Units and Measurement

Accuracy, Precision of Instruments and Errors in Measurement

Repeated measurements often yield different values for the same quantity. What is the accuracy of a measurement and which instrument is the precise one? Let’s answer these questions and wander into the realm of measurements.


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Accuracy and Precision

The accuracy of a measurement is its “closeness” or proximity to the true value or the actual value ( \(a_m\) ) of the quantity. Let \( a_1, a_2, a_3, a_4 \) … \(a_n\) be the ‘n’ measured values of a quantity ‘a’. Then its true value is defined as:

\(a_m\) = [ \( a_1 + a_2 + a_3 + a_4 …. + a_n \) ]/n     …. (1)

Suppose your height is 183 cm. If we measure it with some instrument (measuring tape and a fancy laser beam!) it comes out to be 182.9995 cm. Another measurement (with a meter rod and a 6th grader) yields a result of 195 cm. We can see that the value obtained from the first measurement is closer to the actual value (true value) of your height.  So the first measurement is more accurate in comparison to the second one.

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Let’s suppose that our 6th grader likes to measure heights. He takes three more measurements of your height and gets the following results: 197 cm, 195.3 cm, and 196.1 cm. Are these measurements accurate? Of course not, they are far from the true value of your height. But we see that all these measurements are close to each other i.e. 197 cm, 195.3 cm, 196.1 cm and 195 cm are close to each other. They are precise measurements.

Accuracy and Precision of measurements

Thus, precision is the closeness of the various measured values to each other. Accuracy, on the other hand, is the closeness of the measured values with the true value of the quantity.

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Errors in Measurement

A measurement is reliable if it is accurate as well as precise. The error in a measurement is the deviation of the measured value from the true value, \(a_m\) of the quantity. Less accurate a measured value, greater the error in its measurement.

The error in a measurement is the uncertainty in its value. This is the amount by which the measurement can be more or less than the original value. It is denoted by putting a delta sign before the symbol of the quantity e.g. Δa denotes the error in the measurement of a quantity ‘a’.

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Errors can be classified into the following types:

Systematic errors

In such errors, the measurement deviates from the actual value by a fixed amount. Hence the prediction of these errors can be made. An erroneous instrument, changes in the physical conditions at the time of measurement, human error etc. are the main causes of systematic errors.

Random Error

These errors are due to unknown sources. These type of errors are removed by taking a bunch of readings and finding their mean.

Absolute Error

Absolute error in a measurement \(a_1\) is given by \(Δa_1\) = |\(a_m – a_1\)|

In general we say that the absolute error in the \(a_n\) = \(Δa_n\) = | \(a_m – a_n\) |

For example, the absolute error in the measurement of your height is:

|183 cm – 195 cm| = 12 cm for the 195 cm reading = \(Δa_1\)

|197 cm – 183 cm| = 14 cm for the 197 cm reading = \(Δa_2\)

|183 cm – 195.3 cm| = 12.3 cm for the 195.3 cm reading = \(Δa_3\)

|183 cm – 196.1 cm| = 13.1 cm for the 196.1 cm reading = \(Δa_4\)

Mean Absolute Error

If \(Δa_1\) , \(Δa_2\) , \(Δa_3\) … \(a_n\) are the absolute errors in \(a_1\), \(a_2\), \(a_3\), … \(a_n\), then:

Mean Absolute Error =  [ | \(Δa_1\)| + |\(Δa_2\)| + |\(Δa_3\)| + … + |\(a_n\)| ]/n

For example in the above case, the mean absolute error = (12 cm + 14 cm + 12.3 cm + 13.1 cm)/4 = 12.85 cm

Relative Error

The ratio of the mean absolute error and the true value of the quantity gives the relative error.

Relative error = (Mean absolute error)/ \(a_m\)

In the above case, Relative error = 12.85/183 = 0.07022 cm

Percentage Error

The percentage error is obtained from the relative error by expressing it in terms of percentage i.e. Percentage error = Relative error × 100%

In the above case, Percentage error = 0.07022 cm × 100% = 7.022%

Propagation of Errors

If you are using the measured values in calculations then the errors will also enter your results. Following is the manner in which errors propagate or add up:

Errors in Addition and Subtraction

Let (a ± Δa) and (b ± Δb) be two quantities.

Suppose x = (a ± Δa) ± (b ± Δb)

Then error in x i.e. Δx = ±(Δa+Δb)

Hence errors add up under addition or subtraction.

Errors in Multiplication and Division

Let x = (a ± Δa)×(b ± Δb) or x = (a ± Δa)/(b ± Δb)

Then the relative error in x is given by:

Δx/x = ±[(Δa/a + Δb/b)]

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Solved Examples  For You

Example 1: The resistance of metal is given by V=IR. The voltage in the resistance is V=(80.5±0.1)V and current in the resistance is I=(20.2±0.2)A, the value of resistance with its percentage error is :

A) 3.98% Ω                         B) 2.34% Ω

C) 1.11% Ω                           D) 2.4% Ω

Solution: C)  V = IR [Ohm’s Law]

V = 80.5 Volt and I = 20.2 A

R = V/I

Relative percentage error in R = (0.1/80.5 + 0.2/20.2)×100 = 1.11%

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