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I. SIGNIFICANT FIGURES
What are significant figures?
All measurements have some degree of uncertainty; how great the uncertainty is depends on both the accuracy of the measuring device and the skill of its operator. A usual 30 cm ruler in the market can measure length up to 0.1 mm; length differences less than this cannot be detected by this ruler. Therefore, the length of a pen may be expressed as say 15.25 0.02. Also, the balance you used many times in Phys 101 and Phys 102 labs can measure masses up to 0.01 grams; so a typical reading is like 25.236 0.005. The 0.02 in the ruler and 0.005 in the balance is a measure of the accuracy of the measurement. It is important to have some indication of how accurately any measurement is made; the notation is one way to accomplish this. All of the digits, including the last one, are called significant digits or, more commonly, significant figures. The number 2.2 has two significant figures, while the number 2.2405 has five significant figures. Notice that the last digit (2 in the ruler reading and 5 in the balance reading) is uncertain and might change from one person to another; the important thing is to know how to know the correct number of significant figures that should be there for the reading of a specific device.
How can we determine how many significant numbers a measurement has?
The following rules apply to determining the number of significant figures in a measured quantity:
1. All nonzero digits are significant--457 cm (three significant figures); 0.25 g (two significant figures).
2. Zeros between nonzero digits are significant--1005 kg (four significant figures); 1.03 cm (three significant figures).
3. Zeros to the left of the first nonzero digits in a number are not significant; they merely indicate the position of the decimal point--0.02 g (one significant figure); 0.0026 cm (two significant figures).
4. When a number ends in zeros that are to the right of the decimal point, they are significant--0.0200 g (three significant figures); 3.0 cm (two significant figures).
5. When a number ends in zeros that are not to the right of a decimal point, the zeros are not necessarily significant--130 cm (two or three significant figures); 10,300 g (three, four, or five significant figures). The way to remove this ambiguity is either to add a decimal point (e.g., 400. has 3 significant figures, while we are not sure about 400) or to use the exponential notation (e.g. while we are not sure about how many significant figures in 400, 4102 has certainly one significant figure).
How do we work with significant figures?
In carrying measured quantities through calculations the rule used is that the accuracy of the result is limited by the least accurate measurement. You will see how this is done in the following.
Addition and Subtraction
When adding or subtracting, the number of digits to the right of the decimal point in the answer is determined by the measurement which has the least number of digits to the right of the decimal point.
e.g. adding:
26.46 this has the least digits to the right of the decimal point (2) + 4.123 30.583 rounds off to 30.58 two digits to the right of the decimal point
e.g. subtracting:
26.46 - 4.123 22.337 rounds off to 22.34
Multiplication and Division
In multiplying or dividing, the number of significant figures in the answer, regardless of the position of the decimal point, equals that of the quantity which has the smaller number of significant figures.
e.g. multiplying:
2.61 x 1.2 this has the smaller number of significant figures (2) 3.132 rounds off to 3.1 has 2 significant figures
e.g. dividing:2.61 1.2 = 2.175 rounds off to 2.2
II. USEFUL CONCEPTS IN ERROR ANALYSIS
1) Types of Error:
Random errors
What do they mean? Think about examples of each.
Systematic errors
2) Maximum Error or Maximum Possible Error (MPE):
From the name of this quantity, it represents the maximum possible error. You learned three methods to estimate the MPE in the first lab of this course.
3) Mean (Average):
To reduce random errors, one takes many readings and takes the mean (average) value. If n readings were taken for a quantity x, then the mean value is given by:
EMBED Equation.3
4) Standard Deviation:
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EMBED Equation.3
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