Measuring accuracy of latitude and longitude

Question

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 that are 6 decimal places long.

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

Answer 1

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 11 cm: 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 1.1 cm: 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!

Answer 2

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

Answer 3

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?

Answer 4

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 point

If 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 single table, because this relation change (for E/W distance measurements) with the change of latitude. Here is another table to show the changes:

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)

Answer 5

I think this XKCD is a perfect answer to this question :)

https://xkcd.com/2170/

Answer 6

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

Answer 7

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.