# on my Eos, GPS Speed vs Speedomerter Speed, Off by 10%



## ialonso (Aug 26, 2006)

Returning from a 5 hour trip today, can behaved magnificently, but I noticed that the speedometer is 10% off the GPS speed...
70mph on the GPS is 77 on the speedometer,
80mph on GPS shows around 88mph on the speedometer, The same at lower speeds.
Anybody else ? Any possible adjustment ? 
I understand tolerances, but 10% is a big number at 70 mph...
70mph looks much more like 80mph on the speedometer.


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (ialonso)*

I noticed the same thing on my GPS vs the speedo on my truck. About 10% variance, but in kph instead of mph.
At 110 kph (approx. 70 mph) on the speedo, the GPS was showing about 100 kph (approx. 62 mph).
Not sure which one is accurate, I know my GPS is only accurate to within 3 meters (10 ft) as far as location goes, so maybe the calculated speed is off a bit too.








Either that, or the auto industry has collectively agreed to try and protect us from speeding tickets.








Kevin








The GPS would be calculating speed using several waypoints over elapsed time, Maybe it is because the GPS is showing an average speed, rather than the actual speed at the moment??


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## neweosowner (Nov 25, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (ialonso)*

There was a tech bulletin related to a misaligned speedometer configuration...but that is only a 7% difference in my experience. Depends what size of tires you're running I think, but I have no evidence of that. http://forums.vwvortex.com/zerothread?id=2907493 
I was also watching a (cheap) GPS and its real-time numbers wandered quite a bit (about 10% as I recall) so I wouldn't be surprised if it was GPS-related. I never did any research on the reliability of GPS information to learn more.
If you have a chance to drive at a steady speed on a state highway, you should be able to measure your speed using mile markers. They're not 100% accurate but in my experience they tend to be close enough to see a 5% or greater difference. Grab a stop watch, engage cruise control, and after 3 miles you should be able to do the math. I'd suggest a cruising speed of 60mph to make the math simple, but in Texas that's likely to be accused of Sunday driving.


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## Freund (Jul 3, 2002)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*

If you have a fully operationable GPS the speedoffset is only +/- 1KPH / +/- 1MPH.
GPS speed is very accurate.
Every Car (with the original Wheelsizes) generates a signal for the speedometer that can be up to 10% of the real speed.
If you use your MFA/Tripcomputer and reset your avg. Speed while driving steady (CruiseControl) @ constant speed - the MFA will show the real speed you are going (again +/- 1).
Have Fun driving your EOS!
Freund


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## neweosowner (Nov 25, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (Freund)*


_Quote, originally posted by *Freund* »_
If you use your MFA/Tripcomputer and reset your avg. Speed while driving steady (CruiseControl) @ constant speed - the MFA will show the real speed you are going (again +/- 1).


Thanks for the response. Good to know that a good GPS should provide the correct speed - that's what I was telling my parents just as the GPS-displayed speed starting floating. One place where I would think GPS speed error is possible is while driving around a corner (depending upon the sampling frequency of the GPS and the angle of the corner...not to be confused with the change in vehicle speed that is possible during cornering.) Without doing the math, I'd be shocked if the discrepancy was more than 1% even with a ludicrous 15 sec. sample.
Doesn't your test using the Trip Computer assume that the odometer is correctly calibrated? I look forward to trying this test while my speedometer is misaligned.


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## Freund (Jul 3, 2002)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*

"Your" Odometer is correctly calibrated - trust me.
The "setoff" is programmed into the software.
If you have the MFA there will be a Setting for the adjustment of the odometer in combination with wintertires/-wheels (not shure if this is avail. in Can/US).
Freund


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## PanEuropean (Nov 3, 2001)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (Freund)*

Our experience in the Phaeton forum over the past few years has determined the following:
*1)* The needle on the speedometer is deliberately set to read high - typically about 7% - in order to ensure that it does not under-report speed if larger tires are fitted to the vehicle. This is a regulatory requirement both in Europe and North America.
*2)* The odometer of the car is always accurate if the car is equipped with the same size of tires that it left the factory with.
*3)* The car does know the true (accurate) speed that it is moving at - again, assuming that it is equipped with the same size tires that it left the factory with. This can be observed by looking at the measured value block (MVB) for vehicle speed that is displayed by either the engine controller or ABS controller.
*4)* The offset between true speed (what the MVB reports) and indicated speed (what the needle points to) is usually about +7%. There is no known way of adjusting the needle to make it accurate.
*5)* Some vehicles have an adjustment function in the instrument cluster to allow the speedometer / odometer to be re-calibrated in case tires with a larger or smaller diameter than original fitment are installed. Such adjustments need to be made with a diagnostic scan tool. However, the offset between the MVB speed and needle indication still remains.
*6)* Some vehicles have the capability to store two different wheel circumferences (summer tires and winter tires), and give the driver the ability to switch between the two using controls on the instrument cluster (multi-function indicator). Again, the offset between MVB speed and needle indication remains.
There was a TB (technical bulletin) promulgated concerning speedometer adjustments on the Eos. It can be found at this link: TB: RVU - Instrument Cluster, Update Speedometer Coding (OR) - Includes TB 00-06-10. I am going to guess that this TB arose as a result of cars with all season tires (the North American fitment) being coded to indicate that they had summer tires on them (the European fitment). In any case, this TB is a 'Required Vehicle Update', which means that it will be automatically carried out next time your car is in for service.
There is more discussion about speedometer accuracy at this post in the Phaeton forum: How accurate is the Phaeton speedometer? You could probably apply 99% of the findings to an Eos, because I doubt that the legislation in either Europe or North America has changed in the past few years.
Michael


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (PanEuropean)*

Micheal,
As usual, a very informative and well written post.
You have confirmed what I have long suspected, that the factory speedo settings read higher than actual speed. 
I first started to wonder about this when they began putting those portable, radar equiped, speed warning signs along the roadside, they never seemed to match the speedo. After I purchased a GPS I noticed pretty much every vehicle I was in seemed to indicate inaccurate speed.
Being a consumate cynic, I assumed the auto manufactures were doing it to protect themselves against liability. i.e. better the speedo indicates high rather than low: "honestly judge, I thought I was only doing 100, but the morons that built my car put a crappy speedo in it, that's why the fine officer over there clocked me at 110".
This explains why you can set your cruise at 10km over the posted limit, and the cops will generally never give you a second look. that, and the fact the courts (in Alberta at least) generally give you a little leeway as well.
Kevin








This also explains why the cops always ask "do you know how fast you were going". Next time, I'm shaving 7-10% off my answer!


_Modified by just4fun at 6:23 AM 2-12-2007_


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (PanEuropean)*

This post motivated me to dig the GPS out of the boat and run a check on the speedometer in my current truck, since I hadn't used the GPS in this vehicle yet.
Interestingly, the speedo in my Dodge Ram is only offset by +1% at 100 kph. With the cruise set at 100 kph, the GPS was showing 99 kph.
Kevin


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## jdl (Dec 7, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (PanEuropean)*

I was shown how to display the actual speed and told the difference is intentional and a safety issue. 
You can display a lot of diagnostic information about the car, which I haven't researched but one code is for speed. 19.1 is mph and I think 19.0 is Kilometers. Maybe Michael can be more accurate at what the codes are.
How and where to display these codes: I'm not in front of my car at the moment, so I don't have pictures and may be somewhat inaccurate. But the codes can be displayed on the internal car temp LCD's on the center console.
Hit the Econ switch and the upper air switch at the same time. This will get you in diagnostic mode. Then use the left temp dial to change the diagnostic mode until you get to 19.0. Then you can rotate the center or right dial (don't remember which) to get to 19.1 to dsplay the actual speed. 
Hope this helps and it's ok to post.


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## neweosowner (Nov 25, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (PanEuropean)*

Thanks for the detailed posting. I wish someone at my dealership knew this when I first posed the question.
I did want to address one point in your post regarding the regulatory requirement for North America. I didn't have the discussion w/ the engineers in Dresden, so perhaps you can provide some more information. According to the Car and Driver article from a few years back, it would appear that North-American-built vehicles have needles that tend to be closer to being accurate than their European counterparts. From the regulations and recommendations listed in the article, it's clear that there's much greater tolerance for what the needle can display in North America, versus the rather tight requirement and subsequent implementation of the rules in Europe.
I've rented numerous vehicles over the last seven years, most of them from Ford and G.M. but some Chrysler and Japanese manufacturers too. The vehicles have almost never shown the variance that my Eos does (or, from what I'm reading, many European vehicles show). About half of the time I'll get a 99%+ accurate reading, about half of the time I will get a variance (sometimes up to 5% or 6%, but often much less). And I'm pretty sure you can put larger tires on most of those N.A. vehicles, one of the reasons cited in having a variance in the speedometer needle.
And yes, I have checked the accuracy of the speedometer on most of my rentals.








So this problem appears to be primarily with European-built vehicles to comply with European regulations (even those being shipped to NAR), with less potential for occurance on North-American-built vehicles. Is that your reading of the situation too?
I am actually thinking of buying larger tires in order to get the damn needle to read correctly - it is certainly a pet peeve of mine if I'm going to spend hundreds of dollars on that solution. The other fix would be to purchase one of the roof modules (can't remember which one) that offers a digital speed readout on the MFD.
Is there a speedometer's anonymous chapter I can join?


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## minnvw (Oct 16, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*

neweos, just do like i do wait until he cops pull me over, they have all the latest equipment and its recently calibrated, no seriously i had ours recalibrated at the dealer and it seems real close now, im running 17 in. tires so im sure that was why it was off, it did seem like my milage did drop some , but no biggie , in this frozen tundra i do let it idle alot to warm up, (thats zero mpg ,so throws it all out of wack) thanks for the heated seats, toasty nice, someday well try it with the top down,havent yet, sure is fun car to drive,, doug


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## PanEuropean (Nov 3, 2001)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*


_Quote, originally posted by *neweosowner* »_...So this problem appears to be primarily with European-built vehicles to comply with European regulations (even those being shipped to NAR), with less potential for occurrence on North-American-built vehicles. Is that your reading of the situation too?

I can't say if that statement is correct or not (I don't have the knowledge or authority to do so), but I suspect that there is an element of truth to it.
I do know that the laws in Europe are very, very strict about speedometers: They must not under-read no matter what the circumstances are, which means that most manufacturers set the needle to over-read somewhat. On my 2002 NAR Golf, my 2004 NAR Phaeton, and my 2001 Swiss Golf, the error is a pretty constant 7% over-reading at the needle, even though the speed reported by the ABS controller (as observed when reading the Measured Value Block with a diagnostic scan tool) is dead on the money.
I suspect that the big North American manufacturers (Honda, Toyota, GM, Ford et al) probably put a bit of padding into their speedometers as well, if for no other reason than to protect themselves against hungry American tort lawyers coming after them. There is a SAE spec and a FMVSS spec that deals with speedometer accuracy in automobiles. Light trucks are another story.
The OP's (original poster's) report of a 10% inaccuracy is higher than anything I have seen or heard of, so, it is possible that he or she may have a vehicle that is affected by the TB that I referenced earlier. I don't know what the circumstances behind that TB are, however, I kind of suspect that Eos were going out the door in Portugal coded with the circumference of a commonly used European tire when, in fact, the car was equipped with a NAR spec all season tire that had a slightly smaller circumference - hence the error.
In summary:
*1)* For the OP - see if your car is affected by the aforementioned TB. Your VW dealer will be able to determine this.
*2)* For the rest of us - if the error is +7% or less at highway speeds (e.g. 100 km/h or 60 MPH), then that is normal and we ought to live with it.
Michael


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## Canadian Lurker (Nov 11, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (PanEuropean)*


_Quote, originally posted by *PanEuropean* »_
In summary:
*1)* For the OP - see if your car is affected by the aforementioned TB. Your VW dealer will be able to determine this.
*2)* For the rest of us - if the error is +7% or less at highway speeds (e.g. 100 km/h or 60 MPH), then that is normal and we ought to live with it.
Michael

Regarding point #2 above, when I had the Dealer look at my car shortly after getting it they confirmed that all was operating as expected. In addition to scanning etc., they also road tested it including pacing with another vehicle. I walked out only partially convinced....only a few days later with what I suspected was still an inaccurate speedo did it dawn on me - what vehicle did they use to pace my car with....a VW Demo from a Sales guy of course!! So go figure that when compared to another vehicle with a speedo inaccuracy that it checked out as operating the same or as expected....








So, I've learned to adjust to it and am now quite comfortable with it. Most times I just pace with traffic, other times when solo on the road I make a mental adjustment to see where I'm really at and adjust accordingly. That it's probably gotten me to slow down slightly cannot be a bad thing at the end of the day!!








Perhaps though, VW should just come clean and put it in as a passive safety item since safety seems to help sell cars these days!!








Cheers,
JJ


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (Canadian Lurker)*

Confirmed a speedo offset of approx 4% on our Eos. With the cruise set at 100 kph, my GPS was showing 96.3 kph.
The odometer is off by the same percentage. With the GPS showing 100km traveled, the odometer was registering 96.1km.
Kevin








This assumes my GPS is accurate.


_Modified by just4fun at 2:44 PM 2-20-2007_


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## neweosowner (Nov 25, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (just4fun)*

What size of tires are you running? 
(Ah...separate thread shows 235/45 R17. Thanks.)
Somewhat curious that yours is off 4% while mine is around 6%-7%. Your experiences don't seem to align with the majority wisdom in this thread - that the odometer is accurate and then the needle should read higher than the actual speed. Did you get a chance to reset the trip computer and get a reading from it?
P.S. - my odometer appears to be 99%+ accurate. We've read in previous posts that the odometer is "definitely accurate".










_Modified by neweosowner at 4:32 PM 2-20-2007_


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*

Hi Arylnn,
I noticed that my observations did not jive with the general concensus of the other posts.
I reset both the trip odometer, and the GPS trip "odometer' to zero before proceeding. I also pulled over and rolled slowly to a stop at the 100km travel mark, and compared the two readings. So they are accurate observations.
The question is, how accurate is the GPS? I know another post indicates they should be dead on.
Kevin


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## PanEuropean (Nov 3, 2001)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (just4fun)*


_Quote, originally posted by *just4fun* »_The question is, how accurate is the GPS? 

In theory, the GPS should be pretty accurate, because even with the worst possible position fix (Dilution of Position = or > 100 feet), the GPS error will be zilch if you are measuring over a distance of more than a mile.
However - what you really have to watch is where you are getting the 'distance travelled' information from. If you have a 'pure automotive' design GPS, and it has an odometer function built in, that should be pretty trustworthy.
Michael


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (PanEuropean)*


_Quote, originally posted by *PanEuropean* »_
If you have a 'pure automotive' design GPS, and it has an odometer function built in, that should be pretty trustworthy.
Michael

The GPS has several navigation screens, and I was using the one with a dedicated odometer, speedometer, elapsed time, heading, etc. (no map on this screen)
Kevin


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## neweosowner (Nov 25, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (just4fun)*

The odometer should be "simple" to fix. What I want to know is how the needle got in sync with the reported driving velocity...and how to do that to my car!








Look forward to hearing more about your discrepancy should you decide to pursue it in the future.


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## PanEuropean (Nov 3, 2001)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*


_Quote, originally posted by *neweosowner* »_What I want to know is how the needle got in sync with the reported driving velocity...and how to do that to my car!

Sadly, there ain't no way to get the needle of the speedometer to point at the speed that the guts of the speedometer knows is the true speed of the vehicle. We explored this topic at great length in the Phaeton forum, even to the point of contacting Robert Bosch, who make the speedometers (instrument clusters) for the Phaeton.
Bottom line is this: The car knows, with great accuracy, what speed it is moving at. It uses this 'accurate' speed for both the odometer and for the ABS and various stability control systems such as ESP, as well as any other car system that might inquire about vehicle speed, such as the transmission or even the roof controller. But, the needle has a deliberate 'fudge factor' built into it, to cause it to read a little bit high. There is no way to calibrate the needle to match the (known) accurate speed.
Michael


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## owr084 (Nov 15, 2004)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (Freund)*


_Quote, originally posted by *Freund* »_*"Your" Odometer is correctly calibrated - trust me.*
The "setoff" is programmed into the software.
If you have the MFA there will be a Setting for the adjustment of the odometer in combination with wintertires/-wheels (not shure if this is avail. in Can/US).
Freund








 
Too bad Honda can't say the same...


_Modified by owr084 at 7:56 AM 2-21-2007_


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*

Hi Mike,
I probably won't persue this much further until such time I put new tires on the car. I will probaly re-confirm the readings then, just because there can be slight dicrepancies in tire sizing between manufactures.
I just wanted to know where my offset (setoff?) was on the car so I knew how much to "adjust" for actual speed. Now I know if I set the cruise 3-5 km above my desired speed, I should be +/- 1 km of where I want to be.
Because of the discussion on this thread, I was surprised to find my odometer offset by the same margin.
Kevin


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## just4fun (Oct 29, 2006)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (neweosowner)*


_Quote, originally posted by *neweosowner* »_Look forward to hearing more about your discrepancy should you decide to pursue it in the future.

*NOTE: this response is also posted under First Impressions, but I felt it belonged on this discussion thread as well.*
I'm going to double check it again next time we are going for a drive of more than 100km. I know my initial observations were accurate, but can't hurt to double check.
I'm not overly concerned about the outage, but also don't fully understand the logic behind all the computer assisted functions of the car.
If the sensors themselves are inaccurate, then it could have an adverse affect on ABS, ESP, and ASR. That would not be so good.
If the sensors are accurate, and just the odometer display is offset, then it is no big deal.
If I re-confirm it is out I will ask the dealership to look at it just to be on the safe side.
Kevin


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## jgermuga (Jan 11, 2007)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (just4fun)*

I am surprised it is at all possible to adjust the speedo, and/or the odometer. Apparently the two are no longer connected, but if the odometer is off by a corresponding 7%, this means I am getting bilked out of 2730 miles on a 39K mile lease term.
As for adjusting to snow tires, I am surprised that this allowed given the possibility to cheat the system. 
Accordingly, I can get a pretty decent subsidy on going to a taller set of tires. Given some other discussions on snow tires, it would make sense for me to purposely buy my replacement tires NOW and make sure they are taller than stock. This way I will be guaranteed a return on my investment in the first couple years of the term.
Following this logic, I could borrow a smaller set of wheels (assuming the overall wheel+tire circumference is also smaller), put them on, tell my service tech my reading is high, get him (or her) to adjust it down and put the original wheels back on.
Can this be true??


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## PanEuropean (Nov 3, 2001)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (jgermuga)*

John:
The questions you raise have been addressed elsewhere in the forum recently, however, the discussions seem to be all over the place, so, here's the 'executive summary':
*1)* Normally, the speedometer *needle *reads high - usually between 5% and 7% higher than actual speed. This is a legislative issue and there is nothing that can be done to make the needle point at the exact correct speed.
*2)* Despite the intentional (by design) over-reading of the needle, the car does know the exact correct speed that it is moving at, typically within 1% accuracy. You can confirm this by comparing the measured value block (MVB) readout of vehicle speed from the engine, transmission, or ABS controller with a groundspeed readout from a GPS.
*3)* The odometer uses the true speed (measured value block) speed of the car to calculate distance, *not *the speed displayed by the speedometer needle.
*4)* There was a mistake made at the Eos factory in Portugal last year - a limited number of cars destined for the North American market were coded to indicate that they had smaller diameter tires on them (the size sold in the European market, not the all season tires supplied in North America). VW issued a technical bulletin about this a long time ago (within a few weeks of Eos introduction), and classified the TB as a 'RVU' (Required Vehicle Update). This means that the action needed to correct this mistake will be automatically carried out either before delivery of the new car (for cars delivered after October 18, 2006), or the next time the car comes back to the dealer for any kind of service, (for cars delivered before October 18 2006). There is a post about this TB here: TB: RVU - Instrument Cluster, Update Speedometer Coding (OR) - Includes TB 00-06-10.
*5)* To adjust the measurement of the distance travelled by the car to ensure correct odometer and MVB speed readings when snow tires of a different diameter than the OEM tires are installed, you have to take your car to a VW dealer, who will then measure the tire circumference and make the appropriate software coding changes to your car. This will then allow you to set your odometer / speedometer to indicate whether you are using snow tires or summer tires. See this post for additional detail about this procedure: Eos Tire Pressure Monitoring System (TPMS) - design and theory.
*6)* Regarding your proposal: _"As for adjusting to snow tires, I am surprised that this allowed given the possibility to cheat the system. Accordingly, I can get a pretty decent subsidy on going to a taller set of tires. Given some other discussions on snow tires, it would make sense for me to purposely buy my replacement tires NOW and make sure they are taller than stock. This way I will be guaranteed a return on my investment in the first couple years of the term."_
What can I say to that? VW presumes that the customers are basically honest. The entire customer - dealer - importer - manufcturer relationship is built on that assumption. So, let's not disappoint them, OK? Would your mother be proud of you if you told her that was what you were planning to do?








Michael


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## jgermuga (Jan 11, 2007)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (PanEuropean)*

Seeing how Mom is doing 5-7 at state on grand larceny, she has little room to judge.







JK
I am not saying I would do this and I hope no one who reads this will decide it is a good idea either. Besides being dishonest, from what I recall from my garage days, odometer tampering is a felony. I am just surprised the ability to adjust the odometer is possible at all. 
Although I would not take advantage of this situation myself, I know at least a few people (within the industry in fact, making it that much easier, by perhaps offering this service for a "modest" fee) who would not find this very objectionable. In fact, I decided long ago that working in the auto industry was not for me, mainly because I saw so many shady practices.


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## jgermuga (Jan 11, 2007)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (jgermuga)*

"In fact, I decided long ago that working in the auto industry was not for me, mainly because I saw so many shady practices."
Speaking of which, one doean't have to look far beyond this very forum to find an example, since I have found through direct experience that a certain luxury auto aftermarket product purveyer fits this description quite well.


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## PanEuropean (Nov 3, 2001)

*Re: on my Eos, GPS Speed vs Speedomerter Speed, Off by 10% (jgermuga)*


_Quote, originally posted by *PanEuropean* »_Would your mother be proud of you if you told her that was what you were planning to do?










_Quote, originally posted by *jgermuga* »_Seeing how Mom is doing 5-7 at state on grand larceny, she has little room to judge.






































Michael


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