Here we are discus about Modeling, Error and its measurement.
Please you try to answer the following key question of chapter one
1.Finite precision is decimal representation of a number which has been rounded or truncated. There many cases where this may be necessary or appropriate.
2.
| 35 | 2 | |||||
| -34 | 17 | 2 | ||||
| 1 | -16 | 8 | 2 | |||
| 1 | -8 | 4 | 2 | |||
| 0 | -4 | 2 | 2 | |||
| 0 | -2 | 1 | ||||
| 0 | ||||||
 | ||||||
 | |
| 0 | .625 |
| . | 2 |
| 1 | 25 |
| 2 | |
| 0 | 5 |
| 2 | |
| 1 | 0 |
1 the precision of a numerical quantity is a measure of the detail in which the quantity is expressed. This is usually measured in bits, but sometimes in decimal digits. It is related to precision in mathematics, which describes the number of digits
2 There is no such thing as "human error"! This vague phrase does not describe the source of error clearly. Careful description of sources of error allows future experimenters to improve on your techniques. This long list of common sources of error is meant to help you identify some of the common sources of error you might encounter while doing experiments. If you find yourself stuck for words when describing sources of error, this list may help. The list goes from the common to the obscure.
Incomplete definition (may be systematic or random) - One reason that it is impossible to make exact measurements is that the measurement is not always clearly defined. For example, if two different people measure the length of the same rope, they would probably get different results because each person may stretch the rope with a different tension. The best way to minimize definition errors is to carefully consider and specify the conditions that could affect the measurement.
Failure to account for a factor (usually systematic) - The most challenging part of designing an experiment is trying to control or account for all possible factors except the one independent variable that is being analyzed. For instance, you may inadvertently ignore air resistance when measuring free-fall acceleration, or you may fail to account for the effect of the Earth's magnetic field when measuring the field of a small magnet. The best way to account for these sources of error is to brainstorm with your peers about all the factors that could possibly affect your result. This brainstorm should be done before beginning the experiment so that arrangements can be made to account for the confounding factors before taking data. Sometimes a correction can be applied to a result after taking data, but this is inefficient and not always possible.
Environmental factors (systematic or random) - Be aware of errors introduced by your immediate working environment. You may need to take account for or protect your experiment from vibrations, drafts, changes in temperature, electronic noise or other effects from nearby apparatus.
Instrument resolution (random) - All instruments have finite precision that limits the ability to resolve small measurement differences. For instance, a meter stick cannot distinguish distances to a precision much better than about half of its smallest scale division (0.5 mm in this case). One of the best ways to obtain more precise measurements is to use a null difference method instead of measuring a quantity directly. Null or balance methods involve using instrumentation to measure the difference between two similar quantities, one of which is known very accurately and is adjustable. The adjustable reference quantity is varied until the difference is reduced to zero. The two quantities are then balanced and the magnitude of the unknown quantity can be found by comparison with the reference sample. With this method, problems of source instability are eliminated, and the measuring instrument can be very sensitive and does not even need a scale.
Failure to calibrate or check zero of instrument (systematic) - Whenever possible, the calibration of an instrument should be checked before taking data. If a calibration standard is not available, the accuracy of the instrument should be checked by comparing with another instrument that is at least as precise, or by consulting the technical data provided by the manufacturer. When making a measurement with a micrometer, electronic balance, or an electrical meter, always check the zero reading first. Re-zero the instrument if possible, or measure the displacement of the zero reading from the true zero and correct any measurements accordingly. It is a good idea to check the zero reading throughout the experiment.
Physical variations (random) - It is always wise to obtain multiple measurements over the entire range being investigated. Doing so often reveals variations that might otherwise go undetected. If desired, these variations may be cause for closer examination, or they may be combined to find an average value.
Parallax (systematic or random) - This error can occur whenever there is some distance between the measuring scale and the indicator used to obtain a measurement. If the observer's eye is not squarely aligned with the pointer and scale, the reading may be too high or low (some analog meters have mirrors to help with this alignment).
Instrument drift (systematic) - Most electronic instruments have readings that drift over time. The amount of drift is generally not a concern, but occasionally this source of error can be significant and should be considered.
Lag time and hysteresis (systematic) - Some measuring devices require time to reach equilibrium, and taking a measurement before the instrument is stable will result in a measurement that is generally too low. The most common example is taking temperature readings with a thermometer that has not reached thermal equilibrium with its environment. A similar effect is hysteresis where the instrument readings lag behind and appear to have a "memory" effect as data are taken sequentially moving up or down through a range of values. Hysteresis is most commonly associated with materials that become magnetized when a changing magnetic field is applied.
3 Measurement Error (also called Observational Error) is the difference between a measured quantity and its true value. It includes random error (naturally occurring errors that are to be expected with any experiment) andsystematic error (caused by a mis-calibrated instrument that affects all measurements).
For example, let’s say you were measuring the weights of 100 marathon athletes. The scale you use is one pound off: this is a systematic error that will result in all athletes body weight calculations to be off by a pound. On the other hand, let’s say your scale was accurate. Some athletes might be more dehydrated than others. Some might have wetter (and therefore heavier) clothing or a 2 oz. candy bar in a pocket. These are random errors and are to be expected. In fact, all collected samples will have random errors — they are, for the most part, unavoidable.
5
Precision is a description of random errors, a measure of statistical variability.
Accuracy has two definitions:
In simplest terms, given a set of data points from repeated measurements of the same quantity, the set can be said to be precise if the values are close to each other, while the set can be said to be accurate if their average is close to the true value of the quantity being measured. In the first, more common definition above, the two concepts are independent of each other, so a particular set of data can be said to be either accurate, or precise, or both, or neither.
Precision is a description of random errors, a measure of statistical variability.
Accuracy has two definitions:
In simplest terms, given a set of data points from repeated measurements of the same quantity, the set can be said to be precise if the values are close to each other, while the set can be said to be accurate if their average is close to the true value of the quantity being measured. In the first, more common definition above, the two concepts are independent of each other, so a particular set of data can be said to be either accurate, or precise, or both, or neither.
6 You enter the number: 35.62510 in Decimal number system and want to translate it into Binary.
Converting 35.62510 in Binary system here so:
Whole part of a number is obtained by dividing on the basis new
| 35 | 2 | |||||
| -34 | 17 | 2 | ||||
| 1 | -16 | 8 | 2 | |||
| 1 | -8 | 4 | 2 | |||
| 0 | -4 | 2 | 2 | |||
| 0 | -2 | 1 | ||||
| 0 | ||||||
| ||||||
| |
| 0 | .625 |
| . | 2 |
| 1 | 25 |
| 2 | |
| 0 | 5 |
| 2 | |
| 1 | 0 |
1 is decimal representation of a number which has been rounded or truncated.2, Numerical and non Numerical error
Under numerical there is mathematical modeling & input data
under non numerical there is Truncation error& roundoff error
3,Absolute error& Relative error
absolute error it isthe magnitude of difference between true value & approximate value
Relative error is the ratio of absolute error to the size
4,the process of developing mathematical model is termed Mathematical modeling
5, Accuracy is the closely to the real value
Precision is the value which is close to each other
6, (35.625)10 =(100011.101)2
1Decimal representation of a numerical there is mathematical modeling and input data
2 in simplest terms given a set of data point from repeated
Measurements of the same quantity so A particular set of data
Or precise or both or neither
Rounding off the round off error is used because representi every number as a real number is not possibly
3 absolute error it is the magnitude of difference between true value and approximate value
Relative error is the ratio of absolute error to the size
4 the process of developing mathematical model is termed
5 accuracy is the closely to the real value precision is
The value which is close to each them
6-
1002/10 Muluken Abebe
Answer For Chapter One Forum Questions
1.Finite precision is adecimal representation of anumber which has been rounded or truncated
2. Types of Error
a.Rounding -off
b.Truncation
c.Discretization
d.Numerical
e.Error Benefits
3.Measurement Error (also called observational Error) is the difference between ameasured quantity and its true value.
.It includes Roundom Error (naturally occuring error)
that are to be expected with any experiment.
.Systematic Error(caused by amiscalibrated instrument that affect all measurement).
4.Numerical Modeling is amathematical models that use some sort of numerical time-stepping procedure to obtain.
.The mathematical solution is represented by agenerated table and/ or graph.
5.Accuracy and Precision
.Accuracy refers to the closeness of measured value to astandared or known value.
.Precision refers to the closeness of two or more measurement to each other.
6.Convertong (35.625)10 to Binary
.first converting the integer one by dividing on basis new
(35.625)10 to ( )2
35÷2
3417÷2
1 -168÷2
0 -42÷2
0 -2÷1
0
If the value of integer from down ward to up
(35)10=(100011)2
.The fraction one is
(0.625)10 to ( )2 this one the opposite to the first by multipliying the new basis
0.625×2
125×2
05×2
10
If the value is from up to dowm
(0.625)10=(0.101)2
add together= (100011)2+(0.101)2= (100011.101)2
=(35.625)10=(100011.101)2
CIVIL-A ,ID-074/10
1.Finite precision is decimal representation of a number which has been rounded or truncated. There many cases where this may be necessary or appropriate.
Reminder
35/2= 17 1
17/2=8 1
8/2=4 0
4/2=2 0
2/2=1 0
1/2=0 1
the most significant bit is at the bottom so write the digits from lower to upper =(100011)
The fractional part conversion
Integer part
0.625*2= 1.28 1
0.28*2=0.56 0
0.56*2=1.12 1
the most significant bit is at the top so write the digits from upper to lower =(101)
then combine the integer part conversion & fractional part conversion. The result become (100011.101)2
1, Finite person is decimal represntion of a number of w/c has been rounded or truncated. Ther are mony cases w/hare this may be neccessary or appropriate.
2, Ther are two sorce of error
1, Approximation error
a, Truncation /disctretization error
b, Convergency error
2, Raundoff error
3, Measurement error also called observetion error is the d/ce b/n amesuared quantity and its true vallu it includedes randam error(naturary occuring errore that are to be expected whith any expesurment and systematic errore cuaseded by a miscaiibrate instrument that affects all measurments.
4, Mathematical modeling is the art of describing in symbolic language a real life system so that approximately correct prediction can be mode regarding the behavaour or acuretion or the system under the varied circumstances of interor,
5, Accuary is the characterstic of mesurement and calcultion which is refes of how closely avalue agrees with the true value Precision also the charcterstic of mesurement and cacultion which is refer how closely the values agree is other.
6, Intiger Remender Fraction part Remender
35/2=17 1 625/2=312 1
16/2=8 1 312/2= 156 0
8/2=4 0 156/2=78 0
4/2=2 0 78/2=37 0
2/2=0 0 37/2=18 1
1/2 18/2=9 0
100011. 9/2=4 1
4/2-2 0
2/2=1 0
1/2=1. 1
100011.1000101001 100011.101
0.625*2 =1.29
0.28*2=0.56
0.56*2=1.12 Anser =101