Combination of Errors in a series of Measurements
Suppose the values obtained in several measurement are a1, a2, a3, …, an.
Arithmetic mean, amean = (a1+ a2 + a3+ … + an)/n
o Absolute Error: The magnitude of the difference between the true value of the quantity and the individual measurement value is called absolute error of the measurement. It is denoted by |Δa| (or Mod of Delta a). The mod value is always positive even if Δa is negative. The individual errors are:
Δa1 = amean – a1, Δa2 = amean – a2, ……. ,Δan = amean – an
o Mean absolute error is the arithmetic mean of all absolute errors. It is represented by Δamean.
Δamean = (|Δa1| + |Δa2| +|Δa3| + …. +|Δan|) / n
For single measurement, the value of ‘a’ is always in the range amean± Δamean
So, a = amean ± Δamean Or amean – Δamean< a <amean + Δamean
o Relative Error: It is the ratio of mean absolute error to the mean value of the quantity measured.
Relative Error = Δamean / amean
o Percentage Error: It is the relative error expressed in percentage. It is denoted by δa.
δa = (Δamean / amean) x 100%
Combination of Errors
If a quantity depends on two or more other quantities, the combination of errors in the two quantities helps to determine and predict the errors in the resultant quantity. There are several procedures for this.
Suppose two quantities A and B have values as A ± ΔA and B ± ΔB. Z is the result and ΔZ is the error due to combination of A and B.
|Criteria||Sum or Difference||Product||Raised to Power|
|Resultant value Z||Z = A ± B||Z = AB||Z = Ak|
|Result with error||Z ± ΔZ = (A ± ΔA) + (B ± ΔB)||Z ± ΔZ = (A ± ΔA) (B ± ΔB)||Z ± ΔZ = (A ± ΔA)k|
|Resultant error range||± ΔZ = ± ΔA ± ΔB||ΔZ/Z = ΔA/A ± ΔB/B|
|Maximum error||ΔZ = ΔA + ΔB||ΔZ/Z = ΔA/A + ΔB/B||ΔZ/Z = k(ΔA/A)|
|Error||Sum of absolute errors||Sum of relative errors||k times relative error|
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Types of Errors
Errors which can either be positive or negative are called Systematic errors. For a particular measurement device, it will be either positive or negative. The causes of such errors are known and hence they can be avoided or accounted for. They are of following types:
1. Instrumental errors:
These arise from imperfect design or calibration error in the instrument. Worn off scale, zero error in a weighing scale are some examples of instrument errors.
2. Imperfections in experimental techniques:
If the technique is not accurate (for example measuring temperature of human body by placing thermometer under armpit resulting in lower temperature than actual) and due to the external conditions like temperature, wind, humidity, these kinds of errors occur.
3. Personal errors:
Errors occurring due to human carelessness, lack of proper setting, taking down incorrect reading are called personal errors.
These errors can be removed by:
o Taking proper instrument and calibrating it properly.
o Experimenting under proper atmospheric conditions and techniques.
Removing human bias as far as possible
Errors which occur at random with respect to sign and size are called Random errors. The errors whose actual cause is not known and cannot be accounted for. However, the random errors come on either side, i.e. the repeated measurements result in positive as well as negative errors. Therefore the random error in a measurement can be minimized by repeating the experiment a number of times and taking the mean. Since some of the measurements will have positive errors and the rest negative, taking the mean of all results of measurements will take us closer to the true value. Greater the number of repetitions, greater will be the closeness to the accurate value.
o These occur due to unpredictable fluctuations in experimental conditions like temperature, voltage supply, mechanical vibrations, personal errors etc.
o Smallest value that can be measured by the measuring instrument is called its least count. Least count error is the error associated with the resolution or the least count of the instrument.
o Least count errors can be minimized by using instruments of higher precision/resolution and improving experimental techniques (taking several readings of a measurement and then taking a mean).
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Read Next:: Propagation of Errors
Some Important Conversions
- 1 bar = 106 dyne/cm2 = 105 Nm-2 = 105 pascal
- 76 cm of Hg = 1.013×106 dyne/cm2 = 1.013×105 pascal = 1.013 bar.
- 1 toricelli or torr = 1 mm of Hg = 1.333×103 dyne/cm2 = 1.333 millibar.
- 1 kmph = 5/18 ms-1
- 1 dyne = 10-5 N,
- 1 H.P = 746 watt
- 1 kilowatt hour = 36×105 J
- 1 kgwt = g newton
- 1 calorie = 4.2 joule
- 1 electron volt = 1.602×10-19 joule
- 1 erg = 10-7 joule
Some Important Physical Constants
- Velocity of light in vacuum (c) = 3 × 108 ms-1
- Velocity of sound in air at STP = 331 ms-1
- Acceleration due to gravity (g) = 9.81 ms-2
- Avogadro number (N) = 6.023 × 1023/mol
- Density of water at 4oC = 1000 kgm-3 or 1 g/cc.
- Absolute zero = -273.15oC or 0 K
- Atomic mass unit = 1.66 × 10-27 kg
- Quantum of charge (e) = 1.602 × 10-19 C
- Stefan’s constant = 5.67 × 10–8 W/m2/K4
- Boltzmann’s constant (K) = 1.381 × 10-23 JK-1
- One atmosphere = 76 cm Hg = 1.013 × 105 Pa
- Mechanical equivalent of heat (J) = 4.186 J/cal
- Planck’s constant (h) = 6.626 × 10-34 Js
- Universal gas constant (R) = 8.314 J/mol–K
- Permeability of free space () = 4π × 10-7 Hm-1
- Permittivity of free space () = 8.854 × 10-12 Fm-1
- The density of air at S.T.P. = 1.293 kg m-3
- Universal gravitational constant = 6.67 × 10-11 Nm2kg-2