The result of
every measurement by any measuring instrument contains some uncertainty. This
uncertainty is called error.
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.
For example,
suppose the true value of a certain length is near 5.678 cm. In one experiment,
using a measuring instrument of resolution 0.1 cm, the measured value is found
to be 5.5 cm, while in another experiment using a measuring device of greater
resolution, say 0.01 cm, the length is determined to be 5.38 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.
Least
Count
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.
Estimation of Error
Absolute
Error: Suppose
the values obtained in several measurement of physical quantity a are a1,
a2, ...... an. The magnitude of the difference
between the individual measurement and the true value of the quantity is called
the absolute error of the measurement. In absence of any other method of
knowing true value, arithmetic mean is considered as the true value.
The
arithmetic mean of all the absolute errors is taken as the final or mean
absolute error.
\[\bar{a}=\frac{a_{1}+a_{2}+\cdot \cdot \cdot +a_{n}}{n}=\frac{1}{n}\sum_{i=1}^{n}a_{i}\]
\[\Delta a_{1}=\bar{a}-a_{1},\ \Delta a_{2}=\bar{a}-a_{2}, \ \cdot \cdot \cdot \ \Delta a_{n}=\bar{a}-a_{n}\]
\[where \ \Delta a_{1}, \ \Delta a_{2} \ \cdot \cdot \cdot \cdot \Delta a_{n}\ are \ absolute \ errors\]
\[\Delta a_{1}=\bar{a}-a_{1},\ \Delta a_{2}=\bar{a}-a_{2}, \ \cdot \cdot \cdot \ \Delta a_{n}=\bar{a}-a_{n}\]
\[where \ \Delta a_{1}, \ \Delta a_{2} \ \cdot \cdot \cdot \cdot \Delta a_{n}\ are \ absolute \ errors\]
Relative
Error: The
relative error is the ratio of the mean absolute error to the mean value of the
quantity measured.
\[relative \ errors \ \delta _{a}=\frac{\Delta \bar{a}}{a}\]
Percentage
Error: When
the relative error is expressed in percent, it is called the percentage error.
\[percentage \ errors \ \delta _{a}\times 100 \%=\frac{\Delta \bar{a}}{a}\times 100 \%\]
Combination
of Errors
If an
experiment is done involving several measurements, the errors in all the
measurements should combine to give final errors.
For example, density
is the ratio of the mass to the volume of the substance. If there
are errors in the measurement of mass and of the volume, then use
combination of errors to determine the error in the density of the substance.
\[1. Addition \ Z=A+B\Rightarrow \Delta Z=\Delta A+\Delta B\]
\[2.Subtraction \ Z=A-B\Rightarrow \Delta Z=\Delta A+\Delta B\]
\[3.Division\ Z=\frac{A}{B}\Rightarrow \frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}\]
\[4.Multiplication\ Z=A\times B \Rightarrow \frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}\]
\[5. Power Z=A^{n}\Rightarrow \frac{\Delta Z}{Z}=n\frac{\Delta A}{A}\]
Significant Figures
There are three rules on determining how many significant figures are in a number:
Rule 1: Non-zero digits are always significant.
Example: a number like 26.38 would have four significant figures and 7.94 would have three
Rule 2: Any zeros between two significant digits are significant.
Example : a number like 406 would have three significant figures
Rule 3: A final zero or trailing zeros in the decimal portion ONLY are significant.
Example : In 0.00500 and 0.03040 only boldface digits are significant:
