# Errors in Measurement

Everything experimental research is built on measurement. Many of the great scientific advancements would not have been possible without ever-increasing standards of measuring precision. Quantities are measured using international measurements and are completely accurate as compared to others. Measuring is done in the same way as vegetable vendors do: by comparing an unknown amount of weight to a known amount of weight. Any calculation contains a level of uncertainty, which is referred to as an error. This error may occur during the procedure or even as a result of a failure in the experiment. As a result, no approach can have a 100 % precise calculation.

The aim of every experiment is to determine a physical quantity as accurately as possible. Every measurement, however, consists of some error that may occur due to the observer, or the instrument used or both. Errors may also creep in due to small changes in the condition of the experiment or due to various factors inherent in the experiment. The measured value of a quantity differs somewhat from its true value due to the presence of such errors.

### Error

Both experimental research and technology are built on measurement. Any measurement made with any measuring instrument gives a certain degree of uncertainty. This uncertainty is referred to as an **error**. The difference between the real value and the estimated value of a quantity is known as measurement error. An error may be positive or may be negative.

The deviation of the measured quantity from the actual quantity or true value is called

error.

E = A_{m }– A_{t}where E is the error, A

_{m }is the measured quantity and A_{t }is the true value.

**Different Types of Errors**

The errors are mainly of three types,

**1. Systematic or Constant Errors:**

The type of error which affects the results of the experiment always in the same direction i.e., makes the obtained result always higher or always lower than the true value is known as systematic error. In fact, all instrumental errors are systematic. If the graduations of a meter scale are faulty or if the measurements are carried out with a scale at a temperature other than that at which it was calibrated, a systematic error will be introduced.

So, the systematic errors are of the following types:

(i)Instrumental errorswhose examples are zero error of screw gauge, vernier caliper, end error in meter bridge, etc.

(ii) Personal errorswhich are due to the observer.

(iii) Error due to external causes,due to changes in temperature, pressure, velocity, height, etc.

(iv) Error due to Imperfection.

Systematic errors are usually determinate. So they can be eliminated by taking proper precautions or can be rectified. However, when the source of such errors can not be properly identified, the experiment is repeated by different methods.

**2. Random or accidental errors:** The results of several measurements of the same quantity by the same observer under identical conditions do not show in general exact agreement but differ from one another by a small amount. The instrument may be a very good and sensitive one, the observer may be very careful, yet such small differences in the results generally occur. No definite cause for such errors can be traced; their sources are unknown and uncontrollable. Such errors are, therefore, purely accidental in nature and are termed random or accidental errors. An error that occurs randomly and whose causes are unknown and indeterminate is called **random error.**

**3. Gross errors:** These are large errors and occur due to carelessness or undue haste of the observer which are also termed as mistakes. The wrong recording of some data may be cited as an example. So mistakes obviously do not follow the law and can be avoided only by constant vigilance and careful observation from the observer.

**Errors of Observation by Instruments and Degree of Accuracy**

- In all measurements even after minimizing systematic and random error, errors of observations inherent in the manufacture of the instrument used remain present. The scale of a measuring instrument is divided by the manufacturer only to its limit of reliability and no further. We already know that the smallest output that we can detect clearly from the instrument is called its
**least count**. - This gives the worst possible error which might occur in measurements with that instrument. So in all measurements, the
**degree of accuracy**attainable is limited by the least counts of the different instruments used. For example, a meter scale is usually graduated in millimeters. Hence, the greatest error which might be committed to measuring length with such a scale is 1 mm. - The result of measurement of the length of a rod should therefore be expressed as the length of the rod 22.4 Â± 0.2 cm. This is the scientific method of recording a reading with the limits of error. This means that the length of the rod lies between 22.6 cm and 22.2 cm. The errors are known as
**errors of observation**or**permissible errors**. - Therefore, in general, if the measured value of a
**quantity**is x and the**limits of error**are âˆ†x then the reading should be written as x Â± âˆ†x which means that the value of the quantity lies between x+âˆ†x and x-âˆ†r.

**Proportional Error and Percentage Error**

The ratio of the error of observation to the observed reading is known as the proportional error. If the proportional error is multiplied by 100 or expressed in percentages, then it is called a **percentage error**. Proportional error is also called **relative error** or f**ractional Error**.

The formula to calculate the Proportional error is given by,

Proportional Error = (Error / Observed reading)or

Percentage Error = (Error / Observed reading) Ã— 100%

**Combination or Propagation of Errors**

In general, an experiment in physics involves a number of measurements made with different instruments. The final result is then calculated by performing different mathematical operations. The error in the final result depends on the errors in the individual measurements and on the nature of the required mathematical operations. We should, therefore, know the rules that how errors are combined in different mathematical operations.

**1. Addition and Subtraction:** In these operations, errors are combined according to the following rule: When two quantities are added or subtracted, the net error in the experiment result is the sum of errors associated with those quantities.

So if the observed values of two quantities are x Â± Î”x and y Â± Î”y and the sum or the difference of them is z Â± Î”z then the error Î”z in the value of z is given by Î”z = Î”.x + Î”y.

e.g.,Let the lengths of two rods measured by a meter scale be 22:4 Â± 0.2 cm and 20.2Â±0.2 cm respectively.Then the difference in their lengths (22.4 – 20.2) or, 2.2 cm.

But each reading is erroneous by 0.2 cm. hence the greatest possible error that may occur in the difference is 0.4 cm.

So we write, the difference in their lengths is 2.2 Â± 0.4 cm.

**2. Multiplication and Division:** In these operations, the concerned rule is given by:

When two quantities are multiplied or divided, the proportional error within the outcome is that the sum of the proportional errors in those quantities.

So if z = xy or, z = (x/y), then according to this rule,

**(Î”z/z) = (Î”x/x) + (Î”y/y)**

**3. Powers of Quantities:** When a quantity is raised to a power n, the proportional error in the final result is n times the proportional error in that quantity.

If, **z = x ^{n} **

Then according to this rule

**(Î”z/z) = n(Î”x/x)**

And if,** z = (x ^{n}y^{p}/w^{q})**

Then it can be proved that:

** (Î”z/z) = n (Î”x/x)+p(Î”y/y)+q(Î”w/w)**

**Proportional error in z = m Ã— (proportional error in x) + p Ã— (proportional error in y) + q Ã— (proportional error in w).**

**Sample Problems**

**Problem 1:** **If all measurements in an experiment are performed up to the same number of times, then a maximum error occurs due to which measurement?**

**Solution:**

The maximum error occurs due to the measurement of the quantity which appears with maximum power in the formula. If all the quantities in the formula have the same powers, then a maximum error occurs due to the measurement of the quantity whose magnitude is least.

**Problem 2: If the length of the pencil is given by (4.16 Â± 0.01) cm. What does it mean?**

**Solution:**

It means that the true value of the length of the pencil is unlikely to be less than 4.15 cm or greater than 4.17 cm.

**Problem 3: Two resistances R _{1}=(100Â±5) ohm and R_{2}=(200Â±10) ohm are connected in series. Find the equivalent resistance of the series combination.**

**Solution:**

Since, it is known that,

Equivalent resistance=R= R

_{1}+R_{2}Given that, the resistance is:

R

_{1 }= (100 Â± 5)R

_{2 }= (200 Â±10)Therefore,

R = (100 Â± 5) + (200 Â± 10)

=

(300 Â± 15) ohm

**Problem 4: A capacitor of capacitance C = (2.0 Â± 0.1) ÂµF is charged to a voltage V = (20 Â± 0.2) V. What will be the charge Q on the capacitor?**

**Solution: **

Q = CV

= 2.0Ã—10

^{-6 }Ã— 20 C= 4.0Ã—10

^{-5}Coulomb.Proportional error in C = (Î”C/C)

= (0.1/2)

Percentage error in C = (0.1/2) Ã—100

=5 %

Proportional error in V = (Î”V/V)

= (0.2/20)

Percentage error in V = (0.2/20)Ã—100

=1%

Charge on capacitor,

(Î”Q/Q) = (Î”C/C) + (Î”V/V)

Percentage error in Q = 5%+1%

= 6%

Charge = 4.0Ã—10

^{-5 }Â± 6% Coulomb=

(4.0Â±0.24)Ã—10^{-5}Coulomb

**Problem 5: The centripetal force acting on a body of mass 50 kg moving in a circle of radius 4 m with a uniform speed of 10 m/s is calculated using the equation F = mv ^{2}/r. If the accuracies of measurement of m, v, and r are 0.5 kg, 0.02 m/s, and 0.01 m respectively, determine the percentage error in the force.**

**Solution: **

It is known that,

(Î”F/F) = (Î”m/m) + 2(Î”v/v) + (Î”r/r)

(Î”m/m) = (0.5/50)

= 0.01

(Î”v/v) = (0.02/10)

= 0.002

(Î”r/r) = (0.01/4)

= 0.0025

So, (Î”F/F) = 0.01 + 2(0.002) + (0.0025)

= 0.0165

Thus, Percentage error in force = (0.0165) Ã— 100%

=

1.65 %

**Problem 6: The resistance R = V/I where V = (200 Â± 5) V and I = (20 Â± 0.2) A. Find the percentage error in R.**

**Solution: **

Proportional error in V = (Î”V/V)

= (5/200)

Percentage error in V = (5/200)Ã—100%

= 2.5%

Proportional error in I = (Î”I/I)

= (0.2/20)

Percentage error in I = (0.2/20) Ã—100%

= 1%

So, Percentage error in R = 2.5%+1%

=

3.5%

**Problem 7: The mass and the length of one side of a cube are measured and its density is calculated. If the percentage errors in the measurement of mass and length are 1% and 2% respectively, then what is the percentage error in the density?**

**Solution: **

If the mass of the cube is m and the length of its one side is l, then its density,

d = m/lÂ³

So, (Î”d/d) = (Î”m/m) + 3(Î”l/l)

Thus, Percentage error in density = (1+3Ã—2)%

=

7%