Calculating Chi-Squared


\( 2 \times 2 \) Contingency Table

The following shows the results of a survey of a sample of \( 200 \) randomly chosen adults classified according to gender and sport. These are called observed values or observed frequencies.
$$ \begin{array}{|c|c|c|c|} \hline
& \text{loves sport} & \text{hates sport} & \text{sum} \\ \hline
\text{male} & 72 & 48 & 120 \\ \hline
\text{female} & 18 & 62 & 80 \\ \hline
\text{sum} & 90 & 110 & 200 \\ \hline
\end{array} $$

Expected Frequency Table

$$ \begin{array}{|c|c|c|c|} \hline
& \text{loves sport} & \text{hates sport} & \text{sum} \\ \hline
\text{male} & \dfrac{120 \times 90}{200} = 54 & \dfrac{120 \times 110}{200} = 66 & 120 \\ \hline
\text{female} & \dfrac{80 \times 90}{200} = 36 & \dfrac{80 \times 110}{200} = 44 & 80 \\ \hline
\text{sum} & 90 & 110 & 200 \\ \hline
\end{array} $$

Calculating \( \chi^{2} \)

$$ \chi^{2} = \sum \dfrac{(f_o – f_e)^2}{f_e} $$

\( f_o \) is an observed frequency
\( f_e \) is an expected frequency

$$ \begin{array}{|c|c|r|c|c|} \hline
f_o & f_e & f_o – f_e & (f_o – f_e)^2 & \dfrac{(f_o – f_e)^2}{f_e} \\ \hline
72 & 54 & 18 & 324 & 6.0 \\ \hline
48 & 66 & -18 & 324 & 4.9 \\ \hline
18 & 36 & -18 & 324 & 9.0 \\ \hline
62 & 44 & 18 & 324 & 7.4 \\ \hline
&&& \text{sum} & 27.3 \\ \hline
\end{array} \\
\therefore \chi^2 = 27.3
$$

\( 2 \times 3 \) Contingency Table

The following shows the results of a survey of a sample of \( 500 \) randomly chosen adults classified according to gender and political preferences. These are called observed values or observed frequencies.
$$ \begin{array}{|c|c|c|c|c|} \hline
& \text{Liberal} & \text{neutral} & \text{Democrates} & \text{sum} \\ \hline
\text{male} & 120 & 40 & 90 & 260 \\ \hline
\text{female} & 105 & 50 & 95 & 240 \\ \hline
\text{sum} & 225 & 90 & 185 & 500 \\ \hline
\end{array} $$

Expected Frequency Table

$$ \begin{array}{|r|r|r|r|r|} \hline
& \text{Liberal} & \text{neutral} & \text{Democrats} & \text{sum} \\ \hline
\text{male} & \dfrac{260 \times 225}{500} = 117 & \dfrac{260 \times 90}{500} = 46.8 & \dfrac{260 \times 185}{500} = 96.2 & 260 \\ \hline
\text{female} & \dfrac{240 \times 225}{500} = 108 & \dfrac{240 \times 90}{500} = 43.2 & \dfrac{240 \times 185}{500} = 88.8 & 240 \\ \hline
\text{sum} & 225 & 90 & 185 & 500 \\ \hline
\end{array} $$

Calculating \( \chi^{2} \)

$$ \chi^{2} = \sum \dfrac{(f_o – f_e)^2}{f_e} $$

\( f_o \) is an observed frequency
\( f_e \) is an expected frequency

$$ \begin{array}{|r|r|r|r|r|} \hline
f_o & f_e & f_o – f_e & (f_o – f_e)^2 & \dfrac{(f_o – f_e)^2}{f_e} \\ \hline
120 & 117.0 & 3.0 & 9.00 & 0.0769 \\ \hline
40 & 46.8 & -6.8 & 46.24 & 0.9880 \\ \hline
100 & 96.2 & 3.8 & 14.44 & 0.1501 \\ \hline
105 & 108.0 & -3.0 & 9.00 & 0.0833 \\ \hline
50 & 43.2 & 6.8 & 46.24 & 1.0704 \\ \hline
85 & 88.8 & -3.8 & 14.44 & 0.1626 \\ \hline
&&& \text{sum} & 2.5314 \\ \hline
\end{array} \\
\therefore \chi^2 = 2.5314
$$


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