Adjustment Theory 1

accuracy: how close the measurement is to the true value

precision: how close repeated measurements are to each other

blunders/gross errors:

single measurements that have a completely different value than the other measurements.

Caused by reading mistakes, misidentification of targets

Eliminated by checking measurements

Systematic errors:

errors that influence the result in the same way, affecting the accuracy of measurements

Caused by poor calibration of instruments, one-sided use of instruments, external influences on the instrument such as temperature or pressure

Emilinated by calibration, proper selection of measuring procedure, mathematical compensation

Random errors:

all remaining unknown errors after elimination of blunders and systematic errors, mainly affects precision; measurements are equally likely to be higher or lower than true value. In mathematical statistics they are considered as independent stochastic variables

Caused by limitations of measuring instruments and human senses, uncontrollable changes of the environment or measured object

Eliminated by taking enough measurements -> will average out

discrete (X) and continous (L).

Discrete random variables can only take on a countable (finite) number of distinct values. Example: Rolling a dice, playing cards, roulette, number of children in a family.

Continous random variables can reflect an uncountable (infinite) number of values within certain boundarties (an interval [a,b]). Example: measuring the table length, heights of students in class, change of temperature during a day

In this course, we use continous random variables.

absolute and relative frequency; the relative frequency gives us the frequency function

The interval [a,b] in which the random variable exists is divided into discrete invervals/bins K with a certain width Δx:

The absolute frequency is then the number of observations k inside of the bins.

The relative frequency is the ratio of the number of observations k in a bin to the total number of observations n.

A frequency function is for a finite number of observations, for the probability function n converges to infinity.

Frequency function:

sometimes also represented as cumulative frequency function:

Probability function:

arithmetic mean/empirical mean:

expectation:

For n -> infinity, arithmetic mean converges to expectation.

Often the expectation is unknown, which is why calculations often are taken w.r.t the mean value.

Important! Expectation is NOT the true value.

Standard deviation means that if the experiment is repeated under the exact same conditions, there is a probability of 68,3% that the result is within the standard deviation. It is therefore a measurement of repeatability.

The degree of freedom is an indicator for redundancy; how many observations too much?

variance: value to how much the numbers in a data set vary from the mean

co-variance: measure of dependency between 2 random variables; for stochastic independent variables (random errors!) covariance equals zero, i.e. sin(x) and cos(x) are NOT independent from each other:

Both can be visualized by the Variance-Covariance-Matrix (VCM).

Mathematical/functional/algrebraic correlation: purely mathematical, i.e. sin(x) and cos(x)

Physical correlation: Correlation due to systematic deviations that are not considered in the functional model

Variance: Value to how much the numbers in a data set vary from the mean

standard deviation: how far apart numbers in a data set are from each other

covariances: measure of dependency between 2 random variables; for stochastic independent values (random errors) the covariance equals 0

correlation: dependencyof 2 values, either mathematical or physical

There is the theoretical and the empirical approach.

For the theoretical approach the expectation must be known. If the probability density function is known, the theoretical variance can be directly computed using the expectation. If the probability density function is not known, the theoretical variance is calculated using the error vector/matrix (matrix of random deviations) which is calculated by subtracting the expectation from the observation value.

For the empirical approach the expectation can be either known or unknown. If it’s known, like for the theoretical variance, the error vector is used. For unknown expectations, which is the general case, instead of the expectation the vector of residuals is used, which is calculated by substracing the observation value from the arithmetic mean.