### Measuring accuracy of latitude and longitude?

I have latitude and longitude as

`19.0649070739746`

and`73.1308670043945`

respectively.In this case both coordinates are

`13`

decimal places long, but sometimes I also get coordinates which are`6`

decimal places long.Do fewer decimal points affect accuracy, and what does every digit after the decimal place signify?

There are several charts in this thread and they can appear in almost any sequence. I can speak to the one in my answer: although I made a conscious decision not to include this information (any reference to trigonometry risks scaring off people who otherwise have the background to understand everything else), it's a great point and reminds us to make quantitative statements when we can. +1.

If you consider the two common data types of "float32" and "float64", these give about 7 and 16 decimal digits of precision, respectively, here's a rule of thumb: the former gives you about a meter of precision, the latter give you a nanometer of precision. In other words, using floats with latitude and longitude, with a single precision I can distinguish the lat/lng position of adjacent chairs in a conference room. With double precision, I can distinguish the lat/lng position of adjacent mitochondria in a skin cell on the scalp of the person sitting in one of those chairs.

Plase cite the coordinate system, seems WGS84, the default, but important to be explicit about it.. No discussion make sense without a CRS.

What kind of coordinate measurement accuracy are you asking for? 1) About instrumental accuracy, 2) about cartographic accuracy, 3) about analytical accuracy, etc., etc. : -) ...

whuber Correct answer

9 years ago*Accuracy*is the tendency of your measurements to agree with the true values.*Precision*is the degree to which your measurements pin down an actual value. The question is about an interplay of accuracy and precision.As a general principle, you don't need much more precision in recording your measurements than there is accuracy built into them. Using too much precision can mislead people into believing the accuracy is greater than it really is.

Generally, when you degrade precision--that is, use fewer decimal places--you can lose some accuracy. But how much? It's good to know that the meter was originally defined (by the French, around the time of their revolution when they were throwing out the old systems and zealously replacing them by new ones) so that ten million of them would take you from the equator to a pole. That's 90 degrees, so one degree of latitude covers about 10^7/90 = 111,111 meters. ("About," because the meter's length has changed a little bit in the meantime. But that doesn't matter.) Furthermore, a degree of longitude (east-west) is about the same or less in length than a degree of latitude, because the circles of latitude shrink down to the earth's axis as we move from the equator towards either pole. Therefore, it's always safe to figure that

*the sixth decimal place in one decimal degree has 111,111/10^6 = about 1/9 meter = about 4 inches of precision.*Accordingly, if your accuracy needs are, say, give or take 10 meters, than 1/9 meter is nothing: you lose essentially no accuracy by using six decimal places. If your accuracy need is sub-centimeter, then you need at least seven and probably eight decimal places, but more will do you little good.

Thirteen decimal places will pin down the location to 111,111/10^13 = about 1 angstrom, around half the thickness of a small atom.

Using these ideas we can construct a

**table of what each digit in a decimal degree signifies:**- The
**sign**tells us whether we are north or south, east or west on the globe. - A nonzero
**hundreds digit**tells us we're using longitude, not latitude! - The
**tens digit**gives a position to about 1,000 kilometers. It gives us useful information about what continent or ocean we are on. - The
**units digit**(one decimal degree) gives a position up to 111 kilometers (60 nautical miles, about 69 miles). It can tell us roughly what large state or country we are in. - The
**first decimal place**is worth up to 11.1 km: it can distinguish the position of one large city from a neighboring large city. - The
**second decimal place**is worth up to 1.1 km: it can separate one village from the next. - The
**third decimal place**is worth up to 110 m: it can identify a large agricultural field or institutional campus. - The
**fourth decimal place**is worth up to 11 m: it can identify a parcel of land. It is comparable to the typical accuracy of an uncorrected GPS unit with no interference. - The
**fifth decimal place**is worth up to 1.1 m: it distinguish trees from each other. Accuracy to this level with commercial GPS units can only be achieved with differential correction. - The
**sixth decimal place**is worth up to 0.11 m: you can use this for laying out structures in detail, for designing landscapes, building roads. It should be more than good enough for tracking movements of glaciers and rivers. This can be achieved by taking painstaking measures with GPS, such as differentially corrected GPS. - The
**seventh decimal place**is worth up to 11 mm: this is good for much surveying and is near the limit of what GPS-based techniques can achieve. - The
**eighth decimal place**is worth up to 1.1 mm: this is good for charting motions of tectonic plates and movements of volcanoes. Permanent, corrected, constantly-running GPS base stations might be able to achieve this level of accuracy. - The
**ninth decimal place**is worth up to 110 microns: we are getting into the range of microscopy. For almost any conceivable application with earth positions, this is overkill and will be more precise than the accuracy of any surveying device. **Ten or more decimal places**indicates a computer or calculator was used and that no attention was paid to the fact that the extra decimals are useless. Be careful, because unless you are the one reading these numbers off the device, this can indicate low quality processing!

"...can indicate low quality processing" may be a little unfair. Perhaps "...can indicate low quality _presentation_" is fairer.

@martin That's a good point. But when the presentation is low quality that suggests the analysis may be lacking, too. Let's be careful to take that as intended: an *indication* is only a flag, not a blanket indictment.

Is this accuracy applies to both Latitude and Longitude? I'm aware that the vertical and horizontal circumference of Earth is slightly different.

For Longitude values the accuracy/precision question only starts to make an order of magnitude difference from the answer given here at 85 Degrees Latitude and higher. At 90 Degrees the question is irrelevant. See the Wiki Page at - link

Excellent answer. Regarding "A nonzero hundreds digit tells us we're using longitude, not latitude!" - hopefully it is obvious to the reader that the opposite does not hold. For example, `+/- 0 to 90` degrees is valid for both longitude and latitude, but `+/- 0 to 180 degrees` is valid only for longitude.

We recently bought 24 acres of land and we're having to verify a lot of survey work on it for some boundary locations, wetlands RPA boundaries, and so on. (Once we know about where things are, then we'll be having surveyors do the official work for permits and such - we'll save a lot of time on a lot that large if they don't have to look for pins and nails.) You have no idea what a big help this is for us! For example, we had trouble pinpointing the back boundary and corners. Now we know one marker is within 30-40' of the back corner. (Which makes it likely it is the back corner marker.)

Important to cite the coordinate system (!), all discussion is about WGS84... is it? Please show in the answer.

@Peter (1) I believe you are referring to the *datum* rather than the coordinate system and (2) it doesn't matter: the answer remains completely the same, because the flattening of the earth is tiny (only about 1/300) compared to the order-of-magnitude statements made here.

@whuber sometimes, even in nowadays, we use e. g. UTM-SIRGAS, that is other system and other datum, example... And about "order-of-magnitude statements" , well, you post 6...10.. so meters to microns ;-)

- The
The Wikipedia page Decimal Degrees has a table on

*Degree Precision vs. Length*. Also the accuracy of your coordinates depends on the instrument used to collect the coordinates - A-GPS used in cell phones, DGPS etc.`decimal places degrees distance ------- ------- -------- 0 1 111 km 1 0.1 11.1 km 2 0.01 1.11 km 3 0.001 111 m 4 0.0001 11.1 m 5 0.00001 1.11 m 6 0.000001 11.1 cm 7 0.0000001 1.11 cm 8 0.00000001 1.11 mm`

If we were to extend this chart all the way to

`13`

decimal places:`decimal places degrees distance ------- ------- -------- 9 0.000000001 111 μm 10 0.0000000001 11.1 μm 11 0.00000000001 1.11 μm 12 0.000000000001 111 nm 13 0.0000000000001 11.1 nm`

Its important to also distinguish between accuracy and precision: your device may report any number of digits (its precision) but many of the decimal places might be just erroneous. As Chethan mentions, it's important to check with the instrument which may also provide accuracy information when using the device (typically an error range around the true location).

and to be sure you can use a national control set and find local benchmarks with 1st 2nd or 3rd order coordinates and compare to your results. be sure not to average your results.

These days even a very cheap phone GPS should be perfectly accurate at 4 decimal places (11 meters) if you have a clear view of the sky. The remaining digits will not be accurate but if you collect many values and average it out, then they are still useful to have.

Can I get the same chart for degrees minutes and seconds?

One can use the 1-arc-minute-of-latitude = 1 nautical mile = 1852m definitions to produce your own table.

Here's my rule of thumb table...

Latitude coordinate precision by the actual cartographic scale they purport:

`Decimal Places Aprox. Distance Say What? 1 10 kilometers 6.2 miles 2 1 kilometer 0.62 miles 3 100 meters About 328 feet 4 10 meters About 33 feet 5 1 meter About 3 feet 6 10 centimeters About 4 inches 7 1.0 centimeter About 1/2 an inch 8 1.0 millimeter The width of paperclip wire. 9 0.1 millimeter The width of a strand of hair. 10 10 microns A speck of pollen. 11 1.0 micron A piece of cigarette smoke. 12 0.1 micron You're doing virus-level mapping at this point. 13 10 nanometers Does it matter how big this is? 14 1.0 nanometer Your fingernail grows about this far in one second. 15 0.1 nanometer An atom. An atom! What are you mapping?`

**POINT #1. lets differentiate***Precision*from*Accuracy*As it is clear from the picture we can talk about

*Accuracy*of a measurement (e.g. GPS measurement) if we already know the actual value (exact position). Then we can say how accurate a measurement is. On the other hand if you have some measurements and don't know the actual value you can just talk about the precision of the measurement.**POINT #2. Lets consider the***latitude*of the pointIf you are going to speak in cm or mm scale, it may be better to also consider the earth as an ellipsoid and not a sphere. Then as soon as you model the earth shape as an ellipsoid (two-axis ellipsoid),

**you can not map degree decimals to ground distance with a**, because this relation change (for E/W distance measurements) with the change of latitude. Here is another table to show the changes:*single*table`decimal places degrees N/S or E/W E/W at E/W at E/W at at equator lat=23N/S lat=45N/S lat=67N/S ------- ------- ---------- ---------- --------- --------- 0 1 111.32 km 102.47 km 78.71 km 43.496 km 1 0.1 11.132 km 10.247 km 7.871 km 4.3496 km 2 0.01 1.1132 km 1.0247 km 787.1 m 434.96 m 3 0.001 111.32 m 102.47 m 78.71 m 43.496 m 4 0.0001 11.132 m 10.247 m 7.871 m 4.3496 m 5 0.00001 1.1132 m 1.0247 m 787.1 mm 434.96 mm 6 0.000001 11.132 cm 102.47 mm 78.71 mm 43.496 mm 7 0.0000001 1.1132 cm 10.247 mm 7.871 mm 4.3496 mm 8 0.00000001 1.1132 mm 1.0247 mm 0.7871mm 0.43496mm`

As you can see it is not correct to say e.g.: every 1° is about 100km on the earth because it depends on the latitude (also direction); it is about 40km at 67N/S and 100km at equator (0N/S)

All about WGS84, is it?

yes actually we are talking about a mathematical representation of the Earth however if you try with other ellipsoids, you should get slightly the same approximations

+1 for the visual presentation.

I'll try to explain it in different terms:

- Earth's equatorial circumference is about
`40,000`

kilometers (`25,000`

miles). - A latitude/longitude value breaks that distance up into
`360`

degrees, starting at`-180`

and ending at`180`

.

This means that one degree is

`40,000`

km (or`25,000`

miles) divided by`360`

:`40,000 / 360 = 111`

`25,000 / 360 = 69`

(So, one degree is

`111`

kilometers, or`69`

miles.)For fractions of a degree, you divide it by

`10`

for each decimal place, as @ChethanS's chart nicely demonstrates (in km):`decimal places degrees distance ------- ------- -------- 0 1 111 km 1 0.1 11.1 km 2 0.01 1.11 km 3 0.001 111 m 4 0.0001 11.1 m 5 0.00001 1.11 m 6 0.000001 0.111 m 7 0.0000001 1.11 cm 8 0.00000001 1.11 mm`

Since the Earth is not a perfect shape, are all the degrees equal? Will `x` degree give you twice the length of `2x` degree regardless of what value it is?

@Pacerier the coordinate system is relative to the centre of the earth and is a perfect sphere, so the answer is yes. But obviously if you want to measure the distance between two points and there is a 3,000 foot mountain in between them then you need to take that into account and add the extra distance to climb over the mountain. You need to take into account the shape of the terrain if you want to calculate the distance over land between two points.

- Earth's equatorial circumference is about
The other excellent answers here are primarily about latitude. A degree of longitude shrinks from about 111 km at the equator to 0 at the poles, so the

*nominal precision*of a decimal degree of longitude increases as you get closer to the poles (I am making no comment on the actual precision or accuracy of any given measurement)As an approximation, the length in km of one degree of longitude is

`cos(latitude in DD * pi/180) * 111.321 km`

, where 111.321 is the length of a degree of longitude at the equator and pi/180 converts decimal degrees to radians. Then the nominal precision of a longitude measurement at a given latitude is just determined by moving the decimal point; for example, at 40 degrees N, one degree of longitude is about 85 km and the precision of the first decimal at latitude 40 N therefore has a nominal precision of about 8.5 km.You'll notice that 8.5 km is less than the corresponding distance of 11.1 km for the first decimal for latitude at the equator, and so the nominal precision of the higher-latitude measurement is higher.

You are totally correct. That is why it is safe to say that the error due to imprecision in one degree is _at most_ 111km. That is, if there were trees near the poles, it would not be necessary to reach the fifth decimal in longitudes to differentiate one from the other.

@John, I believe you mean to say the precision of a decimal degree of longitude *increases* as you get closer to the poles, while the distance per degree *shrinks*.

Thanks @Feckmore, I've edited my answer.

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Content dated before 6/26/2020 9:53 AM

Randy W. 5 years ago

The answers here are good. I thought I would add to the answer by relating how the digits in longitude are affected by latitude. The charts given above can have the longitude adjusted by multiplying the value in the table by Cos(latitude)