Summary of Significant Figures

Definitions involving terms under significant figures, use of instants and error analysis.

Accuracy

By using an appropriate device, we can know the measure of a physical quantity.The accuracy of the reading depends on the device and the person taking the measurement. Therefore, accuracy means the extent upto which a measured value agrees with the standard or true value. For example, if the temperature of water is 211.82^oC and the thermometer reads exacts 211.82^oC, then the measured value is accurate.

Least count

The accuracy of a measuring instrument depends upon the least count of the measuring instrument.

For example, length of a body that can be measured using a tailor's tape, cannot be beyond the minimum and maximum length of the measuring tape.

Least count of this tape is the least distance that can be accurately measured using this tape. Various types of measuring instruments like steel tape, ruler, vernier callipers, micrometer, screw gauge, etc., are more accurate compared to each other, i.e., the least count of a screw gauge is 0.01mm, the least count of a vernier callipers is 0.1mm. When we measure the diameter of the wire that is 1mm thick nearest to 0.01mm, using the screw gauge, the accuracy is about 0.01mm in 1.00mm. If this measurement is taken by using the vernier callipers, then the accuracy would have been 1 out of 10. Hence, the accuracy depends on the least count of the measuring instrument.

[Note: However, error cannot be avoided, in spite of using the right instrument]

Significant figures

The significant figures are all those digits which we are absolutely sure of plus those digits that we are not sure of. However, the digits have a meaningful value. This also provides us with information about the extent of uncertainty in measurement.

E.g.,

All the experiments give answer upto 3.14 and the rest of the digits vary due to change in the value adopted for circumference and diameter and hence, it is rounded off to 3 digits.

Sensitivity of the instrument

This is the instantaneous reaction of the instrument, which can indicate the measurement at the instance of use.

Hysteresis

The instrument must be free from any previous recording or reading, and should give the same value for ascending or descending readings.

Precision

The extent to which a given set of measurements of the same quantity, agree with their mean value, which will not be true value.

Errors

All physical quantities can be measured. In spite of using the best and most appropriate instrument or device to measure, we may not get the true value. Hence, the difference between the true value and the measured value is called 'error'. There are different types of errors that are encountered while taking measurements. They are

Systematic errors which are caused due

• instrument error (manufacturing or inbuilt defect or limitation)

• zero error of devices.

• faulty calibration of meters

• inequality of balance arms of physical balance

• backlash error

• end error

Environmental error

Change in temperature, pressure, humidity, magnetic field, wind, etc. Eg., Metal tape expands in summer and at noon and contracts in winter and during night.

Observation or human error

Parallax

Due to imperfection

Conducting experiments imperfectly, without observing all the conditions stipulated for the purpose. E.g., if the experiment is to be conducted in an air-conditioned atmosphere, if the air-conditioning system does not exist during the experiment, this error occurs.

Random errors

These are called accidental errors, which occur irregularly and at random, in magnitude and direction.

(When the experiment is repeated, one may obtain different results each time.)

Gross errors

This is caused due to the carelessness of the person taking measurement.

Absolute error

Absolute error = true value - measured value

Mean absolute error

The arithmetic mean of the absolute error of the different measurements taken is called MAE.

Where is the true value and is the absolute error.

Relative and percentage errors

The ratio of the maximum error to time value of the measured quantity.

Where arithmetic mean is taken as time value.