Error Analysis
Basic Concepts of Error Analysis
1. Significant Figures:
The laboratory usually involves measurements of several physical quantities such as length, mass, time, voltage and current. The values of these quantities should be presented in terms of Significant Figures as follows.
For example, the location of the arrow is to be determined in Fig. 1. It is obvious that the location is between 1 cm and 2 cm. The correct way to express this location is to make one more estimate based on your intuition. That is, in this case, a reading of 1.3 cm is estimated. This measurement is said to contain two significant figures. Note that there should only be one estimated place in any measurement. So, in the example shown in Fig. 1 do not try to locate the position of the arrow as 1.35 cm.
If data are to contain, say, three significant figures, two must be known, and the third estimated.
1
2
3
4
5
cm
Figure 1.
The following rules dictate the handling of significant figures:
(a) Specify the measured value to the same accuracy as the error in the measurement. For example, we report that a physical quantity is x =
3. 45 ± 0. 05 , not 3. 4 ± 0. 05 and not 3. 452 ± 0. 05 ; in other words, the least significant figures in both numbers (the main value and the error) are on the same decimal position;
(b) When adding or subtracting numbers, the answer is only good to the least accurate number present in any of the components: for example, 50.3 +
2.555 = 52.9 and not 52.855;
(c) When multiplying or dividing, keep the same number of significant figures as the factor with the fewest number of significant figures: For example,
5.0 x 1.2345 = 6.2 and not 6.1725.
2. Types of Errors:
Every measurement has its error. In general, there are three types of errors that will be explained below:
(a) Random errors: This type of errors is usually referred to as statistical error. This class of errors is produced by unpredictable or unknown variations
1. Significant Figures:
The laboratory usually involves measurements of several physical quantities such as length, mass, time, voltage and current. The values of these quantities should be presented in terms of Significant Figures as follows.
For example, the location of the arrow is to be determined in Fig. 1. It is obvious that the location is between 1 cm and 2 cm. The correct way to express this location is to make one more estimate based on your intuition. That is, in this case, a reading of 1.3 cm is estimated. This measurement is said to contain two significant figures. Note that there should only be one estimated place in any measurement. So, in the example shown in Fig. 1 do not try to locate the position of the arrow as 1.35 cm.
If data are to contain, say, three significant figures, two must be known, and the third estimated.
1
2
3
4
5
cm
Figure 1.
The following rules dictate the handling of significant figures:
(a) Specify the measured value to the same accuracy as the error in the measurement. For example, we report that a physical quantity is x =
3. 45 ± 0. 05 , not 3. 4 ± 0. 05 and not 3. 452 ± 0. 05 ; in other words, the least significant figures in both numbers (the main value and the error) are on the same decimal position;
(b) When adding or subtracting numbers, the answer is only good to the least accurate number present in any of the components: for example, 50.3 +
2.555 = 52.9 and not 52.855;
(c) When multiplying or dividing, keep the same number of significant figures as the factor with the fewest number of significant figures: For example,
5.0 x 1.2345 = 6.2 and not 6.1725.
2. Types of Errors:
Every measurement has its error. In general, there are three types of errors that will be explained below:
(a) Random errors: This type of errors is usually referred to as statistical error. This class of errors is produced by unpredictable or unknown variations