profile. For example, is the arithmetic average of the roughness profile, is the arithmetic average of the unfiltered raw profile, and is the arithmetic average of the 3D roughness. Each of the formulas listed in the tables assume that the roughness profile has been filtered from the raw profile data and the mean line has been calculated. The roughness profile contains ordered, equally spaced points along the trace, and is the vertical distance from the mean line to the data point. Height is assumed to be positive in the up direction, away from the bulk material. Amplitude parameters Amplitude parameters characterize the surface based on the vertical deviations of the roughness profile from the mean line. Many of them are closely related to the parameters found in statistics for characterizing population samples. For example, is the arithmetic average value of filtered roughness profile determined from deviations about the center line within the evaluation length and is the range of the collected roughness data points. The arithmetic average roughness, , is the most widely used one-dimensional roughness parameter. Here is a common conversion table with also roughness grade numbers: Slope, spacing and counting parameters Slope parameters describe characteristics of the slope of the roughness profile. Spacing and counting parameters describe how often the profile crosses certain thresholds. These parameters are often used to describe repetitive roughness profiles, such as those produced by turning on a lathe. Other "frequency" parameters are Sm, a and q. Sm is the mean spacing between peaks. Just as with real mountains it is important to define a "peak". For Sm the surface must have dipped below the mean surface before rising again to a new peak. The average wavelength a and the root mean square wavelength q are derived from a. When trying to understand a surface that depends on both amplitude and frequency it is not obvious which pair of metrics optimally describes the balance, so a statistical analysis of pairs of measurements can be performed (e.g.: Rz and a or Ra and Sm) to find the strongest correlation. Common conversions: Bearing ratio curve parameters These parameters are based on the bearing ratio curve (also known as the Abbott-Firestone curve.) This includes the Rk family of parameters. Fractal theory The mathematician Benoît Mandelbrot has pointed out the connection between surface roughness and fractal dimension. The description provided by a fractal at the microroughness level may