Chi-squared value

The chi-squared value tells us how different the data we observed is from what we would expect given the column totals and row totals.

A high chi-squared value means that the observed data is very different from what we would expect. Though we do need to consider the degrees of freedom - a high chi-squared value might not necessarily be significant if the degrees of freedom are also high.

Calculating chi-squared for a single cell

If we want to calculate the chi-squared value for a single cell, and we know that the observed value is O and the expected value is E, then we can calculate the chi-squared value for that cell using the formula:

\chi^2 = \frac{(O - E)^2}{E}

Calculating chi-squared for a whole table

We can just do the same formula above (\chi^2 = \frac{(O - E)^2}{E}) for each cell in the table, and then add up all of those values to get the total chi-squared value for the whole table.

If we say that O_r represents the observed value for any cell indexed by r (just a way of saying ‘for any cell’), and that E_r represents the expected value for that cell, then we can write the formula for calculating the chi-squared value for the whole table as:

\chi^2 = \sum_{r} \frac{(O_r - E_r)^2}{E_r}

Example

Let’s take our table from the expected contingency frequency note.

Observed values

Age Group	Pop	Instrumental	Total
Under 18	30	10	40
18-35	50	20	70
35+	20	30	50
Total	100	60	160

Expected values

Age Group	Pop	Instrumental	Total
Under 18	25	15	40
18-35	43.75	26.25	70
35+	31.25	18.75	50
Total	100	60	160

Calculating the chi-squared value

Age Group	Pop	Instrumental
Under 18	\frac{(30 - 25)^2}{25}	\frac{(10 - 15)^2}{15}
18-35	\frac{(50 - 43.75)^2}{43.75}	\frac{(20 - 26.25)^2}{26.25}
35+	\frac{(20 - 31.25)^2}{31.25}	\frac{(30 - 18.75)^2}{18.75}

Calculating those values gives us:

Age Group	Pop	Instrumental
Under 18	1	1.67
18-35	0.89	1.51
35+	4.05	6.75

Adding up all of those values gives us the total chi-squared value for the table:

\chi^2 = 1 + 1.67 + 0.89 + 1.51 + 4.05 + 6.75 = 15.87

We can then compare this to a significance level, considering the degrees of freedom and the chi squared distribution, to determine whether this chi-squared value is significant or not.

flashcards

Question	Answer
What does the chi-squared value tell us about observed data?	It tells us how different the observed data is from what we would expect given the column totals and row totals.
What does a high chi-squared value indicate?	A high chi-squared value means that the observed data is very different from what we would expect.
What factor must be considered when interpreting a high chi-squared value?	The degrees of freedom must be considered; a high chi-squared value might not be significant if the degrees of freedom are also high.
What is the formula for calculating the chi-squared value for a single cell?	\chi^2 = \frac{(O - E)^2}{E}
How do you calculate the chi-squared value for a whole contingency table?	Apply the formula \chi^2 = \frac{(O - E)^2}{E} to each cell and sum all the resulting values.
Give the summation notation formula for calculating the chi-squared value for a whole table.	\chi^2 = \sum_{r} \frac{(O_r - E_r)^2}{E_r}
In a study of age groups and music preference, 30 people under 18 preferred Pop (observed) versus an expected value of 25. What is the chi-squared contribution for this cell?	\frac{(30 - 25)^2}{25} = 1
In a study of age groups and music preference, 10 people under 18 preferred Instrumental (observed) versus an expected value of 15. What is the chi-squared contribution for this cell?	\frac{(10 - 15)^2}{15} \approx 1.67
In a study of age groups and music preference, 50 people aged 18-35 preferred Pop (observed) versus an expected value of 43.75. What is the chi-squared contribution for this cell?	\frac{(50 - 43.75)^2}{43.75} \approx 0.89
In a study of age groups and music preference, 20 people aged 18-35 preferred Instrumental (observed) versus an expected value of 26.25. What is the chi-squared contribution for this cell?	\frac{(20 - 26.25)^2}{26.25} \approx 1.51
In a study of age groups and music preference, 20 people aged 35+ preferred Pop (observed) versus an expected value of 31.25. What is the chi-squared contribution for this cell?	\frac{(20 - 31.25)^2}{31.25} \approx 4.05
In a study of age groups and music preference, 30 people aged 35+ preferred Instrumental (observed) versus an expected value of 18.75. What is the chi-squared contribution for this cell?	\frac{(30 - 18.75)^2}{18.75} \approx 6.75
What is the total chi-squared value for the example table with observed and expected frequencies?	\chi^2 = 1 + 1.67 + 0.89 + 1.51 + 4.05 + 6.75 = 15.87
After calculating a chi-squared value of 15.87, what must be done to determine if it is significant?	Compare it to a significance level, considering the degrees of freedom and the chi-squared distribution.