Introduction to Inferential Statistics
In your research journey, you've already learned how to collect data and describe it using averages (like the mean) or graphs. But how do we know if our results actually mean something? Did our IV (Independent Variable) really cause a change in the DV (Dependent Variable), or was it just down to luck?
Inferential statistics are the tools psychologists use to answer that question. These tests allow us to make "inferences" (logical guesses) about the wider population based on the small sample we studied. They help us decide whether to accept our alternative hypothesis or stick with the null hypothesis.
Think of it like a courtroom: the null hypothesis says the defendant is innocent (nothing happened). Inferential statistics are the evidence we use to see if we can "convict" the null hypothesis and say something interesting actually occurred!
1. Probability and Significance
In psychology, we can never be 100% certain about anything. Instead, we talk about probability (the likelihood of something happening).
Significance Levels
The standard level of significance used in psychology is \(p < 0.05\). This means there is a less than 5% probability that our results happened by fluke or random chance. If our results meet this standard, we say they are statistically significant.
Key Symbols to Know:
- \(=\) Equal to
- \(<\) Less than
- \(<<\) Much less than
- \(>\) Greater than
- \(>>\) Much greater than
- \(\propto\) Proportional to
- \(\approx\) Approximately equal to
Quick Review: If \(p < 0.05\), we are 95% confident the results were caused by our manipulation of variables. This is the "gold standard" for OCR Psychology!
2. Distribution Curves
When we plot our data on a graph, the shape it makes tells us a lot about our participants.
Normal Distribution
A normal distribution is a symmetrical, bell-shaped curve. In a perfect normal distribution, the mean, median, and mode all sit exactly in the middle. Most people score near the average, with fewer people at the extreme high or low ends (e.g., height or IQ scores).
Skewed Distributions
Sometimes data isn't symmetrical. This is called a skew:
- Positive Skew: The "tail" of the graph points to the right (the higher numbers). This happens if a test is very difficult—most people get low scores, and only a few get high scores.
- Negative Skew: The "tail" points to the left (the lower numbers). This happens if a test is very easy—most people get high scores, and only a few get low scores.
Memory Tip: The "tail" tells the tale. If the tail points to the positive (right) side, it's a positive skew!
3. Type 1 and Type 2 Errors
Because we use probability, we sometimes make mistakes in our conclusions.
Type 1 Error (The "False Positive"):
This is when we say our results are significant, but they actually happened by chance. We reject the null hypothesis when we should have accepted it. This often happens if our significance level is too lenient (e.g., \(p < 0.10\)).
Type 2 Error (The "False Negative"):
This is when we say our results are not significant, but there actually was a real effect. We accept the null hypothesis when we should have rejected it. This often happens if our significance level is too strict (e.g., \(p < 0.01\)).
Analogy: A Type 1 error is like a fire alarm going off when there is no fire. A Type 2 error is like a fire raging, but the alarm stays silent.
4. Choosing a Statistical Test
To pick the right test in your exam, you need to know three things: the level of data (Nominal, Ordinal, or Interval), the experimental design (Independent or Repeated), and whether you are looking for a difference or a correlation.
Parametric vs. Non-Parametric
Parametric tests are powerful, but they have strict rules. To use one, you need:
1. Interval or Ratio level data.
2. Data that follows a Normal Distribution.
3. Homogeneity of variance (the spread of scores in each group should be similar).
If your data doesn't meet these criteria, you must use a non-parametric test.
The Five Named Non-Parametric Tests
You need to know when to use these specific tests for the OCR H167 exam:
1. Chi-square (\(\chi^2\))
- Use: Testing for a difference or an association.
- Design: Independent measures.
- Data: Nominal (categories).
2. Binomial Sign Test
- Use: Testing for a difference.
- Design: Repeated measures.
- Data: Nominal.
3. Mann-Whitney U
- Use: Testing for a difference.
- Design: Independent measures.
- Data: Ordinal (ranked/ordered).
4. Wilcoxon Signed Ranks
- Use: Testing for a difference.
- Design: Repeated measures.
- Data: Ordinal.
5. Spearman's Rho (\(r_s\))
- Use: Testing for a correlation (relationship).
- Data: At least Ordinal.
Takeaway Table for Difference:
- Nominal + Independent = Chi-square
- Nominal + Repeated = Binomial Sign
- Ordinal + Independent = Mann-Whitney U
- Ordinal + Repeated = Wilcoxon
5. Using Critical Value Tables
Once a researcher calculates their "Observed Value" (the result of the test), they must compare it to a Critical Value found in a statistical table.
To find the correct Critical Value, you need to know:
1. The significance level (usually \(0.05\)).
2. Whether the hypothesis is one-tailed (directional) or two-tailed (non-directional).
3. The number of participants (often called \(N\) or \(df\) for degrees of freedom).
The Rule of "R":
In your exam, you don't need to do the math, but you might need to know if the Observed Value should be higher or lower than the Critical Value.
Tests with an "R" in their name (Spearman, Wilcoxon, Binomial Sign) require the observed value to be equal to or less than the critical value to be significant. (Note: Chi-square and Mann-Whitney U usually require it to be higher, though Mann-Whitney is a slight exception in some tables—stick to checking the instructions on the specific table provided in the exam!)
6. Handling Data for Statistics
When reporting results, you must follow these conventions:
- Significant Figures: Don't provide a string of 10 decimals. Usually, 2 or 3 significant figures are enough (e.g., \(0.045\)).
- Standard and Decimal Form: You should be able to convert fractions to decimals (e.g., \(1/20 = 0.05\)).
- Estimation: You should be able to look at raw data and estimate if a result is likely to be significant before running the test.
Key Takeaway: Inferential statistics aren't just about math; they are about certainty. They allow psychologists to claim that their findings are "real" and not just a lucky coincidence.
For more on how to design the studies that produce this data, see the chapter on "Planning and conducting research (Section 1.2)".