Inferential testing: Wilcoxon, Spearman and chi-squared (List B)

Welcome to Inferential Testing!

So, you’ve finished your psychology experiment or observation. You have a pile of numbers (data), and you’ve calculated the mean or drawn a graph. But now comes the big question: Are these results actually meaningful, or did they just happen by "fluke" or "chance"?

This is where inferential testing comes in. These tests allow psychologists to make an "inference" (a logical conclusion) about the wider population based on their small sample. In this chapter, we will look at three specific tests you need to know for your IAS Psychology exam: Wilcoxon, Spearman’s Rank, and Chi-squared.

Don’t worry if math isn’t your favorite subject! The exam provides the formulas and the critical value tables for you. Your job is to understand when to use them and how to read the results.

1. The Basics: Probability and Significance

Before we look at the tests, we need to understand the "rules of the game." In psychology, we use probability to decide if our results are significant.

Probability (\(p\))

Probability is written as a decimal. In most psychology research, we use the 5% significance level. This is written as:
\(p \leq .05\)

This means there is a 5% (or less) probability that our results happened by chance. If \(p \leq .05\), we can be 95% confident that our results show a real effect! Occasionally, if researchers want to be even stricter, they might use \(p \leq .01\) (a 1% chance of error).

Observed vs. Critical Values

When you run a statistical test, you get a number called the observed value (also called the calculated value). To see if this number is "significant," you compare it to a critical value found in a statistical table (provided in your exam).

One-tailed or Two-tailed?

This depends on your hypothesis:

• One-tailed: Used for directional hypotheses (e.g., "Students will remember more words in the morning than evening").
• Two-tailed: Used for non-directional hypotheses (e.g., "There will be a difference in the number of words remembered").

Quick Review: To find the right critical value, you need to know three things: 1. Your significance level (usually \(p \leq .05\)), 2. Whether your test is one-tailed or two-tailed, and 3. Your sample size (\(n\)).

2. Spearman’s Rank Correlation Coefficient

When to use it: Use this when you are looking for a relationship or correlation between two co-variables.
Practical Link: You use this for your Topic C (Biological Psychology) practical investigation (e.g., a correlation between two variables like hours of sleep and aggression levels).

The Formula

You don't need to memorize this, but you should recognize it:
\(rho = 1 - \frac{6 \times \sum d^2}{n \times (n^2 - 1)}\)

The Rule for Significance

For Spearman’s Rank, the rule is simple: The observed (calculated) value must be EQUAL TO or GREATER than the critical value to be significant.

Memory Trick: Spearman has a "G" in it (if you stretch your imagination!) for Greater. If your result is bigger than the table value, you've found a significant relationship!

3. Wilcoxon Signed Ranks Test

When to use it: Use this when you are looking for a difference between two sets of data from the same people (Repeated Measures design) or Matched Pairs. The data must be at least ordinal level (data that can be put in order/ranked).
Practical Link: You use this for your Topic B (Cognitive Psychology) practical investigation (e.g., testing the same people's memory with and without interference).

How it works

The test looks at the difference between the scores in Condition A and Condition B, ranks those differences, and gives you a value (represented as \(W\) or \(T\)).

The Rule for Significance

For the Wilcoxon test, the observed value must be LESS THAN or EQUAL TO the critical value to be significant.

Common Mistake: Students often think "bigger is better." For Wilcoxon, it's the opposite! A smaller observed value is usually more likely to be significant.

4. Chi-Squared (\(\chi^2\))

When to use it: Use this when you are looking for an association or difference between categories (Nominal data). This is data where you are simply counting frequencies (e.g., how many people said "yes" vs "no").
Practical Link: You use this for your Topic D (Learning Theories) practical investigation (e.g., observing if boys or girls are more likely to imitate a behavior).

The Formula

\(\chi^2 = \sum \frac{(o - e)^2}{e}\)
(\(o\) = observed frequency, \(e\) = expected frequency)

Degrees of Freedom (\(df\))

To use the Chi-squared table, you don't use \(n\) (sample size). Instead, you calculate Degrees of Freedom using this formula:
\(df = (r - 1) \times (c - 1)\)
(\(r\) is the number of rows and \(c\) is the number of columns in your data table).

Key Takeaway: Chi-squared is for categories. If your data is in a tally chart or a \(2 \times 2\) table, Chi-squared is your best friend!

5. Type I and Type II Errors

Sometimes, statistics can lead us to the wrong conclusion. There are two types of "mistakes" we might make:

Type I Error (The "Optimist's Error")

This happens when we reject the null hypothesis and say our results are significant, but they actually happened by chance. It’s a "false positive." This usually happens if our significance level is too lenient (e.g., using \(p \leq .10\)).

Type II Error (The "Pessimist's Error")

This happens when we accept the null hypothesis and say our results were just chance, but there actually was a real effect. It’s a "false negative." This often happens if our significance level is too strict (e.g., using \(p \leq .01\)) or our sample size is too small.

Summary Table: Which test should I use?

Use this table to help you decide which test fits a scenario in the exam:

Final Tip for the Exam: If a question asks you to "sense check" data, look at the raw numbers first. If the mean scores are very similar or the scatter diagram is a mess, a "significant" result might be a sign of a Type I error!