Measurement Errors

ERROR ANALYSIS
Measurement Errors are not mistakes, but more like a limit of accuracy. There are several common sources for errors - the most obvious is the resolution of the measuring instrument (e.g. to the nearest mm).
Errors in measurements have a carry-through effect in calculations, and this can be analysed to determine the overall error in the final solution.
Symbols: We will use the following symbols:
S = absolute error of a measurement.
x = the measurement itself (the measurand)
S/x = relative error

• The error (S) is never known exactly. If we knew exactly what the error was we could subtract it and get a perfect measurement. Errors are statistical, the measurement is most probably within a certain range. The symbol S is used because it stands for Standard Deviation. (See Statistics) This error can also be called the uncertainty of a measurement.
• It is important to maintain the same method of describing the error throughout the calculations. We usually use +- tolerancing to describe the error.

Sources of Error
Errors can come from various sources. Resolution error is easy to estimate, but the others are usually quite approximate and may have to be estimated by the person taking the measurement.

Random Errors

• Limitation of precision: Resolution of the instrument. Error = Smallest Resolution/2
• Misalignment. Parallax error of needle/scale and eye, misaligned instrument (Eg.The dial gauge is not vertical, the tape measure is at an angle, the caliper is not perpendicular etc).
• Round off or inaccuracy in formula or constant (e.g. Gravity = 9.81)

Systematic Errors: (Inherent in the measurement).

• Errors in the calibration of the measuring instruments. Examples: Stretch of a tape, inaccurate graduations, worn or incorrectly adjusted instruments. The only way to check this is by calibration against a known standard or correct method.
• Incorrect measuring technique: Examples: Incorrect method: pushing too hard on a caliper, parallax error due to viewing at an angle.
• Bias of the experimenter. Examples: The experimenter might consistently read an instrument incorrectly, or might let knowledge of the expected value of a result influence the measurements. 

Systematic Errors do not improve by taking many readings, because the average is not zero. Regular calibration is all about minimising systematic error.

These errors should be added together to give the absolute measurement error;

Absolute error = (Resolution / 2) + (misalignment error) + (systematic error) + (inherent error)
Absolute and Relative Error
Absolute Error is the tolerance of the measurement, or the approximate error of a single measurement. The best estimate is the  standard deviation of the measurement, which can only be determined with many measurements taken. Failing this, an estimation can be made using the error sources above.

Relative Error is the ratio of the size of the absolute error to the size of the measurement being made.
Relative Error = Absolute Error / Value.
Significant figures: Every measurement (or number) is given to a certain number of significant figures (e.g. Gravitational acceleration = 9.8 m/s2). Generally, we take the last digit quoted in a measured value is the one that has some uncertainty. e.g  34.532 kg. This assumes we can measure to the nearest gram. We would write it as 34.532 +-0.001 kg.

When working with constants, it is important to minimise error by using adequate number of significant figures. For example, Pi is known with very high accuracy, so we wouldn't round it off to 3.14. Use the calculator value of 3.1415926535...!

Remember: The % error in the final solution of a complex calculation is always worse than the % errors of all the measurements. So it only takes 1 bad measurement error to damage the accuracy of the solution.

Example:
(a) A length is measured as 120mm using a ruler graduated in mm. Absolute error = 1mm / 2 = 0.5mm. This would be written as +/- 0.5mm
Relative error = 0.5 / 120 = 0.004167  (or 0.4167%)
(b) The same ruler measures 12mm.
Absolute error = 1mm / 2 = 0.5mm. This would be written as +/- 0.5mm
Relative error = 0.5 / 12 = 0.04167  (or 4.167%)
Working with Errors (Error propagation)
Error will usually accumulate with each calculation, depending on the equations. Sometimes the error can reduce, like when we measure the thickness of a piece of paper by measuring a stack of 100 pages.

Dummy Method: When adding/subtracting we add absolute error. When multiplying/dividing we add relative error. This is easy to understand but overly conservative because errors are actually statistical. This method of error propagation overestimates the combined error because of the possibility that errors can cancel when more than one measurement is made. 

Proper Method: If we look at errors statistically, a better approximation of error is given by;

Example:
A rectangle measured as 120mm x 75mm using a ruler graduated in mm.

The steel rule has an error of 1mm / 2 = 0.5mm. This would be written as +/- 0.5mm
(a) Find the error in calculating the perimeter 
Perimeter = 120 + 75 + 120 + 75 = 390 mm
Absolute Error = sqrt (0.52 + 0.52+ 0.52+ 0.52) =  1 mm
Answer: 390 +-1mm. Which means the perimeter is probably between 389 and 391 mm.

(b) Find the error in calculating the area.
Area = 120+-0.5 * 75+-0.5
Rel Error = sqrt ( (0.5/120)2 + (0.5/75)2 ) =  0.00786  (or 0.786%)
Absolute error = (120 * 75) * Rel Error = 70.75 mm2
Answer: 9000 +-70.8 mm2. Which means the area is probably between 8929.2 and 9070.8 mm2.