# Error Analysis

**Topics:**Measurement, Observational error, Statistics

**Pages:**9 (1336 words)

**Published:**January 29, 2013

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 in the measuring process. It always exists even though one does the experiment as carefully as is humanly possible. One example of these uncontrollable variations is an observer’s inability to estimate the last significant digit for a given measurement the same way every time.

(b) Systematic errors: This class of errors is commonly caused by a flaw in the experimental apparatus. They tend to produce values either consistently above the true value or consistently below the true value. One example of the flaw is a bad calibration in the instrumentation.

(c) Personal errors: This type of errors is also called mistakes. It is fundamentally different from either the systematic or random errors stated above; and can be completely eliminated if the experimentalist is careful enough. One example of this type of errors is to misread the scale of an instrument.

3. Mean and Statistical Deviation:

Let's assume that both the systematic and personal errors can be eliminated by careful experimental procedures; then the experimental errors are governed by random or statistical errors. If there are a total number of N measurements made of some physical quantity say x , and the i-th value is denoted by x i , the

statistical theory says that the " mean " of the above N measurements is the best _

approximation to the true value; i.e., the mean x is given by N

_

x=

1

N

∑

xi ≡

1

(x + x2 + x 3 + − − − + x N ) ;

N 1

i=1

where

∑

means summation.

The statistical theory states that the precision of the measurement can be determined by the calculation of a quantity called " standard deviation " from the mean of the measurements, which is defined by the following equation: N

1

σ=

N − 1

∑(

2

xi − x)

;

i=1

≡

1

2

2

2

2

(x − x ) + (x2 − x ) + (x3 − x ) + − − − + (x N − x) N − 1 1

[

]

The statistical theory states that approximately 68% of all the repeated measurements should fall within a range of plus or...

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