The most important piece of demographic data is the population. The law requires districts to have nearly equal populations, and population inequality is what drives the need for redistricting. The quota is the ideal population each district should have. For a geography with k districts and a total population P, the quota for each district is Q = P/k.
The law varies in the extent that districts must all have a population equal to the quota. For example, in federal law congressional districts are held to a higher standard of equality than state or local districts. However, state law may impose stricter equality than required by federal law. Small variations from the quota may be permissible to achieve other neutral redistricting criteria.
The range is the difference between the largest and smallest district expressed as a percentage of the quota. The range reflects the difference in voting power between individuals in the most and least populous districts. Federal law has permitted ranges of congressional districts in a state up to 1% when following other criteria. Even so, the range should be a minimized given the other criteria used. The federal standard for local districts is a range not exceeding 10%.
Another measure is the maximum deviation. This is the maximum absolute percentage difference permitted for a district from the quota Q and is sometimes used by states as the population equality standard. If there is a defined range a maximum deviation equal to half the range guarantees the resulting plan will be within the range. Mathematically for a maximum D and each district population d, |d-Q| <= D.
The average deviation is the average of the absolute deviations from the quota for all the districts. The average deviation correlates to the number of persons in an average district that would have to be shifted to another district to make the district population equal to the quota. Mathematically if there are N districts the average deviation <d> = sum(|d-Q|)/N.