Degrees of freedom
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Degrees of Freedom
Equations are omitted for technical reasons - download the original pdf
Imagine I throw two balls in the air [Diagram goes here &- download the original to see it.] I wish to describe the position of each ball. The balls are travelling independently of one another so I require three coordinates for each ball&- that is six coordinates in all. Now imagine that I attach a rod between the balls. [Diagram goes here &- download the original to see it.] This rod acts as a constraint on the system and reduces the freedom of the balls to move independently of one another, the position of one of the balls is fixed. [Diagram goes here &- download the original to see it.] Then the position of the second ball is such that it must lie on a sphere whose centre is the position of the first ball. So I no longer require three coordinates to describe the position of the second ball &- two will do. The constraint has caused a reduction in the total degrees of the system by one. In statistics there are frequent instances of where a constraint causes a reduction of the number of degrees of freedom of a system Suppose I take a sample of size 100 and construct a grouped frequency table with six intervals. [Table goes here &- download the original to see it.] As we fill up the table the frequency [Equation goes here &- download the original to see it.] does not constrain [Equation goes here &- download the original to see it.]to any particular value, but if we have values for [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] then the constraint that the total frequency is 100. [Equation goes here &- download the original to see it.] means that the value of [Equation goes here &- download the original to see it.] is determined by the other five values. [Equation goes here &- download the original to see it.] So with this grouped frequency table with six columns has only five degrees of freedom, since the requirement that the total frequency = Sample size reduces the degrees of freedom by one. In general a frequency table with n rows or columns will have n&-1 degrees of freedom.
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Contents of
Degrees of freedom
1 Degrees of Freedom
2 Degrees of freedom in a contingency table
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