A random walk is a pattern of change or movement in successive steps, with the change at each step governed by chance, independent of what came before. Time t is the independent variable in random walks and in evolution. Other measures such as displacement d or cumulative displacement s are dependent variables because they change as a function of time. An evolutionary random walk is a time series with successive si drawn from repeated iteration of si + 1 = si ± d0, where d0 is a random variable drawn from a normal distribution with mean μ and standard deviation σ. Rates are ratios. The step rate r0, moving to the left or right, is ±r0 = ±d0 / i0, which defines the relationship of d0 and i0. Individual random walks are by nature unpredictable and idiosyncratic, with a wide range of possible behaviors. Random walks aggregated as Brownian diffusion share some common characteristics: (1) terminal values are normally distributed with most random walks near their starting point and few in each tail; (2) the variance of the aggregate increases as a constant proportion of time; and (3) the standard deviation of the aggregate increases as the square root of time.