Interpreting Contingency Tables

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	$\blue{80}$	$\blue{22}$	$\blue{8}$	$\blue{90}$	$\orange{200}$

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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.

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The absolute frequencies displayed in a contingency table can be transformed into proportions and in turn interpreted as probabilities.
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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.
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	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	$\blue{0.40}$	$\blue{0.11}$	$\blue{0.04}$	$\blue{0.45}$	$1$

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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 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 a positive rhesus factor would be $0.34$ :
$\mathbb{P}(A \cap Rh+) = 0.34$

The following contingency table displays the sample data of $200$ men and women who were interviewed about their pet preference:

	Cat	Dog	Bird	Total
Male	$72$	$40$	$2$	$114$
Female	$30$	$50$	$6$	$86$
Total	$102$	$90$	$8$	$200$

Transform the absolute frequencies in the table into proportions. Round your answers to $3$ decimal places.

To transform the absolute frequencies into proportions, divide each cell in the table by the sample size $n=200$ :

	Cat	Dog	Bird	Total
Male	$0.36$	$0.2$	$0.01$	$0.57$
Female	$0.15$	$0.25$	$0.03$	$0.43$
Total	$0.51$	$0.45$	$0.04$	$1$

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