4 meters.
m = x max − x c = 3 / 2 − 7 / 4 = − 1 / 4 , fail m=x_{\max}\mathbin{-}x_c=3/2\mathbin{-}7/4=-1/4,\quad\text{fail} m = x m a x − x c = 3/2 − 7/4 = − 1/4 , fail Cluster centroid Weighted hits place a centroid inside the span. w=1 w=3 centroid span=1..2 m numerator=7 m totalWeight=4 count centroid=7/4 m span=1..2 m