## Re: gas gauge and mileage

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• ... consumed. ... fillups and ... For example, if ... up to 10% ... tanks or 1% on ... I believe the calculation of measurement error is not quite so simple.
Message 1 of 19 , Feb 2, 2006
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--- In toyota-prius@yahoogroups.com, shrubnews@... wrote:
>
> In a message dated 2/1/2006 9:01:27 PM Eastern Standard Time,
> dldorrance@... writes:
> This sort of observation makes me a bit leary about calculating mpg by
> assuming tank replacement volume reliably equals amount of fuel
consumed.
> ======================
>
> You can't do it on a tank by tank basis, but if you sum several
fillups and
> divide into the total mileage you will get a more accurate measurement,
> because the difference will only be the error in the last fillup.
For example, if
> there is a +- 0.5 gallon error per tankful this can be an error of
up to 10%
> (assuming a 10 gallon fill) on one tank, but is only 2.5% on 4
tanks or 1% on
> 10 tanks.
>

I believe the calculation of measurement error is not quite so simple.
The way to evaluate measurement error is a subject of some length
in Statistics 101. While admittedly I took the course some 40 years
ago, my son took it last semester, so I received a refresher of sorts.

Measurement errors are evaluated by Standard Deviation, which is the
sum of all the errors divided by the number of measurements. It's an
average, but there is a catch. Errors never cancel out, they simply
accumulate. So that, for example, a 2% overestimation on one
measurement does not cancel out a 2% undermeasurement in another
measurement as has been asserted in previous posts on this subject.
*All errors are additive*. While this might not seem reasonable, I
suspect it is based upon probability theory. Perhaps someone here
might elaborate on this?

Now I do not know what all the potential measurement errors are in the
way many people calculate mileage, by assuming that the amount of gas
added is equal to the amount previously consumed. Here is my list:
First there is the error of differential pump shutoff. Then in the
Prius there is the error of the volume of the tank, which is a moving
target, as it is not a tank with a known volume but a
bladder which distends and contracts with its temperature. A
corollary to the bladder issue, I suspect, is the observation
recently discussed on another Prius reflector of how many miles it
takes the first bar's worth of gasoline to be consumed. This varies

I suspect that relying on the Prius MFD calculation probably gives a
more consistent number when comparing one Prius against another, given
some of the aggregious replacement measurement anomolies reported on
this and other Prius reflectors.

Dave '01 Prius
• Dave, Measurement error based on standard deviation is the square root of the mean of the squares of the individual errors. Since the errors are squared, they
Message 2 of 19 , Feb 2, 2006
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Dave,
Measurement error based on standard deviation is the square root of the mean
of the squares of the individual errors. Since the errors are squared, they
can't cancel out, the squares are all positive. Then, IIRC, the estimated
uncertainty of the measurement is twice the calculated standard deviation.
That is what many call the estimated error.
Does any of that ring a bell in your memory?

Jon
----- Original Message -----
From: "dldorrance" <dldorrance@...>
To: <toyota-prius@yahoogroups.com>
Sent: Thursday, February 02, 2006 2:21 PM
Subject: [toyota-prius] Re: gas gauge and mileage

> --- In toyota-prius@yahoogroups.com, shrubnews@... wrote:
>>
>> In a message dated 2/1/2006 9:01:27 PM Eastern Standard Time,
>> dldorrance@... writes:
>> This sort of observation makes me a bit leary about calculating mpg by
>> assuming tank replacement volume reliably equals amount of fuel
> consumed.
>> ======================
>>
>> You can't do it on a tank by tank basis, but if you sum several
> fillups and
>> divide into the total mileage you will get a more accurate measurement,
>> because the difference will only be the error in the last fillup.
> For example, if
>> there is a +- 0.5 gallon error per tankful this can be an error of
> up to 10%
>> (assuming a 10 gallon fill) on one tank, but is only 2.5% on 4
> tanks or 1% on
>> 10 tanks.
>>
>
> I believe the calculation of measurement error is not quite so simple.
> The way to evaluate measurement error is a subject of some length
> in Statistics 101. While admittedly I took the course some 40 years
> ago, my son took it last semester, so I received a refresher of sorts.
>
> Measurement errors are evaluated by Standard Deviation, which is the
> sum of all the errors divided by the number of measurements. It's an
> average, but there is a catch. Errors never cancel out, they simply
> accumulate. So that, for example, a 2% overestimation on one
> measurement does not cancel out a 2% undermeasurement in another
> measurement as has been asserted in previous posts on this subject.
> *All errors are additive*. While this might not seem reasonable, I
> suspect it is based upon probability theory. Perhaps someone here
> might elaborate on this?
>
> Now I do not know what all the potential measurement errors are in the
> way many people calculate mileage, by assuming that the amount of gas
> added is equal to the amount previously consumed. Here is my list:
> First there is the error of differential pump shutoff. Then in the
> Prius there is the error of the volume of the tank, which is a moving
> target, as it is not a tank with a known volume but a
> bladder which distends and contracts with its temperature. A
> corollary to the bladder issue, I suspect, is the observation
> recently discussed on another Prius reflector of how many miles it
> takes the first bar's worth of gasoline to be consumed. This varies
>
> I suspect that relying on the Prius MFD calculation probably gives a
> more consistent number when comparing one Prius against another, given
> some of the aggregious replacement measurement anomolies reported on
> this and other Prius reflectors.
>
> Dave '01 Prius
>
>
>
>
>
>
> To access group's website features such as Files, Photos, Links, Database
> and Polls, go to
> http://autos.groups.yahoo.com/group/toyota-prius
> , Photos, Links, Database and Polls, go to
> http://autos.groups.yahoo.com/group/toyota-prius
>
>
>
>
>
>
>
• Has to do with the design of the bladder vs. a conventional tank. The bladder does not exhale while filling, producing a physical condition at the cutoff
Message 3 of 19 , Feb 2, 2006
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Has to do with the design of the bladder vs. a conventional tank. The bladder does not "exhale " while filling, producing a physical condition at the cutoff mechanism that may trip, depending on the sensitivity of the adjustment of the lever. I filled my Prius up this morning in Bellmead, Tx at a Chevron station on I-35, and noticed when it finally did cut off there was "preasure" at the filler neck. Instead of just pulling the filler nozzle out, I allowed it to "bleed" air through the filler nozzle vent, preventing 87 octane (with Techroline) from spilling on my paint, and their drive (and our air). Fillup with one bar showing after 542 miles was 10.8 us gallons. The vehicle was parked level at the pump.
Second fillup was in South Coffeyville, Ok, and was 412 miles later, 7.8 us gallons. This time the car was slighly nose down, and tilted to the passenger's side. There was fuel visible in the filler neck when the "click" occured, after withdrawal of the nozzle. No preasure present when I withdrew the filler nozzle, and it clicked off when the bladder was full. After leaving South Coffeyville, somewhere north of Topeka the temperature dropped from high 50's to mid 30's and fuel consumption increased. On the southward journey computer was reporting 41-43 mpg, tank measure was nearer to 50 mpg while running into a 45 degree 40 mph cross/head wind. Still had a headwind going north, but temps started out in central Texas this morning around 60, got as high as observed 74 degrees north of Dallas, and as low as 32 degrees near Auburn, Ne. Fuel guage is one bar below half, 340 miles since fill...
Oh, forgot...traveled 758 miles in 12 hours 40 minutes with four stops. Two gas, two food, all at least 10 minutes. And I did not eat in the car...
And to the Oklahoma Highway Patrolman who was "cruising" with your pursuit lights on just north of Mc Alester, its no wonder you didn't make your quota...

Harold Hoffman <haroldh7@...> wrote:
Hi, I went to fill up one day at the gas station. When I went to pump gas it would shut off right away, I tried again and again. So I went to the next pump and it worked fine. It could be that some pumps shut off much easyer than others and it may appear that the tank is full and it is not. Harold--Minnesota

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• All true, but I think you missed my point. If I do not calculate the MPG after each fill, but instead wait until I ve filled it several times, any errors in
Message 4 of 19 , Feb 3, 2006
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All true, but I think you missed my point. If I do not calculate the MPG
after each fill, but instead wait until I've filled it several times, any errors
in all but the first and last fill are irrelevant. All that matters is the
total amount of gas that has gone into the tank(and been consumed) over those
several tankfuls, and the total mileage over that same interval.

best,

Larry

In a message dated 2/2/2006 4:26:46 PM Eastern Standard Time,
dldorrance@... writes:

I believe the calculation of measurement error is not quite so simple.
The way to evaluate measurement error is a subject of some length
in Statistics 101. While admittedly I took the course some 40 years
ago, my son took it last semester, so I received a refresher of sorts.

Measurement errors are evaluated by Standard Deviation, which is the
sum of all the errors divided by the number of measurements. It's an
average, but there is a catch. Errors never cancel out, they simply
accumulate. So that, for example, a 2% overestimation on one
measurement does not cancel out a 2% undermeasurement in another
measurement as has been asserted in previous posts on this subject.
*All errors are additive*. While this might not seem reasonable, I
suspect it is based upon probability theory. Perhaps someone here
might elaborate on this?

Now I do not know what all the potential measurement errors are in the
way many people calculate mileage, by assuming that the amount of gas
added is equal to the amount previously consumed. Here is my list:
First there is the error of differential pump shutoff. Then in the
Prius there is the error of the volume of the tank, which is a moving
target, as it is not a tank with a known volume but a
bladder which distends and contracts with its temperature. A
corollary to the bladder issue, I suspect, is the observation
recently discussed on another Prius reflector of how many miles it
takes the first bar's worth of gasoline to be consumed. This varies

[Non-text portions of this message have been removed]
• Larry, I did understand your point and I was trying to point out that each time you fill up there is an error in the measurement. These errors accumulate,
Message 5 of 19 , Feb 3, 2006
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Larry, I did understand your point and I was trying to point out that
each time you fill up there is an error in the measurement. These
errors accumulate, *they don't cancel out*; there is nothing magic
about an average of many fillups with respect to accuracy (or rather
lack thereof) of the final calculation. The errors make the final
average less accurate. The "slop" in the final average can be
calculated using Standard Deviation.

Dave '01 Prius

--- In toyota-prius@yahoogroups.com, shrubnews@... wrote:
>
>
> All true, but I think you missed my point. If I do not calculate the
MPG
> after each fill, but instead wait until I've filled it several
times, any errors
> in all but the first and last fill are irrelevant. All that matters
is the
> total amount of gas that has gone into the tank(and been consumed)
over those
> several tankfuls, and the total mileage over that same interval.
>
> best,
>
> Larry
>
> In a message dated 2/2/2006 4:26:46 PM Eastern Standard Time,
> dldorrance@... writes:
>
> I believe the calculation of measurement error is not quite so simple.
> The way to evaluate measurement error is a subject of some length
> in Statistics 101. While admittedly I took the course some 40 years
> ago, my son took it last semester, so I received a refresher of sorts.
>
> Measurement errors are evaluated by Standard Deviation, which is the
> sum of all the errors divided by the number of measurements. It's an
> average, but there is a catch. Errors never cancel out, they simply
> accumulate. So that, for example, a 2% overestimation on one
> measurement does not cancel out a 2% undermeasurement in another
> measurement as has been asserted in previous posts on this subject.
> *All errors are additive*. While this might not seem reasonable, I
> suspect it is based upon probability theory. Perhaps someone here
> might elaborate on this?
>
> Now I do not know what all the potential measurement errors are in the
> way many people calculate mileage, by assuming that the amount of gas
> added is equal to the amount previously consumed. Here is my list:
> First there is the error of differential pump shutoff. Then in the
> Prius there is the error of the volume of the tank, which is a moving
> target, as it is not a tank with a known volume but a
> bladder which distends and contracts with its temperature. A
> corollary to the bladder issue, I suspect, is the observation
> recently discussed on another Prius reflector of how many miles it
> takes the first bar's worth of gasoline to be consumed. This varies
>
>
>
>
>
>
>
> [Non-text portions of this message have been removed]
>
• ... You will have to explain it more clearly, then. So far as I am concerned, if I drive 10000 miles, and fill up the tank say 20 times, then the mpg
Message 6 of 19 , Feb 3, 2006
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On 03/02/06, dldorrance <dldorrance@...> wrote:
>
> Larry, I did understand your point and I was trying to point out that
> each time you fill up there is an error in the measurement. These
> errors accumulate, *they don't cancel out*; there is nothing magic
> about an average of many fillups with respect to accuracy (or rather
> lack thereof) of the final calculation. The errors make the final
> average less accurate. The "slop" in the final average can be
> calculated using Standard Deviation.

You will have to explain it more clearly, then. So far as I am concerned, if
I drive 10000 miles, and fill up the tank say 20 times, then the mpg
calculation 10,000miles/total of all fillups is exact, except for:
- incorrect meterage by the pump(s)
- incorrect odometer
- whether the tank was completely full with the first or last fillup

Is there a fault in that reasoning?

Regards

Jerry

Are you a Prius owner?
Put yourself on the MAP!
http://www.frappr.com/priusowners

[Non-text portions of this message have been removed]
• ... concerned, if ... Yes, there is a fault in that reasoning. The assumption that the error occurs only in the first and last fillup is incorrect. The error
Message 7 of 19 , Feb 3, 2006
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--- In toyota-prius@yahoogroups.com, JerryW <jerrywh@...> wrote:
>
> On 03/02/06, dldorrance <dldorrance@...> wrote:
> >
> > Larry, I did understand your point and I was trying to point out that
> > each time you fill up there is an error in the measurement. These
> > errors accumulate, *they don't cancel out*; there is nothing magic
> > about an average of many fillups with respect to accuracy (or rather
> > lack thereof) of the final calculation. The errors make the final
> > average less accurate. The "slop" in the final average can be
> > calculated using Standard Deviation.
>
>
> You will have to explain it more clearly, then. So far as I am
concerned, if
> I drive 10000 miles, and fill up the tank say 20 times, then the mpg
> calculation 10,000miles/total of all fillups is exact, except for:
> - incorrect meterage by the pump(s)
> - incorrect odometer
> - whether the tank was completely full with the first or last fillup
>
> Is there a fault in that reasoning?
>
> Regards
>
> Jerry

Yes, there is a fault in that reasoning. The assumption that the
error occurs only in the first and last fillup is incorrect.

The error occurs with every fillup and those errors do not average out
as you are assuming. The net result is that your final answer is some
figure plus or minus some amount due to accumulated errors involved
with each fillup. The plus or minus part is the "slop" I was
referring to in the post above.

Jerry, I didn't make the rules. The reasoning comes from analysis of
how to deal with the inherent error in measurements of any sort. As I
said earlier this measurement error problem is dealt with in the study
of Statistics, which was developed by mathematicians. These rules are
used on a daily basis by engineers, scientists and social scientists
when they deal with measurements.

Dave '01 Prius
• In a message dated 2/3/2006 12:06:59 PM Eastern Standard Time, dldorrance@hotmail.com writes: Larry, I did understand your point and I was trying to point
Message 8 of 19 , Feb 3, 2006
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In a message dated 2/3/2006 12:06:59 PM Eastern Standard Time,
dldorrance@... writes:
Larry, I did understand your point and I was trying to point out that
each time you fill up there is an error in the measurement. These
errors accumulate, *they don't cancel out*; there is nothing magic
about an average of many fillups with respect to accuracy (or rather
lack thereof) of the final calculation. The errors make the final
average less accurate. The "slop" in the final average can be
calculated using Standard Deviation.
=================================
I'm sorry but you still don't understand.

There are no errors to cancel out, assume we agree that if the pump says
9.124 gallons went into the tank, 9.124 gallons actually went into the tank.

I AM NOT AVERAGING THE FILLUPS. Each time I fill up there is NOT an error in
the measurement; the pump reads that actual amount that went into the tank.

I am summing the fillups rather than averaging. I don't care whether 9.6
gallons could have gone into the tank but only 9.124 actually did. On day 1 I
fill the tank and note the odometer. There is an error in the actual contents
of the tank compared with the amount I put in. This is significant, but put it
aside for the moment.

Several days later I refuel, and the pump reads 8.1 gallons. I don't care
whether that tank would actually have taken 8.5, because all I am going to do
is note the 8.1.

The next time I refuel it the pump reads 7.4 gallons. So I have put 15.5
gallons in the tank now. I really have put that amount in total since I started
the calculation; there is still not error.

Again I refuel, and it takes 9.6 gallons. That's a total of 25.1 gallons
that have gone into the tank.

I now read the odometer, subtract the starting reading, and divide by 25.1.
Say the odometer difference is 1,355 miles. Dividing, I find my fuel economy
is 53.98 MPG. On the last fillup, I have a potential error of, say, .5
gallons. But this is 0.5 over the full 25.1, not over the 9.6 I put in on the last
fill. There is also an error at the start of a similar magnitude.

But the total error is significantly less than the error if I had treated
each tankful separately and averaged them. Not to mention the fact that it is
not a valid operation to average several tankfuls.
• In a message dated 2/3/2006 4:01:43 PM Eastern Standard Time, dldorrance@hotmail.com writes: The error occurs with every fillup and those errors do not
Message 9 of 19 , Feb 3, 2006
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In a message dated 2/3/2006 4:01:43 PM Eastern Standard Time,
dldorrance@... writes:
The error occurs with every fillup and those errors do not average out
as you are assuming. The net result is that your final answer is some
figure plus or minus some amount due to accumulated errors involved
with each fillup. The plus or minus part is the "slop" I was
referring t in the post above.
========================

This is where you are wrong. The only error that occurs with every fillup is
inaccuracy in the pump's meter, which had better be vanishingly small or the
station will be in trouble with the authorities. If it reads 9.4 gallons that
is exactly the amount that went into the tank. It doesn't matter that the
tank is not full.

Larry
• ... No, that is just wrong...if I made a separate calculation every time I filled up you might have a point, but the calculation in my post, total miles/total
Message 10 of 19 , Feb 3, 2006
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On 03/02/06, dldorrance <dldorrance@...> wrote:
>
> The error occurs with every fillup and those errors do not average out
> as you are assuming. The net result is that your final answer is some
> figure plus or minus some amount due to accumulated errors involved
> with each fillup. The plus or minus part is the "slop" I was
> referring to in the post above.

No, that is just wrong...if I made a separate calculation every time I
filled up you might have a point, but the calculation in my post, total
miles/total galllons, is exact because errors in filling any one tank
(except the first and the last) are simply not included in the calculation
and cannot impinge on its accuracy...if one tank was only half filled, it
would make no difference... I'm impressed you can't see this.

Regards

Jerry

Are you a Prius owner?
Put yourself on the MAP!
http://www.frappr.com/priusowners

[Non-text portions of this message have been removed]
• Of course they cancel out. If they didn t your guess guage would say it was empty, and you couldn t put gas in or the other way around. Total miles
Message 11 of 19 , Feb 3, 2006
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Of course they cancel out. If they didn't your guess guage would say it
was empty, and you couldn't put gas in or the other way around.

Total miles driven/total gas put in tank = mpg, where the only error is
really the one error in the last tank put in, and the first tank.

Ted

> -----Original Message-----
> From: toyota-prius@yahoogroups.com
> [mailto:toyota-prius@yahoogroups.com] On Behalf Of dldorrance
> Sent: Friday, February 03, 2006 11:04 AM
> To: toyota-prius@yahoogroups.com
> Subject: [toyota-prius] Re: gas gauge and mileage
>
>
> Larry, I did understand your point and I was trying to point
> out that each time you fill up there is an error in the
> measurement. These errors accumulate, *they don't cancel
> out*; there is nothing magic about an average of many fillups
> with respect to accuracy (or rather lack thereof) of the
> final calculation. The errors make the final average less
> accurate. The "slop" in the final average can be calculated
> using Standard Deviation.
>
> Dave '01 Prius
>
> --- In toyota-prius@yahoogroups.com, shrubnews@... wrote:
> >
> >
> > All true, but I think you missed my point. If I do not calculate the
> MPG
> > after each fill, but instead wait until I've filled it several
> times, any errors
> > in all but the first and last fill are irrelevant. All that matters
> is the
> > total amount of gas that has gone into the tank(and been consumed)
> over those
> > several tankfuls, and the total mileage over that same interval.
> >
> > best,
> >
> > Larry
> >
> > In a message dated 2/2/2006 4:26:46 PM Eastern Standard Time,
> > dldorrance@... writes:
> >
> > I believe the calculation of measurement error is not quite so
> > simple. The way to evaluate measurement error is a subject of some
> > length in Statistics 101. While admittedly I took the
> course some 40
> > years ago, my son took it last semester, so I received a
> refresher of
> > sorts.
> >
> > Measurement errors are evaluated by Standard Deviation,
> which is the
> > sum of all the errors divided by the number of
> measurements. It's an
> > average, but there is a catch. Errors never cancel out,
> they simply
> > accumulate. So that, for example, a 2% overestimation on one
> > measurement does not cancel out a 2% undermeasurement in another
> > measurement as has been asserted in previous posts on this
> subject.
> > *All errors are additive*. While this might not seem
> reasonable, I
> > suspect it is based upon probability theory. Perhaps someone here
> > might elaborate on this?
> >
> > Now I do not know what all the potential measurement errors are in
> > the way many people calculate mileage, by assuming that
> the amount of
> > gas added is equal to the amount previously consumed. Here is my
> > list: First there is the error of differential pump
> shutoff. Then in
> > the Prius there is the error of the volume of the tank, which is a
> > moving target, as it is not a tank with a known volume but a
> > bladder which distends and contracts with its temperature. A
> > corollary to the bladder issue, I suspect, is the observation
> > recently discussed on another Prius reflector of how many miles it
> > takes the first bar's worth of gasoline to be consumed.
> This varies
> > from about 50 to about 200 miles as I recall.
> >
> >
> >
> >
> >
> >
> >
> > [Non-text portions of this message have been removed]
> >
>
>
>
>
>
>
> To access group's website features such as Files, Photos,
> Links, Database and Polls, go to
> http://autos.groups.yahoo.com/group/toyota-> prius
> , Photos,
> Links, Database and Polls, go to
> http://autos.groups.yahoo.com/group/toyota-prius
>
>
>
>
>
>
>
>
>
• Pot. Kettle. Black. Actually, you guys are talking apples and oranges. You are tlling him that the errors don t average out, which they don t if you do it
Message 12 of 19 , Feb 3, 2006
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Pot. Kettle. Black.

Actually, you guys are talking apples and oranges. You are tlling him
that the errors don't average out, which they don't if you do it the way
you are talking about, but that isn't the way he's doing it. He's not
taking each individual tank and mpg average and running standard
deviation etc against it. He's taken total driven and total gas in.
The errors cancel out, at least the error of is it really full or not.
There is no error in the total amount of gas that has entered the tank,
or any error in the amount of miles driven.

> -----Original Message-----
> From: toyota-prius@yahoogroups.com
> [mailto:toyota-prius@yahoogroups.com] On Behalf Of JerryW
> Sent: Friday, February 03, 2006 4:42 PM
> To: dldorrance
> Cc: toyota-prius@yahoogroups.com
> Subject: Re: [toyota-prius] Re: gas gauge and mileage
>
>
> On 03/02/06, dldorrance <dldorrance@...> wrote:
> >
> > The error occurs with every fillup and those errors do not
> average out
> > as you are assuming. The net result is that your final
> > figure plus or minus some amount due to accumulated errors involved
> > with each fillup. The plus or minus part is the "slop" I was
> > referring to in the post above.
>
>
> No, that is just wrong...if I made a separate calculation
> every time I filled up you might have a point, but the
> calculation in my post, total miles/total galllons, is exact
> because errors in filling any one tank (except the first and
> the last) are simply not included in the calculation and
> cannot impinge on its accuracy...if one tank was only half
> filled, it would make no difference... I'm impressed you
> can't see this.
>
> Regards
>
> Jerry
>
>
> Are you a Prius owner?
> Put yourself on the MAP!
> http://www.frappr.com/priusowners
>
>
> [Non-text portions of this message have been removed]
>
>
>
> To access group's website features such as Files, Photos,
> Links, Database and Polls, go to
> http://autos.groups.yahoo.com/group/toyota-> prius
> , Photos,
> Links, Database and Polls, go to
> http://autos.groups.yahoo.com/group/toyota-prius
>