Chapter 3. Probability: Contingency Tables
Interpreting Contingency Tables
Contingency tables
A common way of summarizing the measurements of two categorical variables is with the use of a contingency table.
The contingency table below summarizes the sample data of #200# individuals whose blood was tested in order to determine their blood group and rhesus type:
| A | B | AB | O | Total | |
| Rhesus #+# | #68# | #18# | #6# | #76# | #\blue{168}# |
| Rhesus #-# | #12# | #4# | #2# | #14# | #\blue{32}# |
| Total | #\blue{80}# | #\blue{22}# | #\blue{8}# | #\blue{90}# | #\orange{200}# |
#\phantom{0}#
The row and column totals of a contingency table are located in the margins (edges) of the table and are therefore referred to as #\blue{\textbf{marginal totals}}#.
The total number of observations used to construct a contingency table is called the #\orange{\textbf{grand total}}# and is found in the bottom-right corner of the table.
#\phantom{0}#
The absolute frequencies displayed in a contingency table can be transformed into proportions and in turn interpreted as probabilities.
#\phantom{0}#
Interpreting Proportions as Probabilities
To transform the absolute frequencies of a contingency table into proportions, divide each cell in the table by the total number of observations. The resulting table is commonly referred to as a proportion table.
#\phantom{0}#
| A | B | AB | O | Total | |
| Rhesus #+# | #\purple{0.34}# | #\purple{0.09}# | #\purple{0.03}# | #\purple{0.38}# | #\blue{0.84}# |
| Rhesus #-# | #\purple{0.06}# | #\purple{0.02}# | #\purple{0.01}# | #\purple{0.07}# | #\blue{0.16}# |
| Total | #\blue{0.40}# | #\blue{0.11}# | #\blue{0.04}# | #\blue{0.45}# | #1# |
#\phantom{0}#
The proportions in the margins of a proportion table are called #\blue{\textbf{marginal probabilities}}# and tell us the probability of a single event occurring.
For instance, the probability of randomly selecting a person with blood group B from this sample would be #0.11#:
\[\mathbb{P}(B) = 0.11\]
Similarly, the probability of randomly selecting a person with a positive rhesus factor would be #0.84#:
\[\mathbb{P}(Rh+) = 0.84\]
The proportions located in the center block of a proportion table are called #\purple{\textbf{joint probabilities}}# and tell us the probability of the intersection of two events occurring.
For instance, the probability of randomly selecting a person that has both blood group O and a negative rhesus factor would be #0.07#:
\[\mathbb{P}(O \cap Rh-)=0.07\]
Similarly, the probability of randomly selecting a person that has both blood group A and a positive rhesus factor would be #0.34#:
\[\mathbb{P}(A \cap Rh+) = 0.34\]
| Cat | Dog | Bird | Total | |
| Male | #63# | #45# | #4# | #112# |
| Female | #31# | #54# | #3# | #88# |
| Total | #94# | #99# | #7# | #200# |
| Cat | Dog | Bird | Total | |
| Male | #0.315# | #0.225# | #0.02# | #0.56# |
| Female | #0.155# | #0.27# | #0.015# | #0.44# |
| Total | #0.47# | #0.495# | #0.035# | #1# |
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