[PPT2] Errors and Statics Flashcards
MEASUREMENT
- Used alternatively with observation
- Observations made to determine unknown
quantities - A numerical value as a result of some physical
operations such as preparation, pointing, matching, & comparing - May be classified as:
– Direct: to determine distances and angles
– Indirect: to compute for the station coordinates
ERROR AND CORRECTION
Error - refers to the difference between a given measurement and the “true” or “exact” value of the measured quantity
Correction - the negative of error
MEASUREMENT AND ERROR
It can be stated unconditionally that:
* no measurement is exact,
* every measurement contains errors
* the true value of a measurement is never known, thus
* the exact sizes of the errors present are always unknown
REPEATED MEASUREMENTS
PROBLEM: Variability
* An inherent quality of physical properties
* Must be accepted as a basic property of observations
* Measurements are numerical values for random variables which are subject to statistical fluctuations
* Statistical variations are due to observational errors
SOURCES OF ERRORS
(NIP)
1. Natural errors – Caused by variations in the phenomena of nature such as changes in magnetic declination, temperature, refraction, atmospheric pressure, etc.
2. Instrumental errors – Due to imperfections in the instruments used, either from faults in construction or improper adjustments (e.g., divisions not uniformly spaced)
3. Personal errors – Arise due to limitations of the human senses (e.g., ability to read a micrometer or to center a level bubble). Magnitude are affected by the personal ability to see and by manual dexterity
TYPES OF ERRORS
(MSR)
- Mistake or Blunders
- Systematic Errors
- Random Errors
1. MISTAKE OR BLUNDERS
- Actually not errors because they are usually so
gross in magnitude compared to the other types of errors - One of the most common reasons is simple carelessness on the part of the observer
- An observation with a mistake is not useful unless the mistake is removed
COMMON MISTAKES
(ROTRMIS)
- Reading the wrong graduation on the tape
- Omitting a whole length of tape
- Transposition of figures
- Reading a scale backward
- Misplacing a decimal point
- Incorrect recording of field notes
- Sighting the wrong target
2. SYSTEMATIC ERROR
- So called because they occur according to some deterministic system which, when known, can be expressed by some functional relationship
- Also called a cumulative error
TYPES OF SYSTEMATIC ERROR
a. Constant Error – if its magnitude and sign remains the same throughout the measuring process. e.g. tape “too short” or “too long”
b. Counteracting – if its sign changes while its magnitude remains the same. Perhaps due to personal bias of the observer
3. RANDOM ERROR
- This variation results from observational errors which have no known functional relationship based upon a deterministic system.
- These errors which exhibits random behavior must be treated accordingly.
- Whereas systematic variations are dealt with mathematically using functional relationships or models, random variables must use probability models
CONCETS AND TOPICS
IN STATISTICS
A. General Uses of Statistics
B. Precision versus Accuracy
C. The Concept of Probability
D. Measures of Central Tendency
E. Sample Statistics for Dispersion
F. Measures of Quality
A. GENERAL USES OF STATISTICS
- Statistics aids in decision making
– Provides comparison
– Explains action that has taken place
– Justifies a claim or assertion
– Predicts future outcome
– Estimates unknown quantities - Statistics summarizes data for public use
B. PRECISION AND ACCURACY
Precision - degree of refinement with which a quantity is measured
Accuracy - denotes how close a given measurement is
to the true value of the quantity
C. THE CONCEPT OF PROBABILITY
(RPR)
Random Event – is one whose relative frequency of
occurrence approaches a stable limit as the number of observations/repetitions of an experiment is increased to infinity.
Probability – is the likelihood associated with a random event.
Random Variable – defined as a variable that takes on any of several possible values, with each of which is associated a probability
REPRESENTATIONS OF THE PROBABILITY DENSITY
Frequency Diagrams
* Histogram – constructed to represent the probability density of a single random variable. Bar graphs that show the frequency distributions in the data.
* Stereogram – constructed to represent the probability density of two random variables.
MEASURES OF CENTRAL TENDENCY
- Median
- Mean
- Mode
- Midrange
1. SAMPLE MEDIAN
- positional middle of the arrayed data.
Characteristics:
* Affected by the position of each item but not by the value of each item.
* A stable measure of central tendency.
* For even-numbered data set, median is the average of the 2 items in the middle.
* The number of observations larger than the median equals the number smaller than the median.
2. SAMPLE MEAN
- Sum of all the values of the observations divided by the number of observations
- Most Probable Value; robust and more manipulable
Characteristics:
* Most familiar measure of central tendency used.
* Affected by the value of every observation.
* In particular, it is strongly influenced by extreme values.
* Since it is a calculated number, it may not be an actual number in the data set
3. SAMPLE MODE
- Value that occurs most frequently in the sample
Characteristics:
* Not always exist. If it does, it may not be unique (2 or more sample modes).
* Not affected by extreme values
* Easiest to compute
4. MIDRANGE
- value of observation that is midway along the range
- arithmetic mean of the largest and smallest observations
E. SAMPLE STATISTICS FOR DISPERSION
(RMVS)
1. Range
– the total spread of the sample
– Largest value-Smallest value
2. Mean Deviation
– arithmetic mean of the absolute values of the deviation from any measure of position (usually the mean).
3. Variance
– parameter of dispersion or spread
4. Standard Deviation
– defined as the positive square root of the variance
F. MEASURES OF QUALITY
- Weight – defined as the quantity that is inversely proportional to variance.
- Relative Precision – refers to the ratio of the measure of precision to the quantity being measured or estimated.
- Mean Square Error – used as a measure of accuracy
I. RESIDUAL
- Sometimes called the deviation
- Defined as the difference between any measured quantity and its most probable value
RESIDUAL VS. ERROR
- Error is frequently used when residual is meant
- Residual has the same sense as the “correction” which is the negative of error
- Error is a hypothetical concept while residual does have value in the reduction of redundant observational data
Error
* Negative of correction
* Indeterminate
* Theoretical concept
* When population is used
Residual
* Same sense as the correction
* Takes a particular value
* Not a hypothetical concept
* When sample set is used
II. PROBABLE ERROR
- A quantity which, when added to and subtracted from the MPV, defines a range within which there is 50% chance that the true value lies inside (or outside) the limits thus, set
- A logical estimate based upon:
– methods and equipment used
– experience of the observers
– field conditions existing during the measurement
III. INTER-RELATIONSHIP OF ERRORS
- Since all quantities measured directly contain errors, any values computed from them will also contain errors
- Error propagation
– Intrusion (propagation) of errors that occur in quantities computed from direct measurements
– Examples include the sum and product of the probable errors
IV. RELATIVE PRECISION
- Expressed by a fraction having the magnitude of the error in the numerator and the magnitude of a measured quantity in the denominator
- Used to define degree of refinement obtained
- Both quantities should be in the same units and the numerator is reduced to 1 to provide easy comparison with other measurements
V. WEIGHTED OBSERVATIONS
- Surveying data must usually conform to a given set of geometric conditions
- If not, measurements are adjusted to force that geometric closure
- Weight of an observation is a measure of its relative worth compared to other measurements
- For a set of uncorrelated observations,
– ↑ Precision, ↓ Variance, ↓ correction
– i.e., weights are inversely proportional to variances
– Correction sizes should be inversely proportional to weights - Weights can also be assigned using:
– Judgement of the surveyor (a priori)
– number of measurements taken for a particular quantity
– the assumption that it is inversely proportional to the square of the probable error
– Inverse of lengths and number of setups for differential leveling lines
