Significant Figure and Rounding off

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Measurement Order Magnitude

Very often we find it convenient and sometimes necessary to express quantitative magnitudes only approximately. This is done through the concept of order of magnitude. The order of magnitude of a number N is the nearest power of ten to which it is approximated and is obtained through the following operational definition.
The order of magnitude of a number N is x such that Least Count
The least count of a measuring device is the smallest measurement that can be made accurately with it. E.g., a metering rod usually has numbered centimetre division, subdivided into millimetres. Therefore, the least distance that can be measured accurately with a metering rod is 1 mm. We say that the least count of a metering rod is 1 mm.
The least count of a metering rod is 1 mm or 0.1 cm
The least count of ordinary vernier calliper is 0.1 mm or 0.01 cm.
The least count of an ordinary screw gauge is 0.01 mm or 0.001 cm.
The smaller the least count of measuring devices, the greater the accuracy of measurements. For example, the length of a given rod measured with meter rod may give the result as 3.5 and However, the length of the same rod measured with a vernier calliper may give the result as 3.12 cm. Clearly, the second measurement is more accurate compared to the first measurement.
Accuracy and Errors
Today accurate measurement is an important part of physics. But no measurement is absolutely precise; there is an uncertainty associated with every measurement. The uncertainty in measurement is called error. When giving the result of a measurement, it is a good practice to state the estimated uncertainty or error. For example, if the measured length of a table is 23.2 cm and the uncertainty or error is about 0.1 cm, then length will be recorded as 23.20.1 cm. Errors can be classified as follows:
(i) Instrumental error: We do all measurements with the help of instruments. If we use meter scale for length, it may have graduation at 1 mm interval. The error, in the use of a meter scale with 1 mm intervals is normally taken to be half of the smallest division on the scale of the instrument, that is nearly 0.5 mm. In the case of Calipers, it is 0.05 mm i.e. half of its least count. These errors are called instrumental error.
In addition to the instrumental error, the error may be due to the individual qualities of the experiment depending on his attentiveness and eye sight etc.
(ii) Random error: These errors depends on the individual measuring person and care taken in the measuring process. To minimise the random errors, the measurement should be repeated many times and the mathematical mean’ of all observation of concerned measurement be taken to get the nearly true value of the measured quantity.
Therefore (iii) Absolute error: The magnitude of the difference between the mean value of the quantity and the individual measurement value is termed as an absolute error of the measurement, denoted by symbolically Δa. That means  Actually speaking, there are several causes for errors in measurement listed below:
(i) Least count error: This is the error associated with the resolution of the instrument.
(ii) Resolution of the experiment: Different methods of measurement will have a different magnitude of units and hence reaping the certain error.
(iii) Systematic errors: Systematic error arises due to instrumental defect, imperfection and techniques used etc. They may also occur due to personal reasons of the carelessness of experiment.
(iv) Random errors: These errors occur irregularly and can be accounted by taking careful repeating measurements and an average value be taken.
(v) Gross error: It can be due to improper adjustment of the apparatus.

Significant Figure and Rounding off