Chi-Square Test (AQA A-Level Geography): Revision Notes

Chi-Square Test

What is the chi-square test?

The chi-square test (written as χ²) is a statistical technique that helps you determine whether differences between observed and expected data are genuine patterns or simply due to chance. This test is particularly useful in geographical fieldwork when you want to compare real-world data you've collected with what theory predicts should happen.

The test works by comparing two sets of data:

• Observed data (O): The actual measurements or counts you've collected during fieldwork or obtained from secondary sources
• Expected data (E): The theoretical values you would expect to find if there were no pattern or if a particular theory were correct

Understanding the components

Observed and expected data

When conducting a chi-square test, you're essentially asking: "Does my real-world data match what I expected to find?"

Observed data represents what you've actually measured or counted in the field. For example, if you're studying beach pebble orientations, the observed data would be the actual number of pebbles pointing in each direction.

Expected data represents what you would anticipate finding based on a theoretical assumption. If pebbles were randomly oriented with no preferred direction, you would expect equal numbers pointing in all directions. This becomes your expected distribution.

Formulating a null hypothesis

Before conducting the test, you must create a null hypothesis. This is a statement that assumes there is no significant difference or relationship in your data.

The alternative hypothesis (which you're actually testing for) would be that there is a significant difference, meaning some geographical factor is influencing the pattern.

The chi-square formula

The chi-square statistic is calculated using this formula:

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

Where:

• $\chi^2$ = the chi-square value (the Greek letter 'chi')
• $\sum$ = sum of (add together all the values)
• $O$ = observed frequency
• $E$ = expected frequency

This formula might look complex, but it follows a logical sequence:

1. Find the difference between each observed and expected value $(O-E)$
2. Square each difference $(O-E)^2$
3. Divide each squared difference by the expected value $\frac{(O-E)^2}{E}$
4. Add all these values together to get your chi-square statistic

Working through a calculation

Let's examine a practical example to understand how the chi-square test works in a geographical context.

Calculating degrees of freedom

Before you can interpret your chi-square value, you need to work out the degrees of freedom. This is a statistical concept that relates to the number of categories in your data.

The formula is simple:

$\text{degrees of freedom} = (n-1)$

Where $n$ is the number of categories (or cells) containing observed data.

Interpreting your results

Using critical values

Once you've calculated your chi-square value and degrees of freedom, you need to determine whether your result is statistically significant. This is done by comparing your calculated value against critical values in a standard statistical table.

The critical values table shows the minimum chi-square value needed for significance at different degrees of freedom and significance levels.

Significance levels

There are two standard significance levels used in geographical studies:

• 0.05 significance level (95% confidence): There is only a 5% (1 in 20) probability that the pattern occurred by chance. This means you can be 95% confident that the pattern is real.

• 0.01 significance level (99% confidence): There is only a 1% (1 in 100) probability that the pattern occurred by chance. This means you can be 99% confident that the pattern is real.

Making your decision

To interpret your results, follow these steps:

1. Look up your degrees of freedom in the left column of the critical values table
2. Read across to find the critical values at both significance levels
3. Compare your calculated chi-square value to these critical values

Decision rules:

• If your calculated value is equal to or greater than the critical value: You can reject the null hypothesis. This means the difference between observed and expected data is statistically significant.

• If your calculated value is less than the critical value: You must accept the null hypothesis. This means any pattern in your data could have occurred by chance.

Applying this to the corrie example

In the corrie orientation study:

• Calculated $\chi^2$ value = 8.6
• Degrees of freedom = 3
• Critical value at 0.05 level (3 degrees of freedom) = 7.82
• Critical value at 0.01 level (3 degrees of freedom) = 11.34

The calculated value (8.6) is greater than 7.82 but less than 11.34. This means:

• The result IS significant at the 0.05 (95%) level
• The result IS NOT significant at the 0.01 (99%) level

Therefore, the null hypothesis can be rejected at the 95% confidence level. The students can conclude with 95% confidence that corries in Snowdonia do show a preferred orientation rather than being randomly oriented.

Important considerations and limitations

When using the chi-square test in your geographical investigations, you must keep several important points in mind:

Sample size requirements

The meaning of the chi-square value

The number you calculate is only meaningful when compared to critical values in statistical tables. The chi-square value itself (such as 8.6 in our example) doesn't directly tell you anything - it must be interpreted using the table.

Appropriate application

You should not apply the chi-square test to more than one set of observed data because the mathematical calculations become too complex and unreliable.

Supporting geographical understanding

Making geographical sense

Above all, your investigation should make geographical sense. Demonstrating statistical ability is less important than showing you understand the geographical significance of your findings. Always explain what your results mean in terms of geographical processes, patterns and theories.