# How to measure the accuracy of latitude and longitude?

I am very new to GPS stuff. Suppose 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. Does fewer decimal points affect accuracy? What does every digit after the decimal place signify?

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What does the symbol ' and " means in the measures of latitude and/or longitude means? For example: 10 degrees 11'07" – user5633 Jan 25 '12 at 12:18
Welcome to our site, Jorge! Please start by reviewing our FAQ and follow its guidelines for asking questions. (However, yours is so straightforward it can be answered easily, so maybe we can spare you the effort: ' stands for minutes (1/60 degree) and " stands for seconds (1/3600 degree).) – whuber Jan 25 '12 at 15:06
The approved answer has "cm" which is confusing. Why not show mm instead for all the smaller values? – user25115 Dec 19 '13 at 23:00
I don't know what you mean by the "approved" answer, user25115. In my answer (which does not use centimeters) I took some care to use units that vary by a factor of 1000--kilometers, meters, millimeters, and microns (the latter was used only once, to make a point about overprecision). I employed these units to make the reported numbers of easily comprehensible size. – whuber Aug 2 '14 at 18:47

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!
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"...can indicate low quality processing" may be a little unfair. Perhaps "...can indicate low quality presentation" is fairer. – Martin F Aug 15 '13 at 21:00
Is this accuracy applies to both Latitude and Longitude? I'm aware that the vertical and horizontal circumference of Earth is slightly different. – Baby May 5 '14 at 3:39
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 – user23715 Jun 20 '14 at 23:02
@sepideh I like the idea of encouraging questions about the broader uses and applications of GIS, aka "Geomatics." Perhaps a better place to have such a discussion would be on our meta site. – whuber Aug 30 '15 at 15:01
@sepideh Thank you very much! I am grateful for your help. – whuber Sep 1 '15 at 16:14

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
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Hi Chetan, your answer helped me a lot. This has solved my problem. Thanks a ton. – Saurabh Apr 18 '11 at 7:42
Glad to know that! – Chethan S. Apr 18 '11 at 7:45
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). – scw Apr 18 '11 at 7:56
good point scw. – Jakub Apr 18 '11 at 11:18
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. – Abhi Beckert Jul 7 '13 at 12:40

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 ½ 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?
15               0.1 nanometer      An atom. An atom! What are you mapping?
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This is perfect, we're mapping atoms in a room. (we're not) – William Isted Jan 21 at 14:59

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
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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 Dec 20 '14 at 16:11
@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. – Abhi Beckert Dec 22 '14 at 3:35

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)

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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. – whuber Jun 11 '15 at 21:32

## protected by Community♦Jan 11 '14 at 20:37

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