No measurement is perfect. What matters in surveying is not whether errors occur — they always do — but whether they cancel out or pile up. That single distinction divides all chaining errors into two classes, and it decides how seriously each one has to be treated.
Where Chaining Errors Come From
Errors and mistakes in chaining may arise from any one or more of the following sources:
- Erroneous length of chain
- Bad ranging of intermediate points
- Poor straightening of tape
- Carelessness in holding and marking of points
- Variation of temperature
- Variation of pull
- Displacement of arrows
- Miscounting of chain lengths
- Misreading of tape
- Erroneous booking
Look at this list and you can see the whole chapter reflected in it. Several of these sources get their own correction formula later — chain length becomes the correction for standardization, temperature becomes the temperature correction, pull becomes the pull correction, bad ranging becomes the misalignment correction.
The Two Classes of Error
- Compensating errors — also called random errors
- Cumulative errors — also called systematic errors
1. Compensating Errors (Random Errors)
These are the errors which are liable to occur in both directions and tend to compensate.
Compensating errors are proportional to √L
(the square root of the length of the line)
Why They Partially Cancel
The defining feature is the phrase “in both directions”. A random error is as likely to make a reading too large as too small.
Consider marking the position of an arrow. Sometimes you place it a few millimetres beyond the true point, sometimes a few millimetres short. There is no reason for one to happen more often than the other. Over many tape lengths, the overs and unders partly cancel — hence compensating.
Why the Square Root Appears
This is the part worth understanding rather than memorising.
If the errors cancelled completely, a long line would be as accurate as a short one, which is clearly untrue. And if they added up fully, the error would be proportional to L. The truth sits between the two.
Because the individual errors have random signs, they do not accumulate in a straight line — they accumulate the way a random walk does. Statistically, the total of n random quantities grows as √n, not n. Since the number of tape lengths is proportional to the line length, the total random error grows as √L.
The practical consequence is encouraging: quadruple the line length and the random error only doubles. Random error becomes proportionally less important on longer lines.
2. Cumulative Errors (Systematic Errors)
These are the errors which are liable to occur in the same direction and tend to accumulate. Hence these errors considerably increase or decrease the actual measurements.
Cumulative errors are proportional to L
(the length of the line), and may be positive or negative
Why They Pile Up
The key phrase here is “in the same direction”. A systematic error has a cause, and that cause acts the same way every time.
Suppose the tape is 2 mm too short. Every single tape length you measure is 2 mm too short — never too long, not once. Lay it down a hundred times and you are 200 mm out. Nothing cancels, because nothing ever pushes the other way.
That is why the error grows in direct proportion to L. Every additional tape length adds another full helping of the same error.
Positive or Negative
Note that cumulative errors may be positive or negative — the direction depends on the cause. A tape that is too short makes measurements read long; a tape that is too long makes them read short. Either way, the error is consistent and accumulates.
The Two Compared
| Compensating (Random) | Cumulative (Systematic) | |
|---|---|---|
| Direction | Occurs in both directions | Occurs in the same direction |
| Behaviour | Tends to compensate | Tends to accumulate |
| Proportional to | √L | L |
| Sign | Both, unpredictably | May be positive or negative, but consistent |
| Effect on long lines | Grows slowly; becomes relatively less important | Grows in full proportion; becomes serious |
| Can it be corrected? | No — it has no known cause | Yes — the cause can be identified and a correction applied |
The Most Important Practical Difference
The last row is the one that matters most, and it explains why the whole of the next two topics exist.
A random error cannot be corrected. Since it has no consistent cause, there is nothing to calculate. You can only reduce it by working more carefully or by repeating measurements and averaging.
A systematic error can be corrected exactly, provided you know its cause. If the tape is 2 mm short, you know precisely how much to add. That is what every correction formula in the tape correction topic is doing — identifying a systematic cause and cancelling it out.
Which Sources Fall Into Which Class
| Source | Likely Class | Reason |
|---|---|---|
| Erroneous length of chain | Cumulative | Same error every tape length |
| Bad ranging | Cumulative | A bent path is always longer, never shorter |
| Variation of temperature | Cumulative | The tape is consistently hot or cold on a given day |
| Variation of pull | Cumulative | Consistently more or less than standard |
| Careless holding and marking | Compensating | Equally likely to be over or under |
| Displacement of arrows | Compensating | No preferred direction |
Notice the pattern. Anything with a physical cause acting consistently is cumulative. Anything arising from ordinary human imprecision is compensating.
A Note on Errors versus Mistakes
The source list refers to “errors and mistakes“. These are not the same thing. A mistake — miscounting chain lengths, misreading the tape, erroneous booking — is a blunder, not an error. It does not follow any statistical law and cannot be corrected by formula. It can only be caught by a check, which is exactly why the arrows are counted by the follower.
Quick Revision Notes
- Sources of chaining error include erroneous chain length, bad ranging, poor straightening, careless holding and marking, temperature variation, pull variation, displacement of arrows, miscounting, misreading and erroneous booking.
- Errors are classified as compensating (random) and cumulative (systematic).
- Compensating errors occur in both directions and tend to compensate.
- Compensating errors are proportional to √L.
- Cumulative errors occur in the same direction and tend to accumulate, considerably increasing or decreasing the measurement.
- Cumulative errors are proportional to L and may be positive or negative.
- Cumulative errors can be corrected once the cause is known; compensating errors cannot.
Mistakes Students Commonly Make
- Swapping the two proportionalities. Compensating goes with √L; cumulative goes with L. This is the single most examined point in the topic.
- Saying cumulative errors are always positive. They may be positive or negative — what matters is that they are consistent.
- Confusing the alternative names. Compensating = random; cumulative = systematic.
- Assuming compensating errors cancel completely. They only tend to compensate, which is why a √L residue remains.
- Trying to apply a correction formula to a random error. Only systematic errors can be corrected.
- Treating a mistake as an error. A blunder such as miscounting is caught by a check, not by a formula.
Conclusion
Two classes of error, distinguished by one question — does the error have a consistent direction? Random errors wander both ways and partly cancel, so they grow only as the square root of the line length and become relatively less troublesome on long lines. Systematic errors always push the same way, so they grow in full proportion to length and become serious. But that consistency is also their weakness: because a systematic error has an identifiable cause, it can be calculated and removed, which is precisely what every correction in the next topic sets out to do.
Frequently Asked Questions
How are errors in chaining classified?
As compensating errors, also called random errors, and cumulative errors, also called systematic errors.
What are compensating errors?
Errors which are liable to occur in both directions and which tend to compensate one another.
What are compensating errors proportional to?
The square root of the length of the line.
What are cumulative errors?
Errors which are liable to occur in the same direction and tend to accumulate, thereby considerably increasing or decreasing the actual measurement.
What are cumulative errors proportional to?
The length of the line. They may be positive or negative.
Why do compensating errors follow a square root law?
Because the individual errors have random signs, so they accumulate as a random walk rather than adding directly. The total of n random quantities grows as the square root of n.
Which type of error can be corrected?
Cumulative or systematic errors, because they have an identifiable cause that can be quantified. Compensating errors have no consistent cause and cannot be corrected by formula.
Is an erroneous chain length a compensating or cumulative error?
Cumulative, because the same error occurs in the same direction for every tape length laid down.
What are the sources of error in chaining?
Erroneous length of chain, bad ranging of intermediate points, poor straightening of tape, carelessness in holding and marking points, variation of temperature, variation of pull, displacement of arrows, miscounting of chain lengths, misreading of tape and erroneous booking.