Welcome to the Final Stages of the Statistical Enquiry Cycle!

You’ve done the hard work: you planned your investigation, collected your data, and crunched the numbers (perhaps using ANOVA or a t-test). But a pile of numbers isn't very useful on its own. In this chapter, we look at the Interpretation and Evaluation stages of the Statistical Enquiry Cycle (SEC). This is where you turn "maths" into "meaning" and look back to see how you could do better next time.

Think of this as the "Detective Phase." You’ve found the clues; now you need to explain what they mean for the case and decide if your detective work was up to scratch!

1. Interpretation: Making Sense of the Results

Interpretation is about looking at your diagrams, calculations, and test results to see what they tell you about your original hypothesis. In your exam, this is often where you earn AO2 marks (interpreting information in context).

Analyzing Diagrams and Calculations

Whether you are looking at a Box Plot, a Histogram, or ANOVA output from a computer, you need to describe what you see using statistical language:

  • Summary Measures: Use the mean \( (\bar{x}) \), median, or standard deviation \( (s) \) to describe the "average" and the "spread" of your data.
  • Visual Patterns: Does a scatter diagram show a linear trend? Are there outliers that might be skewing your results?
  • Technology Output: You should be comfortable reading computer printouts. If an ANOVA table shows a very small p-value, it suggests there is a significant difference between the groups you are studying.

Drawing Conclusions on Hypotheses

When you perform a significance test, you must conclude by referring back to your original Null Hypothesis \( (H_0) \) and Alternative Hypothesis \( (H_1) \).

Important Rule: Never say your conclusion is "proven" or "definite." Statistics is about probability, not absolute certainty. Use phrases like:

  • "There is sufficient evidence at the \( 5\% \) significance level to suggest..."
  • "We fail to reject the null hypothesis, as there is insufficient evidence to suggest..."

Communicating Clearly

Your conclusion must be written in plain English that relates back to the real-world context of the question. If you are testing a new fertilizer, don't just say "Reject \( H_0 \)"; say "The evidence suggests the new fertilizer leads to a significant increase in crop yield."

Quick Tip: Always mention the significance level (usually \( 5\% \)) in your final written conclusion to show you understand the risk of a Type I error (rejecting \( H_0 \) when it was actually true).

2. Evaluation: Critiquing Your Own Work

This is the AO3 part of the syllabus. You need to be a "critical friend" to your own investigation. No statistical study is perfect!

Weaknesses in Collection or Display

Look back at how the data was gathered. Could there be bias?
Example: If you used a judgmental sample or a snowball sample, your data might not represent the whole population fairly. Did you use leading questions in a survey? This can "nudge" people toward certain answers.

Limitations of Sample Size

The size of your sample \( (n) \) matters a lot:

  • Small Samples: If \( n \) is small, your test might lack power. This means you might fail to spot a real effect (a Type II error).
  • Effect Size: Remember that "statistically significant" doesn't always mean "practically important." Using Cohen’s d helps you see the size of the difference. A \( d \) value of \( 0.2 \) is small, while \( 0.8 \) or above is considered large.

Reliability and Validity

  • Reliability: If someone else did the same experiment, would they get the same results? (Consistency).
  • Validity: Did you actually measure what you intended to measure? (Accuracy).

3. Review: Improvements and Refinements

The Statistical Enquiry Cycle is a loop. The final stage is thinking about how to improve the process for next time.

Suggesting Improvements

If you found a weakness, how would you fix it?
- Sample Size: "In a future study, I would increase the sample size to improve the power of the test."
- Sampling Method: "Instead of a convenience sample, I would use stratified random sampling to ensure all age groups are represented proportionally."
- Controlling Variables: In an experimental design, could you use blocking or paired comparisons to reduce experimental error?

Refining the Question

Sometimes, your results lead to new questions. For example, if you found that a drug works for adults but your sample had no children, your "Review" might suggest a new study specifically for younger patients.

Summary Checklist for the Exam

When answering questions on Interpretation and Evaluation, ask yourself:

  • Did I mention the context (e.g., the specific names/units used in the question)?
  • Did I avoid using the word "proven"?
  • Did I identify a specific source of bias or a sampling limitation?
  • Is my p-value interpretation consistent with the significance level?
  • Have I suggested a practical improvement (like using a larger random sample)?

Did you know? The SEC is assessed on all three of your exam papers! Whether you are doing a simple t-test or a complex Two-Way ANOVA, you are always expected to think about whether your data is reliable and what the results actually mean in the real world.

Quick Cross-Reference: For more details on the earlier stages of this cycle, see the chapter on "Statistical Enquiry Cycle: planning and data collection". To see how to interpret specific ANOVA results, head over to the "One-way analysis of variance" chapter.