Test of Association: Chi-squared Revision Notes for AQA A-Level Psychology

Test of Association: Chi-squared

What is the chi-squared test?

The chi-squared test is a statistical test that examines whether there is a difference or association between variables. It works specifically with nominal data (categorical data) that is recorded as frequency counts in different categories. The test uses an unrelated design, meaning it compares independent groups of participants.

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The chi-squared test is particularly valuable in psychology and social sciences because it allows researchers to analyse categorical data - such as gender, age groups, or yes/no responses - which are common in behavioural research.

When to use chi-squared

Use chi-squared when you have:

Nominal data presented as frequency counts
Independent groups (unrelated design)
Data organised in categories rather than continuous measurements
At least 2 × 2 categories (though larger contingency tables like 3 × 2 or 3 × 3 are possible)

chatImportant

The key requirement is that each data point represents one person who can only be placed in one category of the contingency table. This ensures data independence, which is crucial for the validity of the chi-squared test.

Setting up hypotheses

Null hypothesis: There is no difference between the groups - any observed differences are due to chance.

Alternative hypothesis: There is a difference between the groups - the observed pattern is unlikely to occur by chance alone.

The alternative hypothesis can be directional (one-tailed) if you predict the direction of difference, or non-directional (two-tailed) if you simply predict a difference exists.

Step-by-step procedure

Step 1: Create a contingency table

Draw a table showing the observed frequencies - the actual data collected in each category. Calculate totals for each row, column, and the overall total.

For example, in a 2 × 2 table examining age groups and ability:

Rows represent one variable (e.g., age: 5-year-olds vs 8-year-olds)
Columns represent another variable (e.g., ability: can decentre vs cannot decentre)
Each cell contains the frequency count for that combination

Step 2: Calculate expected frequencies

Expected frequencies represent what you would expect to find in each cell if there was no difference between groups. Calculate for each cell using:

$\text{Expected frequency} = \frac{\text{Row total} \times \text{Column total}}{\text{Overall total}}$

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Expected frequencies are theoretical values based on the assumption that there is no association between the variables. They provide a baseline for comparison with the observed frequencies.

Step 3: Calculate the chi-squared value

Use the formula: $\chi^2 = \sum \frac{(O-E)^2}{E}$

Where:

$O$ = observed frequency in each cell
$E$ = expected frequency in each cell
$\sum$ = sum of all the $(O-E)^2/E$ calculations

Work through each cell:

Calculate $(O - E)$ for each cell
Square this difference: $(O - E)^2$
Divide by the expected frequency: $\frac{(O - E)^2}{E}$
Add up all these values to get your chi-squared statistic

Step 4: Compare with critical value

Calculate degrees of freedom: $df = (\text{number of rows} - 1) \times (\text{number of columns} - 1)$
Choose your significance level (typically $p = 0.05$ )
Find the critical value from the chi-squared table using your df and significance level
Compare your calculated chi-squared value with the critical value

Interpreting results

If your calculated chi-squared value is greater than the critical value:

Reject the null hypothesis
Accept the alternative hypothesis
The difference is statistically significant ( $p < 0.05$ )

If your calculated chi-squared value is less than or equal to the critical value:

Accept the null hypothesis
The difference is not statistically significant ( $p > 0.05$ )

infoNote

Statistical significance doesn't necessarily mean practical significance. Always consider the real-world importance of your findings alongside the statistical results.

Worked example

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Worked Example: Age and Decentring Ability

A study examined whether age affects children's ability to decentre (see the world from another's perspective). Two age groups were tested: 5-year-olds and 8-year-olds.

Step 1: Observed frequencies

5-year-olds who could decentre: 6
5-year-olds who could not decentre: 27
8-year-olds who could decentre: 28
8-year-olds who could not decentre: 9

Step 2: Calculate expected frequencies and apply chi-squared formula After calculating expected frequencies for each cell and applying the formula $\chi^2 = \sum \frac{(O-E)^2}{E}$ , the calculated value was 23.1.

Step 3: Compare with critical value With $df = 1$ and $\alpha = 0.05$ , the critical value is 3.84.

Step 4: Conclusion Since $23.1 > 3.84$ , the result is significant ( $p < 0.05$ ). This suggests there is a significant difference in decentring ability between 5-year-olds and 8-year-olds.

bookmarkSummary

Key Points to Remember:

Chi-squared tests associations between categorical variables using frequency data
Data must be independent - each person appears in only one cell
Expected frequencies show what would happen if there was no difference between groups
The chi-squared statistic compares observed vs expected frequencies across all cells
Results are significant when the calculated value exceeds the critical value from statistical tables
Always check degrees of freedom and significance level when interpreting results

Test of Association: Chi-squared (AQA A-Level Psychology): Revision Notes