I have already mentioned a few doctrinal opening ranges from WW2 RA anti-tank gunnery policy in a previous post about British "tank destroyer" doctrine.
For historical surveys of the ranges of tank engagements, I can think of Benn and Shephard, Gee, Hardison, and Zaloga and Ness.
WO 291/1212, "Ranges of engagement in the ATk battle" was published in December 1951 by E. Benn and R. W. Shephard. Shephard is one of the authors of "Applied Operations Research" (Plenum Press, New York, 1988, by Shephard, Hartley, Haysman, Thorpe and Bathe) which includes a problem based on this study and a table giving what appears to be the original raw data gleaned from unit war diaries.
The formula
P = 1 – exp (–R/K)
was found to be a good expression for the proportion, P, of engagements that occur at ranges of less than R yards.
For NW Europe, K is about 950 yards.
90% of engagements occur at less than 2200 yards;
80% of engagements occur at less than 1500 yards;
50% of engagements occur at less than 650 yards.
AORG Memo no. C6, "A survey of tank warfare in Europe from D-Day to 12 August 1944", by H.G. Gee, May 1952 (available at the PRO as WO 291/1218), is based on 112 tank vs. tank actions fought by 21 AG. It gives the mean engagement ranges for this sample as 405 yards (standard deviation 380) in close country, and 1204 yards (standard deviation 735) in open country.
BRL Memo no. 798, "Data on World War II tank engagements involving the US Third and Fourth Armored Divisions", by D. C. Hardison, June 1954, is based on 86 tank vs. tank and tank vs. anti-tank engagements fought by the divisions mentioned. It gives the average ranges at which tanks were destroyed, in yards, as follows:
Allied tanks__Enemy tanks__Place
476____________N/A_______Vicinity Stollberg
959____________733_______Roer to Rhein
1000___________833_______Belgian Bulge
1260___________936_______Vicinity Arracourt
1116___________831_______Sarre
731____________915_______Relief of Bastogne
The overall mean casualty ranges are given as 946 yds for Allied and 893 yds for enemy tanks. The distribution of combat ranges was found to fit approximately to a Pearson
III distribution.
Both the negative exponential (Shephard) and Pearson III (Hardison) distributions have long tails, meaning that there are many enegagements at short ranges and very few at very long ranges. There, are, though, rare cases of engagements at extremely long ranges.
The greatest ranges mentioned for any action in those listed by the above studies are:
Shephard: 4,800 yards (118 actions)
Gee: 3,000 yards (112 actions)
Hardison: 3,500 yards (86 actions)
"Red Army Handbook 1939-1945", by S. J. Zaloga and L. S. Ness, 1998 (Alan Sutton, Thrupp) gives on page 179 a table of the ranges in metres at which Soviet tanks and assault guns were knocked out by 75mm and 88mm guns in 1943-44, as follows:
Range_______75mm gun_____88mm gun
100-200_______10.0%_________4.0%
200-400_______26.1%________14.0%
400-600_______33.5%________18.0%
600-800_______14.5%________31.2%
800-1000_______7.0%________13.5%
1000-1200______4.5%_________8.5%
1200-1400______3.6%_________7.6%
1400-1600______0.4%_________2.0%
1600-1800______0.4%_________0.7%
1800-2000______0.0%_________0.5%
The original has omitted the line for 600-800 metres, so I have calculated the entries for that line by subtracting the sum of each column from 100%.
For infantry engagements, work by Rowland and Speight, based if I recall aright on the "Chinese Eye" series of experiments, shows that attacking infantry typically unmask at about 300 metres.
Jane's Infantry Weapons for 1975 contains a couple of graphs showing the expected range of rifle and MG engagements, although the source is not given apart from a vague reference to "US statisticians".
For rifles,
About 30% of engagements take place at 100 metres or less
About 72% of engagements take place at 200 metres or less
About 88% of engagements take place at 300 metres or less
About 97% of engagements take place at 400 metres or less
For MGs,
About 50% of engagements take place at 750 metres or less
About 80% of engagements take place at 1200 metres or less
About 95% of engagements take place at 1600 metres or less
I hope that answers the question.
All the best,
John.