The Drunkard’s Walk
Many dots each take a random step at a time. Watch their typical distance from the start grow like the square root of the number of steps.
0 steps. Typical distance: measured 0.0, theory (√steps) 0.0
x: steps, y: typical distance (solid = measured, dashed = √steps)
Even though every step is random, the mean of the squared distances over many walkers equals the number of steps. So the typical distance (the root-mean-square) is the square root of the steps. Four times the steps only doubles the spread, in one dimension and in two.
How to play
Three hundred dots each take one step at the same time. "Step" moves once; "Auto" keeps walking. The dashed circle (dashed vertical lines in 1D) marks the theoretical typical distance, the square root of the steps. The chart below compares measured and theoretical values.
Learn more
Even though every step is random, the mean of the squared distances over many walkers equals the number of steps. So the typical distance (the root-mean-square) is the square root of the steps. Four times the steps only doubles the spread, in one dimension and in two.