2.6 Accuracy, precision of instruments and errors in measurement

Accuracy, Precision Of Instruments, and Errors In Measurement

Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity which is based on measured values also has an error. We shall distinguish between two terms: accuracy and precision. 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.

Systematic errors:
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 that arise from errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc. For example, the temperature graduations of a thermometer may be inadequately calibrated (it may read \(104^{\circ} \mathrm{C}\) at the boiling point of water at STP whereas it should read \(100^{\circ} \mathrm{C}\) ).
• 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 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. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax.

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

Random errors:
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. For example, when the same person repeats the same observation, it is very likely that he may get different readings every time.

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. For example, a vernier callipers has the least count as 0.01cm; 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.

Absolute Error:

The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. This is denoted by \(|\Delta a|\). In absence of any other method of knowing true value, we considered arithmetic mean as the true value. 

\(
a_{\text {mean }}=\left(a_{1}+a_{2}+a_{3}+\ldots+a_{n}\right) / n
\)
or,
\(
a_{\text {mean }}=\sum_{i=1}^{n} a_{i} / n
\)

Then the errors in the individual measurement values from the true value, are

\(\begin{aligned}
&\Delta a_{1}=a_{1}-a_{\text {mean’ }}\\
&\Delta a_{2}=a_{2}-a_{\text {mean }} \text {, }\\
&\begin{array}{ccc}
\cdots & \cdots & \cdots \\
\cdots & \cdots & \ldots
\end{array}\\
&\Delta a_{n}=a_{n}-a_{\text {mean }}
\end{aligned}\)

The \(\Delta\) a calculated above may be positive in certain cases and negative in some other cases. But absolute error \(|\Delta a|\) will always be positive.

The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity \(a\). It is represented by \(\Delta a_{\text {mean }}\).
Thus,
\(
\Delta a_{\text {mean }}=\left(\left|\Delta a_{1}\right|+\left|\Delta a_{2}\right|+\left|\Delta a_{3}\right|+\ldots+\left|\Delta a_{n}\right|\right) / n
\)
\(
=\sum_{i=1}^{n}\left|\Delta a_{i}\right| / n
\)

If we do a single measurement, the value we get may be in the range \(a_{\text {mean }} \pm \Delta a_{\text {mean }}\)
i.e. \(a=a_{\text {mean }} \pm \Delta a_{\text {mean }}\)
or,
\( a_{\text {mean }}-\Delta a_{\text {mean }} \leq a \leq a_{\text {mean }}+\Delta a_{\text {mean }}\)
This implies that any measurement of the physical quantity \(a\) is likely to lie between \(\left(a_{\text {mean }}+\Delta a_{\text {mean }}\right)\) and \(\left(a_{\text {mean }}-\Delta a_{\text {mean }}\right)\).

Relative Error or Percentage Error:

Instead of the absolute error, we often use the relative error or the percentage error \((\delta a)\). The relative error is the ratio of the mean absolute error \(\Delta a_{\text {mean }}\) to the mean value \(a_{\text {mean }}\) of the quantity measured.

Relative error \(=\Delta a_{\text {mean }} / a_{\text {mean }}\)
When the relative error is expressed in percent, it is called the percentage error \((\delta a)\).
Thus, the Percentage error
\(
\delta a=\left(\Delta a_{\text {mean }} / a_{\text {mean }}\right) \times 100 \%
\)

Example 1:

We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be \(2.63 \mathrm{~s}, 2.56 \mathrm{~s}, 2.42 \mathrm{~s}, 2.71 \mathrm{~s}\) and \(2.80 \mathrm{~s}\). Calculate the absolute errors, relative error or percentage error.
Answer: The mean period of oscillation of the pendụlum
\(
\begin{aligned}
T &=\frac{(2.63+2.56+2.42+2.71+2.80) \mathrm{s}}{5} \\
&=\frac{13.12}{5} \mathrm{~s}=2.624 \mathrm{~s} \\
&=2.62 \mathrm{~s}
\end{aligned}
\)

As the periods are measured to a resolution of \(0.01 \mathrm{~s}\), all times are to the second decimal; it is proper to put this mean period also to the second decimal. The errors in the measurements are
\(
\begin{aligned}
&2.63 \mathrm{~s}-2.62 \mathrm{~s}=0.01 \mathrm{~s} \\
&2.56 \mathrm{~s}-2.62 \mathrm{~s}=-0.06 \mathrm{~s} \\
&2.42 \mathrm{~s}-2.62 \mathrm{~s}=-0.20 \mathrm{~s} \\
&2.71 \mathrm{~s}-2.62 \mathrm{~s}=0.09 \mathrm{~s} \\
&2.80 \mathrm{~s}-2.62 \mathrm{~s}=0.18 \mathrm{~s}
\end{aligned}
\)
Note that the errors have the same units as the quantity to be measured. The arithmetic mean of all the absolute errors (for the arithmetic mean, we take only the magnitudes) is
\(
\begin{aligned}
\Delta T_{\text {mean }} &=[(0.01+0.06+0.20+0.09+0.18) \mathrm{s}] / 5 \\
&=0.54 \mathrm{~s} / 5 \\
&=0.11 \mathrm{~s}
\end{aligned}
\)
That means, the period of oscillation of the simple pendulum is \((2.62 \pm 0.11)\) s i.e. it lies between \((2.62+0.11) s\) and \((2.62-0.11) \mathrm{s}\) or between \(2.73 \mathrm{~s}\) and \(2.51 \mathrm{~s}\). As the arithmetic mean of all the absolute errors is \(0.11 \mathrm{~s}\), there is already an error in the tenth of a second. Hence there is no point in giving the period to a hundredth. A more correct way will be to write
\(
T=2.6 \pm 0.1 \mathrm{~s}
\)
Note that the last numeral 6 is unreliable, since it may be anything between 5 and 7 . We indicate this by saying that the measurement has two significant figures. In this case, the two significant figures are 2, which is reliable, and 6, which has an error associated with it. 

For this example, the relative error or the percentage error is
\(
\delta a=\frac{0.1}{2.6} \times 100=4 \%
\)

Combination of Errors:

If we do an experiment involving several measurements, we must know how the errors in all the measurements combine. 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 sum or a difference: Suppose two physical quantities \(A\) and \(B\) have measured values \(A \pm \triangle A, B \pm \Delta B\) respectively where \(\Delta A\) and \(\Delta B\) are their absolute errors. We wish to find the error \(\Delta Z\) in the sum
  \(Z=A+B .\)
  We have by addition, \(Z \pm \Delta Z\)
  \(
  =(A \pm \Delta A)+(B \pm \Delta B) .
  \)
  The maximum possible error in $Z$
  \(
  \Delta Z=\Delta A+\Delta B
  \)
  For the difference $Z=A-B$, we have
  \(
  \begin{aligned}
  Z \pm \Delta Z &=(A \pm \Delta A)-(B \pm \Delta B) \\
  &=(A-B) \pm \Delta A \pm \Delta B
  \end{aligned}
  \)
  or, \(\quad \pm \Delta Z=\pm \Delta A \pm \Delta B\)
  The maximum value of the error \(\Delta Z\) is again \(\Delta A+\Delta B\).

Example 2: 

The temperatures of two bodies measured by a thermometer are \(t_{1}=20{ }^{\circ} \mathrm{C} \pm 0.5^{\circ} \mathrm{C}\) and \(t_{2}=50{ }^{\circ} \mathrm{C} \pm 0.5^{\circ} \mathrm{C}\). Calculate the temperature difference and the error therein.

Answer: \(t^{\prime}=t_{2}-t_{1}=\left(50^{\circ} \mathrm{C} \pm 0.5^{\circ} \mathrm{C}\right)-\left(20^{\circ} \mathrm{C} \pm 0.5^{\circ} \mathrm{C}\right)\)
\(
t^{\prime}=30{ }^{\circ} \mathrm{C} \pm 1{ }^{\circ} \mathrm{C}
\)

• Error of a product or a quotient: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

  Suppose \(Z=A B\) and the measured values of \(A\) and \(B\) are \(A \pm \Delta A\) and \(B \pm \Delta B\). Then
  \(
  \begin{aligned}
  Z \pm \Delta Z &=(A \pm \Delta A)(B \pm \Delta B) \\
  &=A B \pm B \Delta A \pm A \Delta B \pm \Delta A \Delta B .
  \end{aligned}
  \)
  Dividing LHS by \(Z\) and RHS by \(A B\) we have, \(1 \pm(\Delta Z / Z)=1 \pm(\Delta A / A) \pm(\Delta B / B) \pm(\Delta A / A)(\Delta B / B) .\) Since \(\Delta A\) and \(\Delta B\) are small, we shall ignore their product.
  Hence the maximum relative error
  \(\Delta Z / Z=(\Delta \mathrm{A} / \mathrm{A})+(\Delta \mathrm{B} / \mathrm{B})\).
  You can easily verify that this is true for the division also.

Example 3: 

The resistance \(R=V / I\) where \(V=(100 \pm 5) \mathrm{V}\) and \(I=(10 \pm 0.2) \mathrm{A}\). Find the percentage error in \(R\).

Answer: The percentage error in \(V\) is \(5 \%\) and in \(I\) it is \(2 \%\). The total error in \(R\) would therefore be \(5 \%+2 \%=7 \%\).

Example 4:

Two resistors of resistances \(R_{1}=100 \pm 3  \mathrm{ohm}\) and \(R_{2}=200 \pm 4  \mathrm{ohm}\) are connected (a) in series, (b) in parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation \(R=R_{1}+R_{2}\) and for (b) \(\frac{1}{R^{\prime}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) and \(\frac{\Delta R^{\prime}}{R^{\prime 2}}=\frac{\Delta R_{1}}{R_{1}^{2}}+\frac{\Delta R_{2}}{R_{2}^{2}}\)

Answer: (a) The equivalent resistance of a series combination
\(R=R_{1}+R_{2}=(100 \pm 3) \mathrm{ohm}+(200 \pm 4) \mathrm{ohm}\) \(=300 \pm 7\) ohm.
(b) The equivalent resistance of the parallel combination
\(
R^{\prime}=\frac{R_{1} R_{2}}{R_{1}+R_{2}}=\frac{200}{3}=66.7 \mathrm{ohm}
\)
Then, from \(\frac{1}{R^{\prime}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\)
we get,
\(
\frac{\Delta R^{\prime}}{R^{\prime 2}}=\frac{\Delta R_{1}}{R_{1}^{2}}+\frac{\Delta R_{2}}{R_{2}^{2}}
\)
\(\Delta R^{\prime}=\left(R^{2}\right) \frac{\Delta R_{1}}{R_{1}^{2}}+\left(R^{2}\right) \frac{\Delta R_{2}}{R_{2}^{2}}\)
\(
\begin{aligned}
&=\left(\frac{66.7}{100}\right)^{2} 3+\left(\frac{66.7}{200}\right)^{2} 4 \\
&=1.8
\end{aligned}
\)
Then, \(R^{\prime}=66.7 \pm 1.8 \mathrm{ohm}\)
(Here, \(\Delta \mathrm{R}\) is expressed as \(1.8\) instead of 2 to keep in conformity with the rules of significant figures.)

• Error in case of a measured quantity raised to a power:  The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity. \(\text { Suppose } Z=A^{2} \text {, }\)Then,
  \(
  \Delta Z / Z=(\Delta A / A)+(\Delta A / A)=2(\Delta A / A) .
  \)
  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,
  \(
  \Delta Z / Z=p(\Delta A / A)+q(\Delta B / B)+r(\Delta C / C)
  \)

Example 5:

Find the relative error in \(Z\), if \(Z=A^{4} B^{1 / 3} / C D^{3 / 2}\).

Answer: The relative error in \(Z\) is \(\Delta Z / Z=\) \(4(\Delta A / A)+(1 / 3)(\Delta B / B)+(\Delta C / C)+(3 / 2)(\Delta \mathrm{D} / D)\).

Example 6:

The period of oscillation of a simple pendulum is \(T=2 \pi \sqrt{L / g}\). The measured value of \(L\) is \(20.0 \mathrm{~cm}\) known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of [/latex]g[/latex]?

Answer: \(g=4 \pi^{2} L / T^{2}\)

Here, \(T=\frac{t}{n}\) and \(\Delta T=\frac{\Delta t}{n}\). Therefore, \(\frac{\Delta T}{T}=\frac{\Delta t}{t}\). The errors in both \(L\) and \(t\) are the least count errors. Therefore,
\((\Delta g / g)=(\Delta L / L)+2(\Delta T / T)\)
\(
=\frac{0.1}{20.0}+2\left(\frac{1}{90}\right)=0.027
\)
Thus, the percentage error in \(g\) is
\(
\begin{aligned}
100(\Delta g / g) &=100(\Delta L / L)+2 \times 100(\Delta T / T) \\
&=3 \%
\end{aligned}
\)