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Topics Covered

• Accuracy
• Precision
• Error
• Systematic Errors
• Random Errors
• Least Count Error
• Absolute Error
• Relative Error
• Percentage Error
• Combination of Errors
• Error of a Sum or a Difference
• Error of a Product or a Quotient
• Error in Case of a Measured Quantity Raised to a Power

Accuracy and Precision of Instruments

The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.

Precision tells us to what resolution or limit the quantity is measured.

`"Example:"` Suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, the length is determined to be 3.38 cm.
The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only 0.1 cm), while the second measurement is less accurate but more precise.

Errors in Measurement

• The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.

• In general, the errors in measurement can be broadly classified as (a) systematic errors and (b) random errors.

Systematic Errors

The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are :

`"Instrumental Errors"`
Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument.

`"Imperfection in Experimental Technique or Procedure"`
To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature. Other external conditions (such as changes in temperature, humidity, wind velocity, etc.) during the experiment may systematically affect the measurement.

`"Personal Errors"`
Personal errors that arise due to an individual’s bias, lack of proper setting of the apparatus or individual’s carelessness in taking observations without observing proper precautions, etc.

Systematic errors can be minimised by improving experimental techniques, selecting better instruments and removing personal bias as far as possible.

Random Errors

• The random errors are those errors, which occur irregularly and hence are random with respect to sign and size.

• These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc.

`"Example:"` When the same person repeats the same observation, it is very likely that he may get different readings everytime.

Least Count Error

• The smallest value that can be measured by the measuring instrument is called its least count.

• All the readings or measured values are good only up to this value.

• The least count error is the error associated with the resolution of the instrument.

`"Example"` A vernier callipers has the least count as 0.01 cm; a spherometer may have a least count of 0.001 cm.

• Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors.

• Using instruments of higher precision, improving experimental techniques, etc., we can reduce the least count error.

• Repeating the observations several times and taking the arithmetic mean of all the observations, the mean value would be very close to the true value of the measured quantity.

Absolute Error

Suppose the values obtained in several measurements are `a_1, a_2, a_3...., a_n`. The arithmetic mean of these values is taken as the best possible value of the quantity under the given conditions of measurement as -
`a_(m e a n) = (a_1+a_2+a_3+...+a_n)/n`
or, `a_(m e a n) = sum_(i=1)^n ((a_i)/n)`

The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement. This is denoted by `| Δa |`.
`"Absolute error" = "True value" - "Measured value"`

The errors in the individual measurement values are
`Δa_1 = a_(m e a n) – a_1`
`Δa_2 = a_(m e a n) – a_2`
`.... .... ....`
`.... .... ....`
`Δa_n = a_(m e a n) – a_n`

`"Mean"` `A``"bsolute Error"`
The arithmetic mean of all the absolute error of the different measurement is called as mean absolute error. It is represented by `Δa_(m e a n)`.
`Δa_(m e a n) = (|Δa_1|+|Δa_2 |+|Δa_3|+...+ |Δa_n|)/n`
`Δa_(m e a n) = sum_(i=1)^n|Δa_i|//n`

While calculating mean absolute error the values of absolute error in different observations should be considered with positive sign only, whether it is positive or negative.

Relative Error and Percentage Error

`"Relative Error"`
The relative error is the ratio of the mean absolute error `Δa_(m e a n)` to the mean value `a_(m e a n)` of the quantity measured.
`"Relative Error" = (Deltaa_(m e a n))/(a_(m e a n))`

`"Percentage Error" = (Deltaa_(m e a n))/(a_(m e a n)) xx 100`

Combination of Errors

If we do an experiment involving several measurements, we must know how the errors in all the measurements combine. For example,
(a) Error of a sum or a difference
(b) Error of a product or a quotient
(c) Error in case of a measured quantity raised to a power

Error of a sum or a difference

Suppose two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors. The error ΔZ in the sum,
`Z = A + B`
`Z ± ΔZ = (A ± ΔA) + (B ± ΔB)`
`± ΔZ = ± ΔA ± ΔB`

The maximum possible error in Z
`ΔZ = ΔA + ΔB`

For the difference Z = A – B
`Z ± Δ Z = (A ± ΔA) – (B ± ΔB)`
`± ΔZ = ± ΔA ∓ ΔB`

The maximum possible error in Z
`ΔZ = ΔA + ΔB`

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Error of a product or a quotient

Suppose `Z = AB` and the measured values of A and B are `A ± ΔA` and `B ± ΔB`.

Then, `Z ± ΔZ = (A ± ΔA) (B ± ΔB)`
`Z ± ΔZ= AB ± B ΔA ± A ΔB ± ΔA ΔB`.

Dividing LHS by Z and RHS by AB we have,
`1±((ΔZ)/Z) = 1 ± ((ΔA)/A) ± ((ΔB)/B) ± ((ΔA)/A)((ΔB)/B)`

Since ΔA and ΔB are small, we shall ignore their product.

Hence the maximum relative error
`(ΔZ)/ Z = ((ΔA)/A) + ((ΔB)/B)`

Error in case of a measured quantity raised to a power

Suppose `Z = A^2`

Then, `(ΔZ)/Z = ((ΔA)/A) + ((ΔA)/A) = 2 ((ΔA)/A)`

Hence, the relative error in `A^2` is two times the error in A.

In general, if `Z = A^p B^q//C^r`
Then, `((ΔZ))/Z = p ((ΔA)/A) + q ((ΔB)/B) + r ((ΔC)/C)`