Welcome to the Finish Line: Interpreting Results

You have spent time planning your investigation, collecting data, and creating beautiful charts and calculations. But what does it all actually mean? This stage of the Statistical Enquiry Cycle is where you become a detective. You look at the evidence you have processed and decide if your original ideas were right or wrong. This chapter focuses on how we draw conclusions and, crucially, how much we can trust them.

1. Analysing Diagrams and Calculations

To interpret results, you need to look closely at the "summary statistics" (like the mean or median) and the "measures of spread" (like the range or interquartile range) that you calculated in the previous stage.

  • Comparing Averages: If you are comparing two groups (e.g., test scores in Class A vs. Class B), look at the mean or median. If the mean of Class A is higher, it suggests that, on average, they performed better.
  • Comparing Spread: Averages don't tell the whole story. Look at the Interquartile Range (IQR) or Standard Deviation. A smaller spread means the data is more consistent. If Class A has a huge range but Class B has a small range, Class B is more "reliable" in their performance.
  • Looking at Shapes: When looking at a histogram or box plot, look for skewness. If the data is "bunched up" to the left with a long tail to the right, it is positively skewed. This tells you most values are low, with a few very high exceptions.

2. Tying it Back to the Hypothesis

Every statistical enquiry starts with a hypothesis (a statement you are testing). Your interpretation must answer whether your data supports or rejects that hypothesis.

Example: If your hypothesis was "Tall people have larger feet," and your scatter diagram shows a strong positive correlation, your conclusion would be: "The data supports the hypothesis because as height increases, shoe size also tends to increase."

Important: In GCSE Statistics, you don't need to use formal "null hypotheses," but you must be clear about whether your evidence matches your initial prediction.

3. Inferences and Predictions

Sometimes we use our results to guess what might happen in the future or to describe a whole population based on a small sample. This is called making an inference.

  • Interpolation: Predicting a value inside the range of data you collected. This is usually quite reliable.
  • Extrapolation: Predicting a value outside your data range (e.g., using a trend line to predict sales for next year). Be careful! Extrapolation is risky because trends can change suddenly.
  • Correlation vs. Causation: Just because two things are linked (correlation) doesn't mean one causes the other. For example, ice cream sales and shark attacks both go up in summer, but ice cream doesn't cause shark attacks! This is often due to an underlying factor (like warm weather).

4. Reliability of Findings

Not all conclusions are equally "strong." You must discuss how reliable your results are. Ask yourself these questions:

  • Sample Size: Was the sample big enough? A sample of 5 people isn't enough to represent a whole school. Larger samples generally lead to more reliable results.
  • Bias: Was the sampling method fair? If you only asked your friends, your results are biased and not reliable for the general population.
  • Data Cleaning: Did you find any outliers (extreme values)? If an outlier was a recording error, removing it makes your results more reliable. If it was a genuine piece of data, ignoring it might make your conclusion "cleaner" but less realistic.
  • Replication: Could someone else do the same experiment and get the same result? If the answer is "probably not," your findings lack reliability.

Quick Review: A conclusion is only as good as the data it’s based on. Always check for bias and small sample sizes before you claim your hypothesis is "proven."

5. Quality Assurance (Higher Tier Only)

In industry, statistics are used to make sure products are made correctly. We use control charts to monitor processes over time.

  • Sample Means: We plot the means of small samples. These sample means are usually more closely distributed (less spread out) than individual measurements.
  • Warning Lines: These are usually set at \( \pm 2 \) standard deviations from the target mean. Only 1 in 20 (5%) of samples should fall outside these. If a point hits this line, it’s a signal to "watch out."
  • Action Lines: These are set at \( \pm 3 \) standard deviations. Almost all data (99.7%) should be inside these. If a point falls outside an action line, the process is out of control and you must stop and fix it immediately.

6. Capture-Recapture Reliability (Higher Tier Only)

When estimating the size of a population (like fish in a lake) using the Petersen capture-recapture formula:

\( \text{Population Estimate} = \frac{n_1 \times n_2}{m} \)

(Where \( n_1 \) is the 1st sample size, \( n_2 \) is the 2nd sample size, and \( m \) is the number marked in the 2nd sample.)

For this to be reliable, you must assume:

  • The population didn't change (no births, deaths, or migrations).
  • The marks or tags didn't fall off or hurt the animals.
  • The marked animals mixed back into the population completely.
  • Each animal had an equal chance of being caught (no "trap-shyness").

Common Mistakes to Avoid

  • The "Proof" Trap: In statistics, we rarely "prove" things. We find "evidence for" or "support for" a hypothesis. Avoid saying "This proves..." and use "This suggests..." instead.
  • Ignoring Spread: Don't just compare means. Two sets of data can have the same mean but look completely different because one is much more spread out than the other.
  • Over-extrapolating: Don't assume a trend will continue forever. If a baby grows 10cm in a year, they won't be 3 metres tall by age 20!

Key Takeaway: Interpreting results is about looking at the big picture. Use your calculations to support your words, always keep your original hypothesis in mind, and be honest about the limitations (reliability) of your data.