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Lesson 2 Accuracy and Precision
Retorika (FILI102)
Manuel S. Enverga University Foundation
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Lesson 2:
Accuracy and
Precision
Prepared by: Ma. Christine L. Velez
Jens Martensson
Learning Competencies:
- Define accuracy and precision.
- Write numbers (small and large using scientific notation.
- Perform operations with scientific notation.
Jens Martensson Every reported measurement of a certain property or characteristic of a material is just a best estimate of the correct value. Let us say, for example, that the length of wooden beam was measured and it was reported to be 220± 5 cm. It means that the correct value lies between 215 and 225 cm, and the limit of uncertainty is ± 5 cm.
Jens Martensson This uncertainty of ±5 cm is a probability limit and not an absolute one. On the other hand, if we say that a measurement is accurate, we are stating the degree to which a measurement corresponds to the real value. When the uncertainty is larger, it is reflection of the experimenter’s belief that the measurement has low degree of accuracy. How do we find the percent of uncertainty of a measurement?
Jens Martensson We may not be able to compare the results to the “true value” which is not known, but we can at least know whether there are large random errors in our measurements or not. Precision depends largely on the way the measurements are taken. Repeated careful measurements help us compare results. If the values are close to one another, we are at least assured that there are no large random errors in the measurement. This quality of results is called precision or repeatability (reproduccibility). We increase the precision of measurement through repeated readings of the same quantity. The degree by which these repeated measurements agree with one another is called the precision of the measurement.
Jens Martensson It is possible that the degree of precision is high but the accuracy is low because of the nature of the measuring instrument used and of the object measured. Precision also limited by the skill of the worker in making estimates or in using the measuring intruments (like viewing the meterstick at the same angle or placing the zero reading at the same point of the object).
Jens Martensson
Scientific Notation
10 Small numbers and large numbers are not easy to handle because we are limited by the system of units that we are to use. Since we use the meter as the standard unit of length, we will get into trouble writing the size of the universe as 100,000,000,000,000,000,000,000, meters or the size of the diameter of hydrogen as 0 meters. In each of these measurements, we need to write a number of zeros either at the end of the nonzero digit for the universe or before the nonzero digit for the size of the diameter of hydrogen.
Jens Martensson
Scientific Notation
These zeros indicate where the decimal point belongs. Physicists and students always encounter this problem of writing or reporting large and small numbers because their explorations are always extended to regions both great and small. Such events are of either extremely long duration such as the age of the Earth (estimated to be 100,000,000,000,000, seconds) or extremely short durations as the light pulse used to measure 0 seconds.
Jens Martensson
Scientific Notation
There is a more convenient way of writing these numbers. We call this scientific notation or the power of-ten notation. Writing numbers in this manner requires us to count the number of zeros immediately before or after the nonzero digits.
Jens Martensson
SAMPLE PROBLEMS
- Let us write 1,200,000 using scientific notation. This can be written in the form 𝑁 = 𝑎× 10 - 𝑤ℎ𝑒𝑟𝑒 𝑎 = a number between 1 and 10 𝑏 = is an integer In this case, 𝑎 = 1. 𝑏 = 6 Thus, the number 𝑁 = 1. 2 × 10 ( . This number is the same as 1. 2 × 1 , 000 , 000 𝑜𝑟 10 ( . Note that 1 is between 1 and 10 and the decimal point is placed after the first nonzero digit 1. On the other hand, number 6 or the power of ten is the number of decimal places the decimal point is moved after 1.
Jens Martensson
SAMPLE PROBLEMS
The number 469,000,000,000 is the same as 4×100,000,000,000 and can be written as N = 4× 10 ,, . The number 96,000,000,000,000,000,000 can be written as 9× 10 ,) . The number 0 is the same as 1×0 and can be written as N = 1× 10 *,, . The number 0 is 2× 10 */ The number 10,000 is 1× 10 + or just 10 + . Note that we do not have to write number 1 anymore since 1 multiplied by 10 + is 10 + . The number 0 is 1× 10 * 0 or just 10 * 0 . We do not have to write number 1 anymore
Jens Martensson
SAMPLE PROBLEMS
The number 0 is written as 5× 10 *) . It is very convenient to use the power-of-ten notation or the scientific notation since measurements now are being carried out in nano terms due to the advent of nanotechnology. To summarize, a number can be written using scientific notation by writing it with the decimal point after the nonzero digit and multiplying it by the appropriate power of ten. A few of these powers of ten are the following: 1 = 10 0 1000 = 10 3 0 = 10
- 1 0 = 10
- 4 10 = 10 1 10,000 = 10 4 0 = 10
- 2 0 = 10
- 5 100 = 10 2 100,000 = 10 5 0 = 10
- 3 0 = 10
- 6
Jens Martensson
Operations on Scientific Notation
If in case the exponents of the numbers to be added or subtracted are not the same, we can first shift the decimal point of some numbers to make their exponents the same. Multiplication and Division Let us do some arithmetic using numbers written in scientific notation. If we are going to multiply 2,100,000 by 3,000, we have to first put the numbers in scientific notation. The problem will look like this (2× 10 ( )(3× 10 / ). All we have to do is multiply 2 and 3 in the usual way and then add the exponents of ten. The answer to this is 6× 10 ) .
Jens Martensson
Operations on Scientific Notation
20 If the exponents are negative like in this one (1× 10 *+ )(4× 10 */ ), we first multiply 1 and 4 in the usual way and then add the negative exponents of ten algebraically. The answer to this problem is 4× 10
- 0 . When we do division in the usual way and then subtract the exponents of ten in the divisor (denominator) form that in the dividend (numerator). Let us do the following example. Divide 4× 10 ) by 9× 10 / . The solution is as follows: +. 2 ×,' ! )×,' "
= 0. 5 × 10
( However, we have to write the answer in the correct form, so we have to shift the decimal point to its proper position which is one place to the right, so that it becomes 5 × 10 2 .
Lesson 2 Accuracy and Precision
Course: Retorika (FILI102)
University: Manuel S. Enverga University Foundation
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