Significant Figures
The significant figures in Addition and Multiplication
The significant figures of the measure of a physical quantity are all those digits about which we are absolutely sure plus last one digit that has a little doubt or uncertainty. The larger the number of significant figures in a measurement, the higher is the accuracy of the measurement. Significant figures express the precision of a measuring tool. The number of significant digits increase as the measured value gets more precise and the range of uncertainty gets smaller. When multiplying or dividing measured values, the final answer should contain only as many significant figures as the least
1.1.3 Error in the Measurement
When a physical quantity is measured by an instrument, the result contains some uncertainty. Every calculated quantity which is based on the measured values also has an error. The discrepancy between true value and measured value of a physical quantity is called error or uncertainty. Lesser the error more is the accuracy in the measurement of a physical quantity. There are several factors contributing to error in a measurement, such as limitations of the measuring device, irregularities in the object being measured, the skill of the person making the measurement etc. Some errors occurring in
measurements are
i. Instrumental error: This error occurs due to faulty calibration and imperfection in design of instrument. Zero error is the instrumental error. Suitable corrections are applied to experimental values to eliminate this error.
Systematic error: This error occurs due to defective setting of the instrument. While measuring electrical quantities using an analog multimeter, the zero setting of instrument should be properly done otherwise systematic error occurs in each reading. This error is minimized by detecting their causes.
m. Personal error: This error occurs due to lack of proper knowledge about the setting of apparatus precautions, or carelessness, faults in observation of reading by human beings.
iv. External cause error: This error is caused when external factors like temperature, pressure etc. change during experiment. These changes to be noted and correction to formula should be applied to minimize this error.
V. Imperfection error: This error arises due to imperfection in experiment technique or procedure.
vi. Random error: If condition of instrument used in beyond one's control then random error occurs. For example, when we are performing experiment related to electrical circuit due to fluctuation in the mains supply this error occurs. Also while performing experiment related to heat and if temperature of surrounding drops considerably then error in temperature reading occurs. Such errors are random errors. These errors are minimized but cannot be completely eliminated.
vii. Least count error: Even under the ideal conditions in which other errors are somehow absent there is another type of error which creeps into observation because of the limitation put on the accuracy of the measuring instrument by their least count. The least amount of the quantity which can be correctly measured by measuring instrument is called the least count (LC) of the instrument. All the readings or measured values are good only up to this value. This error is associated with the resolution of the instrument. Smaller the least count of the measuring device, the higher is the accuracy of measurement. For example, to measure length of object meter scale with 1mm division interval is used causes error upto 1 mm, a Vernier calipers of 10 division causes error upto 0.1 mm, general micrometer screw gauge causes error upto 0.01 mm.
Precision error is random error of a data. It is also called human error. It can be reduced by taking multiple measurements and averaging them. It suggests that to minimize errors (a) more number of readings are taken and mean value of it is estimated, (b) instruments with minimum least count are used for measurement, and (c) magnitude of reading taken should be as large as possible.
Experimental determination of any physical quantity (y) may involve the measurement of a large number of various physical quantities such as a, b, c etc. with the help of different instrument. The measurement of each physical quantity carries some error and these errors get propagated through the expression of y involving various mathematical operations. During calculation of y, the errors in the individual measurement of physical quantity combined and error in the final result depends on the nature of mathematical operation apart from errors in the individual measurement.
Suppose Aa, Ab and Ac be the absolute errors in measurement of three quantities a, b and e respectively. Let m, n and r be the numbers, then the rules to determine the error in a calculated quantity y from the errors in each of the quantities used in the calculations are as follows:
i. The absolute error is the sum or difference of the quantities is equal to the sum of the absolute errors in the individual quantities. If y = a b then Ay = Aa Ab.
Approved by Curriculum Development Center, Sanothimi, Bhaktapur