A "Formula" for Comparing Recoil Stats

[probaddie]

So it seems that there's been some latent interest for some time in having an objective way to compare the recoil statistics for CoD's weapons. I remember it starting when the original stats chart came out for MW3, in which the author included his own -- and with all respect, very unscientific -- gun ranking charts, and very recently someone attempted to pound out a "balance formula" for the weapons; ultimately, the inability to compare recoil stats was the coup de grace for that idea.

In the hopes that this interest still lingers, I've developed a method for creating a statistic that does just what the title says: grades recoil plots. More than simple, unguided arithmetic, this method uses probability distributions representing the weapons' behavior to calculate the amount of "spread" and "deviation" from the original point of aim to form a statistic, which can be compared between guns to rank their efficacy.

The basic idea is that I create a pdf (probability density function) to represent the likelihood of the point of aim landing in a given spot (on each axis) after the first shot. I then use these functions to calculate the "origin-centered" variance on each axis, and use those in turn to create an overall statistic that represents the "spread" and "deviation" of the recoil plots we have. This statistic is not perfect; in fact, it is considered impossible to represent the variance of a bivariate distribution with a single number (see en.wikipedia.org/wiki/Covariance_matrix for more on that). But I was generally impressed with the results and I think most of them agree with my own intuition about where the guns should rank compared with each other.

Here is a link to my Mathematica code (8.0.1) which I wrote to develop the statistic.

Here are the latest results for MW3:
Unbiased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dEhubGRTUXpBNmtHVmFLSGVFdWdLd0E#gid=0
Biased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dDVNNTdDWUUxT0NzRUVrT3FCdXcxdnc&pli=1#gid=0

Here are results for BO using the same statistics:
Unbiased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dGNZNkNtWTdMWFJfRi1LVWdEXzZIcFE#gid=0
Biased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dHN2T01Pb25zdm5naEM3YWY2RTdUZFE#gid=0

Results for CoD4:
Unbiased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dE9QcEdBanZZUC1wSDJ5bnlIZzJaTEE#gid=0
Biased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dFdJdnFkV3pRN1hwMTJuaERCbnJvMXc#gid=0

And for MW2:
Unbiased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dHMzYi1KdVpNeWtTa3c2X19DSjIzUGc#gid=0
Biased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dElMNDltNVFHMWNBNm9jNV9KS0I3dnc#gid=0

And finally W@W:
Unbiased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dFRWdm85MWRvcGRnWWdwaUFpNUl0bVE#gid=0
Biased: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dHpuczl3SV9PX0hLWmt4cmt4SFpLaWc#gid=0

And psijaka's recoil plots to compare with:

I look forward to your feedback and to answering any questions you may have.

You are by bro, bro.


Update: I've compiled a spreadsheet of inaccuracy scores for Advanced Warfare. Note that these are not at all comparable to the statistics I used in my older work. I guess this is the tenth draft...?



Edit: Added link to psijaka's recoil plots.
Edit: Removed the Mathematica code, replacing it with a link to the Google Docs version of the same.
Edit: Removed first draft-specific questions.
Edit: Removed link to product metric (obsolete), replaced it with link to most recent draft.
Edit: Added link to Black Ops results.
Edit: Updated latest results to contain, well, the latest results.
Edit: Fixed broken link
Edit: Removed link to psijaka's recoil plots due to an inconsistency between his simulation method and my calculation methods which make the two works incomparable
Edit: Added links to CoD4 results
Edit: Added link to MW2 results
Edit: Added link to W@W results, as requested by Marvel4
Edit: Added recoil scores for Advanced Warfare

[cmck]

It seems like the rankings favor the weapons with only vertical kick. I was kind of surprised where the acr ended up in the class ranking. And I wouldn't guess that the pp90 is better than the ump.

Is it possible to make vertical only recoil less dominant? I don't understand code so I'm not sure if its possible, but I wouldn't say that the G36C is the best and the acr is lower tier in its class for recoil.

[probaddie]

May 22, 2012 19:35:50 GMT -5 cmck said:

It seems like the rankings favor the weapons with only vertical kick. I was kind of surprised where the acr ended up in the class ranking. And I wouldn't guess that the pp90 is better than the ump.

Is it possible to make vertical only recoil less dominant? I don't understand code so I'm not sure if its possible, but I wouldn't say that the G36C is the best and the acr is lower tier in its class for recoil.

It's definitely possible, but it doesn't really address the root of the problem. The formula will favor guns with strong horizontal kick as well. When I run the program using numbers for a stock Black Ops Skorpion I get a value of 0.999526, or essentially 100%! It has to do with the way the variance is calculated. It is essentially measuring the size of the "area" in which the bullets can land and how far from the origin the shots lie. When a recoil plot is a straight line you can effectively say the recoil plot has no area -- a line only has one dimension -- and so that plot receives a near-perfect score. All my workarounds to this have been unsuccessful thus far.

Furthermore, I think I would be in favor of a statistic that favored vertical recoil weapons, just for the fact that a man-shaped target is taller than it is wide. I won't tweak with that just yet until I get more feedback.

I understand your point about the ACR. Mind you, most people will run it with Kick, which makes a significant difference to it's recoil plot. Maybe it's just me, but I definitely see the non-Kick ACR losing out to the non-Kick M4, FAD, G36C and maybe the SCAR-L (I'm looking at psijaka's recoil plots for comparison: denkirson.proboards.com/index.cgi?board=general&action=display&thread=3256). With Kick, the ACR is much more competitive. I ran it though the program and got a value of 0.978 (97.8) which has it beating all stock assault rifles and narrowly losing out to the stock MP7, which seems reasonable.

It's another close call between the UMP and the PP90-M1. For me, the PP90-M1 plot looks just as wide but shorter than the UMP's. It's a counter-intuitive case where the gun with the ridiculous firerate has a (somewhat) compact plot because it fires its shots off while in the middle of it's upward climb, rather than when it reaches its natural peak.

(That's another math-laden thread I could start showing how that works. It turns out that there is a "worst-case" firerate for each gun (for given ViewKick and CenterSpeed values) that maximizes the dispersion of the shots. Once you exceed that value, the spread reduces as you increase firerate. It seems wrong, but think about it: if you had a rifle with a 30 round magazine and an infinite firerate (i.e. zero firetime), all 30 bullets would have to land in exactly the same spot no matter how large the recoil numbers are! Because if you don't allow any time for the gun to climb, it can't deviate from its original point of aim before expending its entire magazine.)

Also, I'm adding a link to psijaka's plots, since, y'know, that's what my statistic is supposed to be grading

[Marvel4]

May 22, 2012 20:38:54 GMT -5 probaddie said:

It turns out that there is a "worst-case" firerate for each gun (for given ViewKick and CenterSpeed values) that maximizes the dispersion of the shots. Once you exceed that value, the spread reduces as you increase firerate.

I think it would be very interesting if you could calculate that rate of fire for some different recoil values.

[probaddie]

May 22, 2012 20:48:18 GMT -5 Marvel4 said:

May 22, 2012 20:38:54 GMT -5 probaddie said:

It turns out that there is a "worst-case" firerate for each gun (for given ViewKick and CenterSpeed values) that maximizes the dispersion of the shots. Once you exceed that value, the spread reduces as you increase firerate.

I think it would be very interesting if you could calculate that rate of fire for some different recoil values.

Actually, I have a formula for that (for a single direction). The worst-case rate-of-fire for a given cardinal direction is r = 24*c/v, where v is the maximum ViewKick in the given direction, and c is the CenterSpeed. For example, take the AK47 in the left direction. The relevant values are v = 60 and c = 1400. We get a worst-case firerate of r = 24*1400/60 = 560 rpm. So, if the damn thing fired any slower it would have a worse recoil plot (not taking the other 3 directions into account.) Getting a result that takes all four directions into account is something I never worked out but I could try harder!

Do you care for the derivation? It requires first-year undergrad math (or IB) to understand.

[cmck]

I didn't just mean vertical. I should have said one directional. I wouldn't say that pure vertical is always better. If the circle of where the bullets can land is small it could get more shots in than the pure vertical gun. I'm just making sure the g36c isn't winning due to its vertical nature.

[b0xr]

P90 isn't on the chart? #2 spot is missing for SMG's

[Indy_Bones]

As soon as you start 'scoring' weapons, it becomes opinion as against facts (something I was reminded of when I did my MW2 guide).

Player A may prefer weapons which only have vertical recoil
Player B may prefer weapons which only have horizontal recoil
Player C may not care which direction the recoil goes as long as it's predictable
Player D may not care at all as they only engage in extreme CQB where it will pretty much hit anyway

Etc, etc.

I do think there is a use for the stuff you're doing, but I think part of the approach is flawed, especially when guns with very high recoil in a particular plane naturally 'score' better than those that have more 'rounded' recoil.

I also think there needs to be the semi-auto and burst weapons on there, as how can you score a class then leave weapons out of it, especially when at least 2 of those weapons can be extremely dangerous?

As for do the current stats fit my perception of the guns - unfortunately not, there's an awful lot of those ranks I'd disagree with, and there has to be better separation of the 'tied' weapons, as for example the G18, Skorpion and FMG's do not share the same level of recoil in-game, so how they are tied is beyond me...

It's a project that has some promise, but needs pulling back to the drawing board first

[Marvel4]

May 23, 2012 7:31:47 GMT -5 Indy_Bones said:

as for example the G18, Skorpion and FMG's do not share the same level of recoil in-game, so how they are tied is beyond me...

They all have shit recoil, so it doesn't matter anyway.

[Indy_Bones]

May 23, 2012 7:43:01 GMT -5 Marvel4 said:

They all have doo-doo recoil, so it doesn't matter anyway.

So we're going to simply ignore the stats in some cases then?

[wwaa]

people are not crocodiles, they are vertical, not horizontal, so it is obvious that vertcal "ecoil = no recoil" many times.

[probaddie]

May 22, 2012 23:08:30 GMT -5 cmck said:

I didn't just mean vertical. I should have said one directional. I wouldn't say that pure vertical is always better. If the circle of where the bullets can land is small it could get more shots in than the pure vertical gun. I'm just making sure the g36c isn't winning due to its vertical nature.

It's definitely getting help from it. I'm going to try running it again with a modified statistic later on today to see if we can help out those "rounded" plots get a leg-up.

[probaddie]

May 23, 2012 2:50:43 GMT -5 b0xr said:

P90 isn't on the chart? #2 spot is missing for SMG's

Fixed. Thanks for spotting that.

As soon as you start 'scoring' weapons, it becomes opinion as against facts (something I was reminded of when I did my MW2 guide).

Player A may prefer weapons which only have vertical recoil
Player B may prefer weapons which only have horizontal recoil
Player C may not care which direction the recoil goes as long as it's predictable
Player D may not care at all as they only engage in extreme CQB where it will pretty much hit anyway

Etc, etc.

Always. Remember, the score is just for the recoil plots themselves. The overall effectiveness of the weapon is completely down to the user and the situation in which it's being used.

I also think there needs to be the semi-auto and burst weapons on there, as how can you score a class then leave weapons out of it, especially when at least 2 of those weapons can be extremely dangerous?

This is where I need some input. I could easily do this for the semi-automatics (pistols and MK14) but at what rate of fire? Is the average player capable of maximum rate of fire? (PC bro here, by the way, so I'm not familiar with all this "modded controller" business.) The burst-fire weapons are going to be a problem, though. My statistic has no way of accommodating the fact that you're forced to wait between busts: it will grade them as though they were automatics. I can't see an easy workaround to this so far.

As for do the current stats fit my perception of the guns - unfortunately not, there's an awful lot of those ranks I'd disagree with, and there has to be better separation of the 'tied' weapons, as for example the G18, Skorpion and FMG's do not share the same level of recoil in-game, so how they are tied is beyond me...

It had to do with rounding errors. The statistic is graded along an inverse exponential curve, and once the score gets so low it just rounds down to zero. This I could definitely fix, but...

They all have doo-doo recoil, so it doesn't matter anyway.

I'll wait until that argument is settled before acting further

And sorry for the double post.

[Marvel4]

May 23, 2012 12:28:26 GMT -5 probaddie said:

This is where I need some input. I could easily do this for the semi-automatics (pistols and MK14) but at what rate of fire? Is the average player capable of maximum rate of fire?

It's possible to shoot at more than 750 RPM, but only one or two shots. My average is about 450-600 RPM. The firecap on Black Ops pistols (625 RPM) is not noticeable.

[cmck]

There's a typo in the chart. The P90 is ranked second with 17 and the PP90 is ranking third in class with 20.6.

[banana]

625 is a good rpm but make it known on the graph that you did it at 625

[probaddie]

Update time. First things first:

May 23, 2012 18:20:15 GMT -5 cmck said:

There's a typo in the chart. The P90 is ranked second with 17 and the PP90 is ranking third in class with 20.6.

Fixed. Thanks for spotting that.

Now, I've done a second chart that uses two new metrics. This time, rather than multiplying the two variances in each direction -- I'm retroactively christening that the "product" metric -- I'm using what are called the "taxicab" and "Euclidean" metrics. Both these metrics will not give a gun with unidirectional recoil a perfect score: only a perfect laser beam can achieve that with these metrics. Also, I've included all guns (except sniper rifles and shotguns) with the following assumptions:

• The pistols fire at a rate of 625 rpm
  
• The Mk14 fires at its stock maximum of 545 rpm
  
• The burst-fire weapons (QBZ-95 and M16A4) are rated just like the others (i.e. as though they were automatics)
  

Both metrics are featured on the same spreadsheet. Here it is: docs.google.com/spreadsheet/ccc?key=0AnFeORnwotj_dHFKbGw4ejFjRW0zNTdEQ0dFS0JxVkE#gid=0

Some notes about the metrics:

• As mentioned, the ideal gun with these metrics is a gun with no appreciable recoil (i.e its plot is a single dot in the center.)
  
• The taxicab metric essentially adds the two variances together. This will slightly favor unidirectional plots, but not nearly as much as the product metric.
  
• The Euclidean metric treats the two variances like the two coordinates of a point in the plane and finds its "distance". This metric will be much nicer to "round" plots than the product metric was.
  
• The final statistics are still incomparable by direct means. They're only good for ranking.
  
• Both metrics are symmetric: both x and y directions are weighted equally.
  

Some interesting notes from my point of view:

• The G36C is getting shafted by both metrics here. 7th or 8th in the AR class is too low for me. Any dissenting opinion?
  
• The MP7 is not first overall with the Euclidean metric; it loses out to the L86. As soon as that number came out I opened up the game, slapped on the Thermal on both guns and fired them before finishing the test. To me it makes sense. When I fire the L86, the recoil will actually return to center most times whereas the MP7 will always climb, albeit slowly.
  
• CM901 vs. MK14 (@ 545 rpm): That surprised me a bit. Maybe it just seems easier to control the Mk14 because you usually only have to fire at most 3 shots to get a kill, whereas we all -- well, those of us silly, brave, stupid, manly, retarded, challenge-seeking or curious enough to use the CM901 (take your pick) -- spray full-tilt with the CM901. Your thoughts?
  

Bring up anything else you want to about the results. Thanks again for your feedback.

P.S.: Any chance we could band together and threaten coerce bribe kidnap any offspring/family members of plead with Stuart Brown (XboxAhoy) for those gosh darn golly gee whiz GunKick numbers?

[Indy_Bones]

I'd say that the 'Euclidean' set of numbers more closely matches my own experience than anything else.

There's still a couple that are maybe a place or two that I'd switch round, but overall I think it's a very accurate representation of what I 'feel' with these weapons.

I always knew the M4 was better than many give it credit for, shame you hardly see them amidst the sea of ACR's, PP90M1's and MP7's...

[sh58]

For me I refuse to use the m4 without a scope as I hate the irons.

[cmck]

Both are closer to how I feel like the recoil should be. Personally I think the taxicab matches my rankings better for assault. The g36c is a little too far down the list now, but I can see why things like the scar might beat it. Overall both match up fairly well with how I perceive them in game.

[probaddie]

May 24, 2012 8:15:18 GMT -5 Indy_Bones said:

I always knew the M4 was better than many give it credit for, shame you hardly see them amidst the sea of ACR's, PP90M1's and MP7's...

Indeed; its damage profile is all that holds it back. Even from the recoil plots it was clear the M4 was better than most recoil-wise. I'm glad the statistics reflect that.

I'm generally happy with the statistics. I'm leaning toward the Euclidean statistic myself. Nonetheless, there are still issues to iron out. I'm still of the opinion that the statistic should somehow incorporate a fair vertical bias to reflect the fact that our targets are taller than they are wide -- as someone already mentioned, people are not crocodiles! Starting out with a statistic that doesn't have that bias to begin with should make that easier, whereas such bias was/is inherent in the product and taxicab metrics. So I need help with this in two ways:

• Should it even be done? Or should we leave the statistics dimensionally unbiased?
  
• What are the measurements of a man-sized target? If I am going to introduce a bias I need to know the ratio of height to width in order to adjust the statistics fairly.
  

Also, I think I'll try to "linearize" (if that's a word) the statistics in the next draft. This will not affect the overall ranking, just the raw number each weapon receives. Hopefully this will allow some sort of direct comparison. But even here I need help. To do this, we need to find COD's worst gun, recoil-wise, for which we have the recoil parameters (RPK? MW2's L86? Barrett? Deagle?). I haven't played every game in the series so I'll need some input. From there, I'll assign that gun a score of 0 and grade everything from there.

[Marvel4]

May 24, 2012 12:54:07 GMT -5 probaddie said:

Should it even be done?

Yes, definitely. The current statistics don't show the strength of the G36C and ACR.

To do this, we need to find COD's worst gun, recoil-wise, for which we have the recoil parameters

The W1200 in CoD4 has the worst view kick and centerspeed. But since it's rate of fire is only 80 RPM, the Barrett is probably worse.

[Indy_Bones]

May 24, 2012 13:18:09 GMT -5 Marvel4 said:

Yes, definitely. The current statistics don't show the strength of the G36C and ACR.

I'm not sure I'd agree there.

What these stats show is that other guns in general have less recoil.

My opinion is that this is backed up by looking as Psijaka's recoil plots and through in-game experience.

The ACR doesn't have much recoil when you throw kick on, but without it, it's a lot less controlled and one of the worst AR's from that perspective, which is exactly what the stats are saying. The other attributes it has may well more than make up for this, but the recoil without kick isn't great at all.

The G36C has extremely high recoil, this doesn't mean it's not controllable and can help at times, but other times even a couple of shots can leave you above their head leading to either having to burst fire, or to try to make quick adjustments from the recoil which you may be killed during the process.

Both guns have their strengths and be extremely dangerous, but in terms of pure recoil analysis, I'd suggest that the stats are pretty accurate here. Throw kick into the mix however and it's a very different story in some cases...

[cmck]

I'd also like to see how the results change with and without kick. It sounds interesting.

[j1000]

May 22, 2012 18:57:09 GMT -5 probaddie said:

More than simple, unguided arithmetic, this method uses probability distributions representing the weapons' behavior to calculate the amount of "spread" and "deviation" from the original point of aim to form a statistic, which can be compared between guns to rank their efficacy.

This sounds fun and I encourage it, but in reality the deviations need to be weighted based on personal preference, which throws quantitative analysis out the door again. For instance, some people are more comfortable with a spread that lingers around the point of origin, while others prefer a spread that moves predictably in one direction. It's hard to beat recoil plots for this type of info. But like I said, go for it! It will be interesting regardless.

[TheHawkNY]

May 24, 2012 14:42:05 GMT -5 j1000 said:

May 22, 2012 18:57:09 GMT -5 probaddie said:

More than simple, unguided arithmetic, this method uses probability distributions representing the weapons' behavior to calculate the amount of "spread" and "deviation" from the original point of aim to form a statistic, which can be compared between guns to rank their efficacy.

This sounds fun and I encourage it, but in reality the deviations need to be weighted based on personal preference, which throws quantitative analysis out the door again. For instance, some people are more comfortable with a spread that lingers around the point of origin, while others prefer a spread that moves predictably in one direction. It's hard to beat recoil plots for this type of info. But like I said, go for it! It will be interesting regardless.

Excellent point.

[probaddie]

This sounds fun and I encourage it, but in reality the deviations need to be weighted based on personal preference, which throws quantitative analysis out the door again. For instance, some people are more comfortable with a spread that lingers around the point of origin, while others prefer a spread that moves predictably in one direction. It's hard to beat recoil plots for this type of info. But like I said, go for it! It will be interesting regardless.

May 22, 2012 18:57:09 GMT -5 probaddie said:

I understand that none of this can replace intuition and actually using the weapons to find what works for you. I just thought there might be interest in finding a way to compare the guns scientifically.

I know, and that's all this really is: a bit of fun. (Well, for me as a math nerd it is!) No matter what I do there's always going to be dissent and disagreement, and I'm okay with that.

I'd also like to see how the results change with and without kick. It sounds interesting.

It is. And what about RF and the Grip attachment? I'd like to see how that all pans out and I eventually hope to do a chart including all possible combinations (within each game that's been sufficiently charted). But for now I'm focusing on getting the core statistic ironed out. The ACR will have its day (not that I'm personally rooting for any gun here ).

Now, about the third draft. By absence of its mention, I assume the product metric (the first spreadsheet) is now obsolete in the eyes of most; we all appear to agree that the taxicab and the Euclidean metrics are superior. However there seems to be differing opinion as to whether the statistic should include a vertical bias. Said Marvel4 when asked about induced vertical bias:

Yes, definitely. The current statistics don't show the strength of the G36C and ACR.

And indybones:

I'm not sure I'd agree there.

What these stats show is that other guns in general have less recoil.

My opinion is that this is backed up by looking as Psijaka's recoil plots and through in-game experience.

The ACR doesn't have much recoil when you throw kick on, but without it, it's a lot less controlled and one of the worst AR's from that perspective, which is exactly what the stats are saying. The other attributes it has may well more than make up for this, but the recoil without kick isn't great at all.

The G36C has extremely high recoil, this doesn't mean it's not controllable and can help at times, but other times even a couple of shots can leave you above their head leading to either having to burst fire, or to try to make quick adjustments from the recoil which you may be killed during the process.

Both guns have their strengths and be extremely dangerous, but in terms of pure recoil analysis, I'd suggest that the stats are pretty accurate here. Throw kick into the mix however and it's a very different story in some cases...

But I think the problem here is we don't know just how much the G36C and MP7 are underrepresented with a dimensionally-unbiased statistic. The G36C might, with its heavy vertical recoil, still lose out after a fair adjustment, or become a contender in the AR class and overall; we don't know either way. That's why I want to go ahead and try a vertically-biased version of both the taxicab and Euclidean metrics, to compare with the non-biased versions of the two metrics.

But I really want to do this fairly. In essence, what I want is for the statistic to reflect the ratio of a player model's height to its width. In other words, if we know the ratio of height to width is a:b, then the variance in the x and y directions should affect the overall statistic by the same ratio. Absent that, I would be forced to just make up a ratio, which would introduce an unwanted subjective measure into the formula.

Long story short: I need to know the height and width of the player model(s). Depending on when/if I or someone else can obtain the measurements -- the search feature has not been my friend -- I'll do the third draft containing four metrics: the unbiased taxicab and Euclidean metrics, and the vertically-biased versions of the same. If those measurements cannot be found, I'll go to plan B, which is to find CoD's worst weapon and "linearize" the raw scores. This will not affect any rankings (I promise .

Thanks again for your feedback.

[cmck]

Does it have to be the player model or can you use the average human height to width ratio? Its probably easier to find that than a confirmed player model ratio. I'm going on the assumption that IW based the look of the players on the average person and not a random number.

[probaddie]

May 24, 2012 22:43:45 GMT -5 cmck said:

Does it have to be the player model or can you use the average human height to width ratio? Its probably easier to find that than a confirmed player model ratio. I'm going on the assumption that IW based the look of the players on the average person and not a random number.

I guess I have no choice but to make that assumption, too. I've started looking at data for these figures. There's a plethora of academically reliable info on human height, but much less with regard to human width. I'm getting around 1.71m for average height and 0.51m for width, so a height-to-width ratio of 3.353:1 (approximately 7:2). I'll use 3.353 as the bias ratio for the biased stats, figure out how to incorporate that into the metric tonight, and then churn out the third draft tomorrow.

[Indy_Bones]

It's actually closer to 1.77m for average if we're basing it around UK and US stats, with an average of approx 50 cm shoulder width dropping to around 32cm for abdominal width.