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.