Interpreting Bivariate Data and Drawing Conclusions

• Pattern: As the distance from the source increases, the river velocity generally increases.
• Range: The distance ranges from $0.5\text{ km}$ to $11.5\text{ km}$ (a spread of $11\text{ km}$).
• Anomaly: Site 3 is an anomaly; the velocity drops to $0.38\text{ m/s}$ despite being further downstream than Site 2.

Describing Relationships

To describe a relationship, you must state three things: direction, strength, and whether a correlation exists.

• Positive Correlation: Both variables increase together (points slope upwards from bottom-left to top-right).
• Negative Correlation: One variable increases while the other decreases (points slope downwards from top-left to bottom-right).
• No Correlation: Points are scattered randomly with no discernible pattern or "path".
• Strength: A strong relationship has points tightly clustered around a central path; a weak relationship has points widely scattered but still suggesting a direction.

Correlation vs Causality: It is vital to distinguish between correlation (a mathematical link) and causality (where one variable directly causes the other to change). For example, there may be a strong positive correlation between ice cream sales and shark attacks, but ice cream does not cause shark attacks. Instead, a third factor—warm weather—causes both to increase.

Sketching Scatter Plots and Lines of Best Fit

A line of best fit (or trend line) represents the "mean path" of data points. When you are asked to sketch a scatter plot, it must be an approximate diagram showing key features with labels.

Key Features of a Scatter Plot Sketch:

1. Axes Labels: Both axes must have clear titles and units (e.g., Velocity ($m/s$)).
2. Plotted Points: Crosses ($x$) representing the bivariate data.
3. Line of Best Fit: A single, continuous solid line (straight or curved) with an equal balance of points above and below.
4. Anomaly Label: Any point clearly away from the trend should be circled and labelled.

Predictions and Reliability

• Interpolation: Estimating an unknown value inside the range of your data (e.g., predicting velocity at $6\text{ km}$ when you have data for $5\text{ km}$ and $8\text{ km}$). This is generally reliable.
• Extrapolation: Extending the line of best fit to predict values outside your data range. This is unreliable because the geographical trend may change (e.g., a river might reach the sea, stopping the increase in width).

Evaluating Statistical Presentations and Data

Statistical presentations and calculations often have limitations that reduce their reliability:

• Truncated axis: The y-axis does not start at zero, which exaggerates small differences.
• Data bias: Small sample sizes or unrepresentative sampling times (e.g., measuring river depth only after a storm).
• Mean vs Median: The mean (average) is highly sensitive to anomalies. In the river velocity table above, the anomaly at Site 3 would drag the mean velocity down, making it unrepresentative. The median (the middle value when data is ordered) is often more reliable when outliers are present because it is not skewed by extreme values.

Drawing and Evaluating Conclusions

To evaluate a conclusion or hypothesis, you must provide arguments for and against it using numerical evidence, ending with a balanced judgement.

Hypothesis: "River bedload size decreases as distance from the source increases."

Evidence For the Hypothesis	Evidence Against the Hypothesis
At Site 1 ($1\text{ km}$), the bedload was $40\text{ mm}$, decreasing to $14\text{ mm}$ at Site 5 ($15\text{ km}$).	At Site 3 ($7\text{ km}$), the bedload increased to , contradicting the downstream trend.

Balanced Concluding Judgement: The hypothesis is mostly supported because the overall percentage change was a $65\%$ decrease in size. However, the anomaly at Site 3 suggests that local factors can interrupt the trend. In this case, using the median bedload size across all sites would provide a more reliable "typical" value than the mean, as it would ignore the impact of the Site 3 outlier.

Worked Example: Percentage Change

Formula: $\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100$

• Step 1: $\text{New} (14) - \text{Old} (40) = -26$
• Step 2: $-26 \div 40 = -0.65$
• Step 3: (A decrease).