Accuracy precision and error measurement:

Error: There is a difference between the true value and measured value when a physical quantity is measured by an instrument as the measuring instrument is not ideal. This difference is known as error.

Accuracy: Accuracy of the measurement of a physical quantity is the closeness of the measured value to the true value of the measurement.

Precision: Precision is the limit to which the measurement of a physical quantity is done. It depends on the least count of the measuring instrument. If the least count is small then the precision is more.

Example: True value of the length of a pen is 8.524cm. A student A measures this length by a measuring instrument which has a least count of 0.1 cm, and records its length as 8.4 cm. Another student B measures this length by a measuring instrument which has a least count of 0.01 cm and records it as 8.14 cm. The measurement taken by student A is more accurate (as it is closer to the true value), but is less precise (as its resolution is 0.1 cm only), whereas the measurement taken by B is less accurate, but has more precision.

Types of error:

(A) Systematic error: This error occurs according to a certain pattern or system.

(i) Instrumental error: This error occurs due to the faulty instrument. Example â€“ zero error of vernier callipers or screw gauge.

(ii) Personal error: This error occurs due to the carelessness of the observer. Example â€“ During measurement of time period of pendulum the readings are different as the stop watch is not properly used. Student might stop the watch a little bit earlier or little bit later than the exactly on the completion of number of full oscillations.

(iii) Natural error: The error generates due to the change in the conditions of environment. Example â€“ The readings of the experiments on electrostatics may vary if it is performed in rainy season instead of winter season. Or in Ohmâ€™s law experiment the temperature should remain constant.

(iv) Imperfection in experimental technique: This error occurs if the imperfect technique or procedure is used for measuring the physical quantity. Example â€“ The observer should measure the length seeing perpendicularly on the meter scale otherwise error generates.

(B) Random error: This error generates due to unknown reasons when the physical quantity is measured and it is beyond the control of observer. This can be minimized by taking large number of observations and taking the mean value of those observations. Example: To find the time period of simple pendulum the amplitude should be small and the motion is simple harmonic but the air resistance makes the variation in the measurement.

Absolute, relative and percentage of error: Let us consider a physical quantity is measured for n times and the corresponding readings are respectively x_{1}, x_{2}, x_{3} – – – x_{n .}

If x_{mean} is the corresponding arithmetic mean of the measurement, then x_{mean }= = .

The difference in the measured value from the true value is known at absolute error.

The absolute error in the measurement of 1^{st} reading is x_{1} = |x_{mean} – x_{1}|

The absolute error in the measurement of 2nd reading is x_{2} = |x_{mean} – x_{2}|

Similarly, the absolute error in the measurement of n^{th} reading is x_{n} = |x_{mean} – x_{n}|

The mean absolute error in the measurement is x_{mean = }Â = .

Relative error is the ratio of mean absolute error to the mean or true value of the measurement.

Relative error = .

Percentage error = Relative error 100 = 100.

Example: The time period of a pendulum is measured as 2.42, 2.47, 2.49, 2.45 and 2.48 (in second). Find % of error of time period in the measurement.

T_{mean}Â = Â = 2.462

T_{1} = |2.462 â€“ 2.42| = 0.042

T_{2} = |2.462 â€“ 2.47| = 0.008

T_{3} = |2.462 â€“ 2.49| = 0.028

T_{4} = |2.462 â€“ 2.45| = 0.012

T_{5} = |2.462 â€“ 2.48| = 0.018

T_{mean} = Â = 0.0216 = 0.022

% of error of time period is 100Â = 0.89%.

Combination of errors: Consider two quantities A and B which have measured values (A Â± âˆ†A) and (B Â± âˆ†B) respectively. Where âˆ†A and âˆ†B are the absolute errors in A and B respectively.

Addition: Let us now calculate the absolute error âˆ†Z in Z when Z = A + B.

Now, Z Â± âˆ†Z = (A Â± âˆ†A) + (B Â± âˆ†B)

Or, Z Â± âˆ†Z = A Â± âˆ†A + B Â± âˆ†B

Or, Z Â± âˆ†Z = A + B Â± âˆ†A Â± âˆ†B

Or, Z Â± âˆ†Z = (A + B) Â± (âˆ†A + âˆ†B)

Comparing the both sides of the equation we get, âˆ†Z = (âˆ†A + âˆ†B). Therefore, in addition the sum of individual absolute error is the resultant absolute error.

Example: : A = 5 Â± 0.2 and B = 3 Â± 0.4 if Z = A + B then, Z = 5 + 3 = 8 and âˆ†Z = (âˆ†A + âˆ†B) = 0.2 + 0.4 = 0.6

Therefore, Z = 8 Â± 0.6.

Subtraction: Let us now calculate the absolute error âˆ†Z in Z when Z = A – B.

Now, Z Â± âˆ†Z = (A Â± âˆ†A) – (B Â± âˆ†B)

Or, Z Â± âˆ†Z = A Â± âˆ†A – B Â± âˆ†B

Or, Z Â± âˆ†Z = A – B Â± âˆ†A Â± âˆ†B

Or, Z Â± âˆ†Z = (A – B) Â± (âˆ†A + âˆ†B)

Comparing the both sides of the equation we get, âˆ†Z = (âˆ†A + âˆ†B). Therefore, in subtraction the sum of individual absolute error is the resultant absolute error.

Example: A = 5 Â± 0.2 and B = 3 Â± 0.4 if Z = A – B then, Z = 5 – 3 = 2 and âˆ†Z = (âˆ†A + âˆ†B) = 0.2 + 0.4 = 0.6

Therefore, Z = 2 Â± 0.6.

Multiplication: Let us now calculate the absolute error âˆ†Z in Z when Z = A B.

Now, Z Â± âˆ†Z = (A Â± âˆ†A) (B Â± âˆ†B)

Or, Z(1 Â± ) = A(1 Â± ) B(1 Â± )

Or, Z(1 Â± ) = AB(1 Â± )Ã—(1 Â± )

Or, Z(1 Â± ) = AB(1 Â± Â± Â± )

Or, Z(1 Â± ) = AB(1 Â± Â± ) [neglecting the term as it is very small]

Or, Z(1 Â± ) = AB[1 Â± ( + )]

Comparing the both sides of the equation we get, = ( + ). Therefore, in multiplication the sum of individual relative error is the resultant relative error.

Example: A = 5 Â± 0.2 and B = 3 Â± 0.1 if Z = AÃ—B then, Z = 5Ã—3 = 15 and = ( + ) or, = ( + ) or, âˆ†Z = 1.1. Therefore, Z = 15 Â± 1.1.

Division: Let us now calculate the absolute error âˆ†Z in Z when Z = .

Now, Z Â± âˆ†Z =

Or, Z(1 Â± ) =

Or, Z(1 Â± ) = (1 Â± )(1 Â± )^{-1}

Or, Z(1 Â± ) = (1 Â± )(1 Â± ) [Using binomial theorem (1 Â± a)^{x} = (1 Â± xa) when aâ‰ª1]

Or, Z(1 Â± ) = (1 Â± Â± Â± )

Or, Z(1 Â± ) = (1 Â± Â± ) [neglecting the term as it is very small)

Or, Z(1 Â± ) = [1 Â± ( + )]

Comparing the both sides of the equation we get, = ( + ). Therefore, in division the sum of individual relative error is the resultant relative error.

Example: A = 10 Â± 0.2 and B = 4 Â± 0.1 if Z = then, Z = = 2.5 and = ( + ) or, = ( + ) or, âˆ†Z = 0.1125. Therefore, Z = 2.5 Â± 0.1125.

Power Let us now calculate the absolute error âˆ†Z in Z when Z = A^{P}.

Now, Z Â± âˆ†Z = (A Â± âˆ†A)^{P}

Or, Z(1 Â± ) = A^{P} (1Â± )^{P}

Or, Z(1 Â± ) = A^{P} (1Â±P ) [using binomial theorem (1 Â± a)^{x} = (1 Â± xa) when aâ‰ª1]

Comparing the both sides of the equation we get, = P.

Example: A = 10 Â± 0.2 and Z = A^{2} then = P or, = or, âˆ†Z = 4.

Therefore, Z = (100 Â± 4).

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