Measurement Theory Accuracy, Precision, Error, Repeatability, Measurement scale Gauri S. Shrestha, M.Optom, FIACLE
Measurement is the acquisition of information about a state or phenomenon (object of measurement)
It can be assessed associating numbers with physical quantities and phenomena
The process is accomplished through the comparison of a measured value with some known quantity (standard) of the same kind.
Quality of measurement
It must be descriptive (observable) with regard to that state or object being measured.
It must be selective : it may only provide information about what we wish to measure.
It must be objective . The outcome of measurement must be independent of an arbitrary observer.
Empirical space Source set S s i States, phenomena Reference:  Image space Abstract, well-defined symbols Image set I i i Transformation
What’s the bottom line
Measurement theory shows that strong assumptions are required for certain statistics to provide meaningful information about reality.
Measurement theory encourages people to think about the meaning of their data.
It encourages critical assessment of the assumptions behind the analysis.
Fundamental theory of measurement
It allows a safe acquirement and reproducibility of measuring characteristics
The object of the theory are (physical) magnitudes/quantities and are quantitative features
The method (mean) of measurement is a comparison.
The aids of measurement are the standards and the scales. The units are not measured, but rather fixed by definition.
The unit of a measurable quantity (the comparison factor = measurement unit) has to be appropriately defined by degree of mathematical representation and by the accuracy, reliability and constancy of its realizations.
Let us define ‘pure’ science as science that has sole purpose of describing the world around us and therefore is responsible for our perception of the world.
In ‘pure’ science, we can form a better, more coherent, and objective picture of the world, based on the information measurement provides.
The information allows us to create models of (parts of) the world and formulate laws and theorems.
We must then determine (again) by measuring whether this models, hypotheses, theorems, and laws are a valid representation of the world.
This is done by performing measurements to compare the theory with reality.
We consider ‘applied’ science as science intended to change the world
It uses the methods, laws, and theorems of ‘pure’ science to modify the world around us.
The purpose of measurements is to regulate, control, or alter the surrounding world, directly or indirectly.
The results of this regulating control can then be tested and compared to the desired results and any further corrections can be made.
checking the tire pressure can be described in the above terms:
1) a hypothesis: we fear that the tire pressure is abnormal;
2) perform measurement; 3) alter the pressure if it was abnormal.
States are only named NOMINAL States can be ordered ORDINAL Distance is meaningful INTERVAL Abs. zero RATIO Abs. unit ABSOLUTE
Two things are assigned the same symbol if they have the same value of the attribute.
Permissible transformations are any one-to-one or many-to-one transformation.
Examples: numbers assigned to religions in alphabetical order, e.g. atheist=1, Buddhist=2, Christian=3, etc.
Gender: male=1, female=2
Things are assigned numbers such that the order of the attribute.
Two things x and y with attribute values a(x) and a(y) are assigned numbers m(x) and m(y) such that if m(x) > m(y), then a(x) > a(y).
Examples: Moh's scale for hardness of minerals; grades for academic performance (A, B, C, ...); rank the students in swimming
Things are assigned numbers such that differences between the numbers reflect differences of the attribute.
If m(x) - m(y) > m(u) - m(v), then a(x) - a(y) > a(u) - a(v).
In the Farenheit temperature scale, the distance between 20 degrees and 40 degrees is the same as the distance between 75 degrees and 95 degrees.
With Interval scales, there is no absolute zero point.
it would not be appropriate to say that 60 degrees is twice as hot as 30 degrees.
Things are assigned numbers such that differences and ratios between the numbers reflect differences and ratios of the attribute
difference between three hours and five hours is the same as the difference between eight hours and ten hours (equal intervals), but we can also say that ten hours is twice as long as five hours (a ratio comparison)
Examples: Length in centimeters; duration in seconds; temperature in degrees Kelvin.
Things are assigned numbers such that ratios between the numbers reflect ratios of the attribute.
If m(x) / m(y) > m(u) / m(v), then
a(x) / a(y) > a(u) / a(v)
Examples: density (mass/volume);
Fuel efficiency in miles per gallon
Things are assigned numbers such that all properties of the numbers reflect analogous properties of the attribute.
The only permissible transformation is the identity transformation.
Examples: number of children.
Relation Among Measurement Scales
These measurement levels form a partial order based on the sets of permissible transformations:
Weaker <-------------------> Stronger
Nominal -- Ordinal - > Ratio -- Absolute
Measurement scale and statistical test
There are certain statistical analyses which are only meaningful for data which are measured at certain measurement scales
it is generally inappropriate to compute the mean for Nominal variables.
Suppose you had 20 subjects, 12 of which were male, and 8 of which were female. If males = '1' and females = '2‘
Is it possible to compute mean value?
Mean sex 1.4?
Measurement scale and statistical test
Examine the relationship or association between two variables, there are also guidelines concerning which statistical tests are appropriate.
the relationship between student gender (a Nominal variable) and major field of study (another Nominal variable). In this case, the most appropriate measure of association between gender and major would be a Chi-Square test .
Measurement scale and statistical test
the relationship between undergraduate major and starting salary of students' first job after graduation. In this case, it is a ratio level variable. The appropriate test of association between undergraduate major and salary would be a one-way analysis of variance (ANOVA), to see if the mean starting salary is related to undergraduate major.
the relationship between undergraduate grade point average and starting salary. In this case, both grade point average and starting salary are ratio level variables. Now, we would look at the relationship between these two variables using the Pearson correlation coefficient
The extent to which the method of measurement reflects the true meaning of a concept being investigated
The external validity depends on how well the variables designed for the study represent the phenomena of interest.
The internal validity depends on how well the actual measurements represent these variables.
Content validity : a scale should measure the true meaning of the concept being studied
example: GCS (General Contentment Scale) should measure depression, not self esteem
face validity (representational validity) : whether a measuring instrument appears to be valid to the persons completing it.
predictive validity : can a measure predict future behavior?
example: Does the Suicide Probability Scale accurately predict which adolescents are likely to attempt suicide?
criterion validity : how well a measure predicts another established criterion
example: Do a person’s reported income predict his or her credit score?
example: “need for cognition” versus “argumentativeness”
construct validity : a measure should fit well with other measures of similar theoretical concepts.
example: scores on a “marital satisfaction” scale should be negatively related to spouse abuse.
The extent to which results are consistent over time and an accurate representation of the total population under study is referred to as reliability
If the results of a study can be reproduced under a similar methodology, then the research instrument is considered to be reliable
Precise measurement is one that has nearly the same value each time it is measured (repeatability)
There are three main sources of random error in making measurements affecting Precise:
Observer variability (choice of words in an interview, or hand-eye coordination in using a mechanical instrument).
Subject variability (biologic variability: fluctuations in mood, circadian rhythms, and time since last medication).
Instrument variability (fluctuating environmental
factors: temperature and background noise).
High precision = small random error
Accuracy and resolution
Accuracy . This is a measure of reliability, and is the difference between the True Value of a measured quantity and the Most Probable Value which has been derived from a series of measures
High accuracy =small systematic error
Resolution. This is the smallest interval measurable by an instrument
XXXXXX Low accuracy high precision X X X X X X X X X X medium accuracy low precision
These may occur at any time, and are caused by carelessness on the part of the observer
e.g. pointing to the wrong light, misreading an instrument, incorrect booking, incorrect computer input, etc.
Blunders will always occur sooner or later, but must never be allowed to occur undetected.
Measurement error- constant error
Errors of constant magnitude and sign,
e.g. standardization error of a tape.
They can sometimes be eliminated, but sometimes they cannot be completely eliminated, e.g. standardization error.
Measurement error- systematic error
Errors of varying magnitude but constant sign,
An error that can be predicted and hence eventually removed from data
The measurement of a weight on a scale with marking in kg is 79 kg, whereas measurement of the same weight on a different scale having further divisions in hectogram is 79.3 kg
It occurs due to
faulty instrument, faulty measuring process and personal bias
Measurement error- instrument error
Examples of measurement equipment systematic errors include calibration errors, input zero drift and gain drift.
instrument error occurs due to faulty design of the instrument.
We can minimize this error by replacing the instrument or by making a change in the design of the instrument
A faulty measuring process may include inappropriate physical environment, procedural mistakes and lack of understanding of the process of measurement.
if we are studying magnetic effect of current, then it would be erroneous to conduct the experiment in a place where strong currents are flowing nearby.
while taking temperature of human body, it is important to know which of the human parts is more representative of body temperature.
A personal bias is introduced by human habits, which are not conducive for accurate measurement.
the habit of reading scales from an inappropriate distance and from an oblique direction. The measurement, therefore, includes error on account of parallax.
Sensor cross sensitivity errors
Few measuring systems respond only to the parameter being measured.
All sensors have a degree of sensitivity to other parameters.
For example a temperature sensor's output may change with pressure, humidity and/or ionizing radiation
Measurement error- periodic error
Errors of varying magnitude and sign, but obeying some systematic law
Being of varying sign, they tend to be reduced or eliminated by repetition of observations under different conditions
Measurement error- random error
These are all the remaining errors, are of varying sign and magnitude, and do not obey a systematic law.
They are usually small, equally likely to be positive or negative, and are numerous
They will always be present in any observations and are caused by imperfections in the observer and instrument, and by varying conditions for the observations
It is not biased.
because of the limitation of the instrument in hand and the limitation on the part of human ability. No human being can repeat an action in exactly the same manner
Some more errors
Quantizing error: All measuring equipment has a resolution limit, input variations below which can not be detected or measured, leading to unrecoverable information loss.
Rounding and truncation errors: In processing the measuring system's readings, the precision of calculation (number of significant digits) can compromise results.
Sampling errors: The frequency and sample window time can impact accuracy, especially for changing or noisy quantity.