**Analysis of variance**

**Analysis of variance**

If at least one component of a trial result is randomly distributed, then a statistical model can be formulated, e.g. as follows:

*x*_{i}* **= **m **+ e*_{i} ,

where

*x*_{i}* *is the value at observation *i*

*m* the expectation value

*e*_{i} the random deviation from the expectation value at observation *i*.

The component *e*_{i} is assumed, for example, to be Normally distributed with expectation value = 0 and standard deviation = *s*.

The above can be a model of repeated measurements of a test object under repeatability conditions (see ISO 5725, part 1), that is, measurements made during a short period of time at the same laboratory and operator, instrument and other equipment. If one or more of these conditions are varied, then often the scatter will increase. If measurements were to be made on different days, it is possible that the expectation value varies from day to day. If one or more of these conditions vary, then the total scatter is often going to increase. This situation could be described with the model

*x*_{ij}* **= **m + **d*_{i}* **+ e*_{ij} ,

where

*x*_{ij}* *is the value of observation *j* on day *i*

*m* the overall expectation value

*d*_{i}* *the difference between the expectation value of day *i* and the overall expectation value

*e*_{ij} the difference between observation *j* day *i* and the expectation value of day *i*.

Here it is assumed that both components *d*_{i}* *and *e*_{ij} vary randomly with expectation values = 0 and standard deviations *s*_{d} and *s* respectively. If more identifiable conditions are varied, then additional error components can be introduced in the model. In order to estimate the standard deviations of these components and to test whether they are non-zero, a statistical method is used called variance analysis (ISO5725, parts 2 and 3).

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