Consistency
If the class frequencies are observed in a certain sample data and all class frequencies are recorded correctly then there will be no error among them and they will be called consistent. But sometimes the class frequencies are not recorded correctly and their column total and row total do not agree with the grand total. If there is an error in any class frequency, then we say that the frequencies are inconsistent. If one class frequency is wrong, it will affect other frequencies as well. A simple test of consistency is that all frequencies should be positive. If any frequency is negative, it means that there is inconsistency in the sample data. If the data is consistent, all the ultimate class frequencies will be positive.
Example:
Given the frequencies $$n = 115,{\text{ }}\left( B \right) = 45,{\text{ }}\left( A \right) = 50$$ and $$\left( {AB} \right) = 50$$, check for the consistency of the data.
Solution:
The data is called consistent if all the ultimate class frequencies are positive. Let us calculate some frequencies of order two:
We know $$\left( A \right) = \left( {AB} \right) + \left( {A\beta } \right)$$
Here $$\left( A \right) = 50$$ and $$\left( {AB} \right) = 50$$
Thus $$50 = 50 + \left( {A\beta } \right)$$ or $$\left( {A\beta } \right) = 0$$
It does not include inconsistency because some frequencies can be zero.
We know $$\left( B \right) = \left( {AB} \right) + \left( {\alpha B} \right)$$
$$45 = 50 + \left( {\alpha B} \right)$$ or $$\left( {\alpha B} \right) = – 5$$
The data is consistent, which means the given frequencies are wrong. If we make a table of (2 x 2), we get
$$A$$

$$\alpha $$

Total


$$B$$

$$\left( {AB} \right) = 50$$

$$\left( {\alpha B} \right) = – 5$$

$$\left( B \right) = 45$$

$$\beta $$

$$\left( {A\beta } \right) = 0$$

$$\left( {\alpha \beta } \right) = 70$$

$$\left( \beta \right) = 70$$

Total

$$\left( A \right) = 50$$

$$\left( \alpha \right) = 65$$

$$n = 115$$

One frequency $$\left( {\alpha B} \right)$$ is negative in the table. Thus the sample data is inconsistent.