Error is the deviation of the measured value from the exact value.
Physics is the subject that deals with measurements of physical quantities such as mass, length, time, temperature, and electricity to mention but a few. The instruments which are used in measuring that quantity are varying depending on the nature and magnitude of the quantity being measured.
For instance, the instrument requires to measure the height of man may not necessarily use the same instrument for measuring the waist or diameter of his hair strand. The scales on instruments are varying degrees of accuracy.
This type of length measured by the meter rule is not the same as that measured by the micrometer screw gauge. The scale on-meter rule is less accurate than the scale on a micrometer screw gauge.
When measuring time, the scale on stop clock is not accurate as that on the digital stopwatch
Accuracy of number
There are different categories of numbers, Our interest will be on two types of numbers, the counting numbers and decimal numbers. These numbers are used more often.
The numeral like 1, 2, 3 etc are used denote number of complete objects which naturally do not exist in parts or fractions. The numbers such as 1, 2, 2.5, 3.0 etc are known as decimal numbers because they consists some fractions.
All scientific measuring instruments bear scales that cater for
decimal number and for this matter all values read from measuring
instrument must be recorded in decimal form and the answer after
calculations using decimal numbers should be written in standard form A
×
Where A lies between 1−9 (1
and n can be positive or negative integers.
Below we have a 5 numbers written in various accuracies:
5 (accurate to nearest unit)
5.0 (accurate to nearest tenth)
5.00 (accurate to nearest hundredth)
5.000 (accurate to nearest thousandth)
Etc.
The more decimal places the more the accurate is the number.
ERROR ANALYSIS
Errors are involved in all measurements. There is a need to know the effects of errors in final results. When one has obtained a result, it is important to have some indication of its accuracy.
For instance error of length measured by a meter rule may be given as x± 0.5 mm where x is the measured value and 0.5mm, is half of the smallest unit in measuring x which is considered as an error in measurement.
There are two main types of errors;
- Systematic errors
- Random errors
SYSTEMATIC ERRORS
These are errors due to experimental apparatus e.g an incorrect zero adjustment of the measuring devices (beam balance, voltmeter, galvanometer) Starting or stopping the clock, scale calibration, use of incorrect value of constant in calculation.These types of errors have a tendency of affecting reading in the same direction.
MINIMIZATION OF SYSTEMATIC ERROR
These cannot be corrected by arranging a large number of reading. They must be recognized in advance by means of careful survey.
These errors can be determined by suitable treatment of observation or careful setting of apparatus over the zero reading check up before using the instruments.
RANDOM ERRORS
These errors gives a spread in answer of repetition equally, likely to be in either direction. These have equal chance to be –ve or +ve. They are caused due to;
i. Fluctuation in the surrounding e.g temperature and pressure..
ii. Lack of the perfection of the observer due to parallax.
iii. Insensitivity of the instrument/apparatus.
MINIMIZATION OF RANDOM ERROR
These errors can be minimized by repeating observation of a particular quantity.
Experiments should be carefully designed, use of highly sensitive instruments and finding mean of the measured value,
An experiment is accurate when the systematic error is relatively small.
MISTAKE
A mistake is done when the measurement is carried out in wrong way and as a result an error is introduced in value recorded. For example using instrument without checking for zero error or arrange instruments without following instructions properly can cause unnecessary errors which can otherwise be avoided.
PRECISION OF ERROR
This indicate the closeness with which measurements agree with one another quite independent to any systematic error.
BLUNDER
This is a mistake done several times.
How to avoid or minimizing errors
It is possible to avoid or if not to minimize by the first reading the instructions of a given experiment and undertaking them.
Apparatus should be checked thoroughly before put them into uses. In most cases diagrams are given to show how apparatus must be arranged and connected, so follow the instructions.
The apparatus must be securely connected to avoid accidents in the middle of the experiment. The following are some of precautions to be taken in minimizing the error in each case:
(i) Instrumental errors
In this case of errors that arise from measuring devices themselves, the precautions should be:
Regular maintenance and repair of apparatus
Proper storage of apparatus in special ductless rooms
Careful handling of apparatus when transferring them between places
Regular dusting, cleaning and oiling
(ii) Observation errors
The precautions to be taken to minimizing these kinds of errors during experiment are as follow:
Avoid parallax by reading the value from the scale perpendicular from position
Study the scale of instrument before operating it
Fix the apparatus according to the instructions and securely as possible to avoid unnecessary movement during experiment
Write the values read from the scale within the accuracy of instrument
Record the values actually made but not imaginary or cooked ones
(iii) Adjustment errors
It is good practice to check apparatus including measuring devices before put them into use. Together with checking the observer must:
Adjust the measuring device to remove zero error and where it is not possible record it somewhere for further reference
Fix each item of the apparatus in the right position and more importantly for instruments like meter rules, thermometers etc must kept vertical or upright
(iv) Random errors
In situation where every trial gives a different value, it is advisable to take as many as measurement as possible and find the average. A good example is determination of the diameter of wire.
It good to take the measurement in different positions along the wire and the average value calculated. This is because the wire may not be uniform, so taking only value may not give the best results.
Accuracy of measuring instruments
(a) The meter Rule
The accuracy of the measuring instrument depends on the smallest unit it can possibly measure. We shall use the instruments for measuring lengths to explain this concept of accuracy.
Take an example of a meter rule in fig 1.1 the total length it can measure is 1m which is subdivided into 1000 partitions each 1mm long. Thus the smallest length a meter rule can possibly measure is a millimeter. This means that the accuracy of a meter rule is 1 part out of 1000 parts that makethe whole.
(b) The Vernier calipers
The Vernier caliper is the measuring device for determine inner and outer diameter of hollow objects like test tubes, pipes etc. It has got two scale the main scale and Vernier scale.
The main scale reads up to one decimal place where as Vernier scale reads up to the second decimal point of centimeter. Fig 2.2 represents the schematic diagram of Vernier calipers.
How to operate it;
Before taking the measurements, the gap between fixed outer jaw and movable jaw and movable jaw is closed by pushing a roller. When the gap is closed, zero mark (first line) on the vernier scale must coincide with zero mark (first line) on the main scale not seen in the diagram.
To measure inner diameter of the hollow objects, the inner jaws are pushed inside the object and the roller used to move the jaws apart until they touch the inner wall of the object. The screw is tightened to avoid accidental change in distance. The scales are read and values are recorded as;
= value on the main scale + Value on vernier scale
For the outer diameter, the outer jaws are opened and the object placed in between. After tightened the screw the scales are read once again and value recorded as;
=Value on main scale + Value on vernier scale
The smallest unit the Vernier calipers can possibly measure is 0.01 cm or 0.1 mm. Thus the error that can arise as a result of using the device is therefore
OR
D =3.8 + 0.05
=3.85 cm
To read this value, the units cleared by zero mark of the Vernier scale are first counted i.e. 3.8 and the value between 0.8 and 0.9 is obtain by looking for the line on the Vernier scale that coincide with the line on the main scale and that is line number 5 on the Vernier scale. Because this value is supposed to be the second decimal of a centimeter, it is recorded as 0.05 cm
(c) The micrometer screw gauge
Another very important measuring instrument is a micrometer screw gauge. It measures the length of the magnitude of 1 mm and less. There two scales on the device the sleeve scale and thimble scale as shown in fig 1.4
A micrometer screw gauge is used in measuring diameters of the wires thickness of the metal sheets, diameter of ball bearings and other tiny lengths. Before using it the gap between anvil and spindle has to be closed to check for zero error.
An object to be measured is placed between anvil and spindle. By means of ratchet to make sure the object is gently held. Stop screwing when the ratchet makes the crackle sound. The value of the diameter obtained is the sum of the readings from the sleeve scale and thimble scale i.e.
Figure 1.5 shows the procedure for reading and recording the value from two scales. The carries a millimeter scale along horizontal line on the upper side of line each interval represents 0.5 mm.
The scale on the thimble has 50 units all round such that when the thimble turn once, it either advances or retreats by 0.5 mm along the sleeve. This means that;
50 divisions = 0.5 mm
I.e 50 × 1 division = 0.5 mm
1 division = 0.5/50 = 0.01mm
Thus the smallest unit that a micrometer screw gauge can possibly measure is 0.01 mm
From the fig 1.5 the reading can be recorded as follow:
d =
× 0.5
mm
= (6.00 + 0.09) mm
= 6.09 mm
This means that the diameter of wire
d = reading on sleeve scale + reading on thimble scale
The reading on the sleeve scale is found by counting the interval cleared by edge of the thimble and these are 6, meaning 6.00 mm. The value on the thimble is obtained by looking for the line on the thimble that coincide with the horizontal line on the sleeve which happen to be line number 9. These are 9 unit outs of 50 units round the thimble that make 0.5 mm
i.e.
of 0.5 mm.
Absolute error
The absolute error is the magnitude of an error regardless of the sign. If d is the quantity then
is an error positive or negative such that d = d0
where d0 is an actual value. The absolute error in d therefore is written as |
Relative error
The absolute error alone shows the size of error but does not tell how serious the error is in relation to the actual value. By taking the ratio of error to the actual value we can see how many times an error is as big as the actual value. This is known as relative error is expressed in decimal number.
Relative error =
Percentage error
The relative error expressed in decimal form multiplied by 100 gives percentage error. It is more convenient to express in percentage rather than as fraction or decimal number. The percentage error is written as
× 100
Calculation of errors
In experiment the measurement is done to more than one parameter and the values it obtained are substituted in a mathematical expression the gives the relationship between the parameter involved.
The parameters concerned may require the same measuring device or different devices such as a meter rule and a micrometer screw gauge or a thermometer and a stopwatch.
As we have seen above these instruments have differed accuracies and therefore contributing different errors in the values obtained. Using the values in a formula, the result is likely to contain a compounded error.
Therefore, it is important to find the ways of calculating the error in the result. We are going to look; the operations such as addition, subtraction, multiplication and division of errors. Fin; we shall deal with compounded error in expressions involving indices.
(a) Addition of errors
Consider two quantities x and y given as x = x0
and y = y
respectively.
Their sum s is s = x + y. We would like to find an error
.The procedures are as follow:
From the results, the absolute error in the sum of two quantities is S the individual error in those quantities. Therefore the sum
S = S0
Where S0 = (x0 + y) the sum of the actual values of x and y.
We can easily find the relative error and the percentage error in S as
and ⃒
× 100
(b) Subtraction of error
To subtract is to find the difference. Taking the same quantity x and y, let the difference between them be d = x-y; the error in d is found as follows
In far as addition and subtraction of quantities are concerned, error are always added not subtraction.
(c) Multiplication of errors
For quantities x and y consisting of errors
and
respectively multiplied together, the product p contains an error
found as follows:
The product of x and y is
The absolute error in p therefore is
=
(d) Division errors
If y is divided by x, a quotient q is formed, that is q =
The results is supposed to be represented as q =
The error in q is found by the following procedures
The range of x and y are
an
The maximum possible quotient,
The minimum possible quotient,
The absolute error in q is therefore,
⃒ =
+
Example 1: Two quantities x and y have values (10
and (5
0.01) respectively. F int the absolute error, relative error and percentage error in their;
(a) Sum S
(b) Difference d
©Product p
(d) Quotient q = y/x
Solution
(a) Given x =
For the sum, ⃒
= ⃒
and
=0.02
The relative error in s is ⃒
⃒ where
= (
+
),
= 10 and
= 5
⃒ = 0.02/15 =0.0013
The percentage error is
⃒
⃒ ×100 = 0.0013 ×100
= 0.13%
(b)For the difference ⃒
=
= 0.01 + 0.01
= 0.02
Relative error in d is ⃒
⃒ where
= (
⃒
0.02/5 =0.004
Percentage error is
⃒
⃒ × 100 = 0.4%
(c) For the product p =
The absolute error ⃒
=
= 10(0.01) +5(0.01)
= 0.10 +0.05
= 0.15
The relative error in p is
⃒
= 0.003
The percentage error in p
⃒ × 100 = 0.003 × 100
= 0.3%
(d) For the quotient
q =
where
/
The absolute error in q is ⃒
= 10(0.01) +5(0.01)/
=0.015/100 = 0.0015
The relative error in q is
⃒
= 0.0015/0.5 = 0.003
The percentage errors in q is therefore
⃒
⃒ × 100 =0.003 × 100
=0.3%
Use of natural logarithms in error analysis
The calculations of errors we have seen above are used in simple cases with only two parameters to deal with.
There are situations where the mathematical expressions into more than two variables raised to some powers in which case using the method above. may not be adequate to produce the answer required.
Application of logarithms may / quicken the process towards the answer. As an example, consider the following problem:
Example 1.2: A quantity Q is connected to another quantity p, r, and η by the expression
Q = ðœ«p
/8lη . Obtain an expression for
(i) Relative error in Q
(ii) Absolute error in Q
Solution
Given that Q = ðœ«p
/8lη
Take natural logarithms on both sided of the equation
Differentiating this we ge
(i) The relative error in Q is
=
(ii) The relative absolute error in Q i
The absolute error in Q is
Exercise
1. State the accuracy of each of numbers below
i/7 ii/3.2 iii/5.00 iv/11.03
2. (a) give the difference between error and mistake.
(b)Mention the types of errors and their sources.
(c) Explain how you would minimizing the magnitude of random error in an experiment.
3. (a) State the smallest unit each of instruments below can possibly measure and mention possible errors
(i) The meter rule
(ii) The vernier caliper
(iii) The micrometer screw gauge
(b) A box P contain smaller packages
and
weighing (12
0.2) kg, (20
0.35) kg and (16
0.25) kg respectively.
(i) State the range off mass of each package
(ii) Calculate the absolute error in the mass P
(iii) What is relative error in mass P
(v) Determine the percentage error
4. (a) Imagine you are doing an experiment on a simple pendulum to determine the value of acceleration due to gravity g at your location. What are possible errors are like to affect your result and what precaution will you take?
(b) Find the maximum possible error in the measurement of force on
the object of mass M moving with velocity V ALONG THE CIRCULAR PATH OF
RADIUS r given that M, V, and r are (3.5
0.1) kg, (20
1) ms-1, and (12.5
m respectively.
5. During the experiment to determine acceleration due to gravity by
using simple pendulum, the length l of the pendulum and periodic time T
were measured an recorded as (120
0.1) cm and (2.25
0.01) s respectively. Given the relationship between l, g and T is; T = 2
Calculate;
(i) Absolute error in g
(ii) The percentage error in g
6. (a) What is the meant by relative error?
(b) A quantity Q is expressed in terms of other quantities F, A, v and
by the equation
Q =
Where F= 5
0.21, A= 0.05
0.005,
and
Calculate the percentage error in Q
7. (a) Distinguish between random error and systematic error
(b) Four trials in experiment of the diameter of the wire gave the diameter values 0.2 mm, 0.25 mm, 0.23 mm and 0.22 mm.
(i) What is the diameter of wire if each reading contains an error of 
0.001?
(ii) Calculate the absolute relative error in the cross-section of the wire.
(c) Fig 2.6 shows parts of vernier calipers adjusted during the experiment
(i) Read and record the value registered by the instrument
(ii) State the range and the accuracy of the value you have recorded
8. A rectangular sheet of metal measures 47.21 cm × 23.79 cm. If the error in each dimension is
0.01 cm
(a) State the range of each dimension
(b) Calculate the percentage error in its
(i) Perimeter (ii) Diagonal (iii) Area
9. (a) In an experiment to determine velocity of liquid, the data bellow were collected:
Pressure at inlet = (4000.2) pa
Pressure at outlet = (2000.2) pa
Length of capillary tube L = (100.001) m
Diameter of capillary tube d = (40.1) cm
Discharge Q = (5000.2) liter/sec
Assume Poiseulli’s formula apply find relative error in viscosity.
(b) (i) What is the meaning by the physical quantity?
(ii) The maximum velocity of the particle moving with simple harmonic motion can be determined from……
Where A, M, and K are amplitude, mass and constant, find dimension of K
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