Decimal places
 

I see people here often quoting data to many decimal places. Sometimes, even (as Aerohead did recently), increasing the number of decimal places after doing a calculation.

So what's the issue?

The number of decimal places is indicative of the accuracy of the measurement, with the more decimal places, the higher the degree of assumed accuracy.

Two points.

1. You cannot increase the number of decimal places that was present in the original measurement. So for example, a 9 per cent reduction in a drag coefficient of 0.32 cannot become 0.2912 - there's no basic of validity for the last two decimal places (ie there was no such resolution in the original measurement) and so it becomes 0.29.

2. As textbook I have says of the use of too many decimal places: "They imply a very precise result from imprecise data." Therefore, the number of decimal places should reflect the uncertainty in the original measurements. If I do fuel economy measurements over a relatively short distance (i.e. not thousands of km) and get 3.2 litres/100 km, and then make a change and get 2.9 litres/100km, the improvement is 9.375 percent. But realistically, taking into account the uncertainties involved, it's better to say "about 10 per cent".

As soon as someone starts using lots of decimal places, you know they either have incredibly precise measurements - or they don't have a good feel for the data.

You're not wrong.

en.wikipedia.org/wiki/Accuracy_and_precision

This is an important point. First semester of an engineering or sciences degree, multiple professors will spend class time on this (in the degree I'm finishing now, we covered this in Chemistry 101, Physics 141, and Engineering Science 201. Yes, it was duplicative. Yes, it's that important).

This is the concept of "significant figures," or, as we referred to it in school, "sig figs." The basic rules are:

1) The last digit is uncertain--it's an estimate. Measure something from a tape measure marked in centimeters, and you estimate the decimal point (between the marks), eg 23.7 cm. Measure it with a tape marked in millimeters, and the decimal point is again estimated, eg 236.7 mm. Accuracy depends on the measuring device, but in all cases the last digit--just beyond the resolution of the device--is estimated. It's uncertain.*

2) Addition and subtraction: The uncertain digit is taken from the smallest significant figure of the two numbers, eg 0.067 + 1.40 = 1.467.

3) Multiplication and division: The answer is rounded to the smallest number of significant figures of the input data, eg 0.067 * 1.40 = 0.094 (not 0.0938; one number has only two significant figures, so the answer is rounded to two significant figures).

*Note that this says nothing about the calibration of the device. If you use a tape marked in millimeters but each mark is actually 1.1 mm, you're going to be way off regardless of correct rounding.

Significant figures are significant.

You can add all the decimal places you want so long as they're all zeros.

Only if you mean zeros before other digits, eg 0.000937 has three significant figures but 0.937000 has six.

Others have covered it pretty thoroughly already, but I'd like to put out there that, after a calculation, you can have more decimal places, even while following the rules for significant figures. It isn't the number of decimal places that matters, but rather, maintaining the precision of measurement through the calculations.

resolution
 

Another point is the resolution of the original measurements.

" If I do fuel economy measurements over a relatively short distance (i.e. not thousands of km) and get 3.2 litres/100 km, and then make a change and get 2.9 litres/100km, the improvement is 9.375 percent. But realistically, taking into account the uncertainties involved, it's better to say "about 10 per cent".
"

If you read 3.2, you are really reading 3.2 +/- 0.05, and 2.9 +/- 0.05. While the reading says 3.2, it could actually be 3.24999, or it could be 3.1500. So your change could be anywhere between 6.4% and 12.4%, (if you assume 3.25 and 2.95, and compare to 3.25 and 2.85) so the margin of error on your 9% is plus minus 33%.

Yep, I was trying to make it all easy!

For example, the textbook I am using has three different rules for sig figs and uses no less than 10 examples, with sig figs up to 6.

What I was attempting to do is address the most egregious examples of people quoting data and calculations here as if they were quite precise, when clearly they are not.

But what about when the number is 937000?

I'm going sociology here: more than 2 places past the decimal is meaningless (mostly because you cant half a person and expect meaningful results)

I doubt that the homebuilt state of the art can actually measure to that accuracy without compensation or under closed controlled atmospheres. Go ahead and prove me wrong with actual data and not massaged findings.

No I think that is right. I can’t think of any aero measurements at all that can be measured to that degree of accuracy on the road.

When you are typing in your results it is common to do .37 for example. I think most of the excess in figures comes from copy pasta. It is a tasty but redundant dish. :D

I agree 99.99999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 999%

When I do my own measurements I write it down to the most decimals I can on my own personal piece of paper. (e.g. 4.7823, 4.2901, 4.9563) But then I usually realize that the fluctuations in my measurements are so big that there's no way I'm even close to precise. So I prefer something a little more generalized (e.g. ~4.6).

I wonder where I could get my seat-o-pants meter properly calibrated.

Suprisingly they come calibrated directly from the factory, but the calibration standards are different for each model.

6- decimal places
 

An article about the Cd 0.31 FIAT 500e in Motor Trend, mentioned that, compared to the FIAT 500 Pop, the 500e had 6.3% lower drag.
In order to 'solve' for the 500 Pop drag coefficient ( not provided ), required running out to six -decimal places to achieve a 6.3% fit.

Only if the 6.3 per cent weren't a rounded number, as obviously it was.

If you think that any wind tunnel is working to 6 decimal places, you know very little about wind tunnels (or maybe decimal places?).

obviously
 

It's not obvious to me at all. I have no way to vet what Motor Trend publishes, one way or another.
I've said nothing about wind tunnel working numbers.
I simply stated that, in order to satisfy 6.3% higher drag than the value which was provided, required 6-decimal place accuracy.
Do you want this thread turned off also? When I tell you what I think about your knowledge of wind tunnels?
------------------------------------------------------------------------------------
PS Thanks for page-17, # 162 permalink.
After 28-days, 17-pages, and 161 permalinks worth of grief, we finally manage to get what ought to have been in the preamble to your first permalink, on March 22.
The talking comes first, then the thinking.

Well, we can now add that you don't know much about car magazines, too! Motor Trend would simply have been quoting Fiat, and Fiat would have provided the 6.3 per cent as a rounded number.

To even try to pretend that the figure was exactly 6.3 per cent, and therefore solving for the Cd of the 500 pop requires calculations to 6 decimal places, is typical of the sort of mental knots you tie yourself in when trying to prove your point. It's pretty sad.

FIAT
 

And now your psychic powers have discerned that 6.3% is a FIAT rounded number. Certainly you have an SAE Paper which clearly certifies the truth of the matter.
Don Sherman was courteous enough to tell us that he 'rounded' his Drag Queens data. Pretty rare.
Perhaps you'd like to rewrite all written correspondence in your superior hand. Thanks in advance!
Those who perhaps, have access to only the popular press, will also have only the values these sources provide in attempting to navigate aerodynamics.
Working with default levels of accuracy, derived from this literature might offend only those who've never attempted the voyage.
Please limit criticism to constructive criticism.

Um, to you it might need psychic powers. To everyone else it's pretty obvious that the change in drag wasn't 6.300000%.

:rolleyes:

Compounded error bars?

Margin of error
 

Car manufacturers almost always release drag coefficient data rounded to two decimal places, so the 0.31 is actually 0.31 +/- 0.005.

Maximum possible value:
0.315x0.937=0.295155

Minimum possible value:
0.305x0.937=0.285785

The error in the measurements even after the 6.3% drag decrease is thus 0.01, there is no way that 6 decimal places is in any way scientifically appropriate.

It is completely irrelevant whether it is 6%, 6.3% or 6.3000%, the margin of error from the original Cd is way too large to get anything remotely accurate.

I think the people it may offend are scientist and engineers who know the limitations of margin of error.

everyone else
 

Speculation?
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Here's an example of a reverse-engineering challenge involving Tesla's Model S
* Julian Edgar reports that the Model S has a 8% cooling-drag system.
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The Tesla Model S has been reported with:
Cd 0.26
Cd 0.247
Cd 0.24
Cd 0.225
Cd 0.208
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SAE 2012-01-0178 provides us with a frontal area of 2.4-meters-squared ( 25.8333 sq-ft )
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This gives us a range for the Model S drag factor, from CdA 6.7166 sq-ft ( 0.6239 meters-squared ), to, 5.3733 sq-ft ( 0.49919 meters-square )
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This range offers a spectrum of at least five-different cooling-system drags.
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Since Julian Edgar has not provided any specificity as to what ' 8% ' actually means, ' 8%' remains an unknown quantity to any of the 136,000 EcoModder.com members who might wish to 'engineer' cooling systems.
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I'm often uncertain what underlies the motivation for some of Julian Edgar's posts. Certainly he must understand that members actually have a desire to 'engineer.'
And since the 'signal-to-noise ratio' of some mods complicates the ability to even discern an actual 'change' in drag, the need of highest specificity cannot be overemphasized.

If someone publishes a number, it is universally agreed that number is the observed data point. What you should have said is that actually the number observed could be between these two points, which would be odd unless the sensors were not calibrated to a standard.

scientifically appropriate
 

Probably depends upon the type of science one is doing.
When I'm 'noodling', I simply leave whatever is on the display, or move it into memory, for future re-insertion, for the duration of the calculations.
It would be 'more' work, and less time-efficient to do otherwise. Not to mention accuracy issues.
When I began formal studies, rolling force coefficients for tires easily ran to ten decimal places. Observed wind tunnel values, up to 4-places.
If you're looking for that 0.005 difference from a radiator shutter, you want as 'tight' an accounting as possible.
You know that it requires extremely advanced statistical analysis tools to even identify some 'trends' which are under scientific examination?

I have no doubt that the measured value in the wind tunnel was to 4 or even 5 decimal places. However, the published number was to two decimal places.

Agreed, on the tiny change from a radiator shutter, they will do measurements down to 3,4 or 5 decimal places. However, that doesn't necessarily reflect in the published data, if the cD was actually 0.304 and then a radiator shutter was added that reduced the drag to an actual 0.299, the published number is still going to be 0.30.

The difference may be 1.6%, but it would be erroneous to claim that because the initial figure was 0.30 and there is a 1.6% reduction, that the new drag coefficient is 0.295.

Because as we see in the above example the actual figure is 0.299.

You are confusing accuracy of measurements with precision, read about it here https://en.wikipedia.org/wiki/Accuracy_and_precision.

Thanks. I gave it a shot at Permalink #2.

decimal places
 

Here are some automotive entities who publish drag coefficients to three decimal places:
Aston Martin
Audi
BMW
Bochum University
CALTY
Cambridge University
Chrysler
Citroen
Coventry University
Robert Cumberford
Daihatsu
Daimler-Benz Mercedes-Benz
Gary Eaker, A2 / Aerodyne
Eindhoven Technical University
Ferrari
FIAT
Ford
Fuji Heavy Industries ( Subaru )
G.A.C.
GALCIT ( Cal Tech )
General Motors
Holden
Honda
Hoxan
Hyundai
Ital Design
JEEP
Konigsegg
Lotus
MIRA
Mazda
NASA
Nissan
Opel
Penske Racing
Peugeot
Pininfarina
Polestar
Porsche
Renault
Shelby Super Cars ( SSC )
Tatra
Tesla
Trabant
Triumph
Toyota
Antoine Volanis
Volkswagen

two
 

please see my # 28 permalink below

If the original data is three decimal places, then there is no issue deriving data to a higher level of precision (than you could with two decimal places), but even then it is still +/- 0.0005 and so no further precision can be gained.

Indeed you did, I didn't re-read the thread. It is very important and one many people on here don't fully understand, see below for one example, perhaps unfairly cherry picked, but certainly, and unfortunately, representative of many people here on ecomodder.

fifteen decimal places for Jet Propulsion Laboratory
 

I peeked over at Cal Tech.
For pi, they're using 15-decimal place accuracy in celestial mechanics.
I won't freak over using 0.00238 for (rho)

Good example of Aerohead deliberately trying to create confusion.

The reference I used was the third post in my thread that introduced the topic:

Tesla Model S -

Palin, R., Johnston, V., Johnson, S., D'Hooge, A. et al., The Aerodynamic Development of the Tesla Model S - Part 1: Overview, SAE Technical Paper 2012-01-0177, 2012

I even asked members to read the paper and check my calculation for themselves!

I can't find your third thread, but for everyone else, selected numbers are.

Total intake block cD change = -0.020

Baseline with 19" wheels = 0.249
19" aero wheels = 0.223
21" style wheels = 0.257

Depending on options specified, such as wheels, it is impossible to say 8% is the same for every option. But this test was done on the baseline model with 19" wheels and that, with all intakes blocked, reduced drag by 8%.

Would the same 0.02 change be found with the other wheels? Who knows? would the same 0.02 change be found on the myriad of numbers listed by aerohead, again we don't know.

Unfortunately true, I think it is a superiority issue, and good points/arguments get drowned out by the never ending gish gallop of aerohead.

Reading the referenced paper generally helps here.

Gish gallop - "technique in which a debater attempts to overwhelm an opponent by excessive number of arguments, without regard for the accuracy or strength of those arguments."

I'd never heard that term before - it's certainly apt for Aerohead.

But around here, we'd simply say: "Trying to blind with BS".

Unfortunately I see problems with significant figures all the time. For example, nutrition students I evaluate sometimes like to estimate calorie requirements to the tenth or even one hundredth. It is pointless to do that because I would say most people would have trouble reporting within 100 calories of their actual intake.

The number of decimal places used should always correspond to practicality of being so specific as well as the sensitivity of the data collection tools. I would argue that it is pointless reporting FE averages to the tenth. There is too much variation in fuel pump cutoffs and odometer accuracy for the tenth decimal place to matter in FE averages. Also the more data points you have the less individial variation matters.

However, if you can measure FE in a somewhat controlled environment, then I can see justifying going to the tenth decimal place.

Yeah, I am amazed that's its even a topic for debate - shows how much some people here lose the wood for the trees.

At minimum, it certainly shows how little contact some people here have with any science, physics, etc. Or even, just car modification generally?

It wasn't until the Original Poster started.

Debate, not discussion.

deliberately
 

1) you make a comment without caveats ( those can take up to 28-days later )
2) it can be days before I return to the computer.
3) what I post is predicated upon your original post before leaving for the interim period.
4) in the meantime, you may finally make the qualifying remarks on a later page, which were originally absent.
5) as long as you 'post first' and 'think later' this sort of thing is going to continue.
6) slowing way down, and providing all conditions and caveats from the get-go, would be a real improvement in communicating your themes.

two significant figures nets me a 16-count discrepancy
 

* Just for giggles, I ran a set of aerodynamic road load calculations for a Cd 0.247 Tesla Model S at top speed ( 155-mph ). In U.S. units.
* In one, I used whatever was in the computer to complete the string.
* In the other, I truncated the values throughout, to two significant figures, regardless of what the computer 'said.'
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* Between the two, the truncated figures resulted in a 6.5% deviation in power, 152.27- hp vs 162.2-hp actual.
* 'Truncated' drag coefficient = Cd 0.2309 vs Cd 0.247 actual ( delta- 16 counts)
* 'Truncated' per-mile energy: 732.9 Wh /mi vs 780.6 Wh / mi actual.
* With zero changes to the car, numerical truncation leads to an implied 16-count drag deficit.
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It appears that truncation leads to an uncertainty which exceeds the spectra of some aero. modifications we might choose to investigate.