Scatter Graphs & Correlation

Purpose and Definition: A line of best fit is a straight line drawn through the center of the data points to model the underlying trend. It serves as a visual average, allowing for the estimation of values not explicitly present in the dataset.

Drawing Methodology: When drawing by eye, the line should follow the general direction of the points and aim to have an equal number of points above and below it. It does not necessarily need to pass through the origin $(0,0)$ unless the context dictates that both variables must be zero simultaneously.

The Mean Point: Mathematically, the most accurate line of best fit must pass through the mean point $(\bar{x}, \bar{y})$, where $\bar{x}$ is the average of all x-coordinates and $\bar{y}$ is the average of all y-coordinates. Using this point as an anchor ensures the line is centered within the data distribution.

Interpolation: This is the process of estimating a value within the range of the existing data points. Because the prediction is supported by surrounding data, interpolation is generally considered reliable and accurate.

Extrapolation: This involves estimating a value outside the range of the observed data (either much higher or much lower). Extrapolation is risky because the established trend may not continue indefinitely; for example, a biological growth trend might level off or reverse over time.

Reliability Factors: The accuracy of any prediction depends on the strength of the correlation and the sample size. A strong correlation from a large dataset provides much higher confidence in predictions than a weak correlation from a small sample.

The Golden Rule: Correlation does not imply causation. Just because two variables show a mathematical relationship does not mean that one causes the other to change.

Lurking Variables: Often, a third, unobserved variable (a confounding factor) may be influencing both variables simultaneously. For instance, ice cream sales and shark attacks both increase in summer due to the temperature, but ice cream does not cause shark attacks.

Precision in Drawing: When asked to draw a line of best fit, always use a ruler and ensure it extends across the full range of the data. Avoid 'joining the dots' or forcing the line through the first and last points if they are outliers.

Identifying Outliers: Look for points that lie significantly far away from the general cluster. In exams, you may be asked to identify these or explain how they might skew the line of best fit if included in calculations.

Contextual Interpretation: Always relate your findings back to the real-world scenario. If a graph shows a negative correlation between 'Hours of Practice' and 'Errors Made', the conclusion should be stated as: 'As practice increases, the number of errors decreases.'