Scatter plots show the relationship between two numerical variables by graphing paired data as points. This cheat sheet helps students identify patterns, describe association, and decide whether a line of best fit is reasonable. These skills are useful for making predictions from data in science, business, sports, and everyday comparisons.
Key Facts
- A scatter plot graphs ordered pairs to show how two numerical variables may be related.
- A positive association means that as increases, tends to increase.
- A negative association means that as increases, tends to decrease.
- No association means the points do not show a clear upward or downward pattern.
- A line of best fit is often written as , where is the slope and is the -intercept.
- The slope of a line is , which represents the rate of change between two points.
- The -intercept is the predicted value of when .
- Interpolation predicts a value inside the data range, while extrapolation predicts a value outside the data range.
Vocabulary
- Scatter plot
- A graph that displays paired numerical data as points on a coordinate plane.
- Association
- The overall relationship or pattern between the two variables in a scatter plot.
- Correlation
- A description of the direction and strength of the relationship between two numerical variables.
- Line of best fit
- A straight line that closely follows the overall trend of the data points in a scatter plot.
- Slope
- The rate of change of a line, calculated by .
- Outlier
- A data point that is far away from the general pattern of the rest of the data.
Common Mistakes to Avoid
- Confusing positive and negative association is wrong because the direction depends on whether tends to increase or decrease as increases.
- Drawing a line of best fit through the most points is wrong because the line should balance the data with about the same number of points above and below it.
- Using only one data point to make a prediction is wrong because predictions should be based on the overall trend, not a single value.
- Ignoring outliers is wrong because an outlier can strongly affect the position and slope of a line of best fit.
- Extrapolating too far beyond the data is risky because the trend may not continue outside the observed range.
Practice Questions
- 1 A line of best fit passes through and . Find the slope .
- 2 A line of best fit is . Predict when .
- 3 A scatter plot comparing hours studied and test score shows points rising from left to right. Describe the association and explain what it means.
- 4 Why is a prediction using interpolation usually more reliable than a prediction using extrapolation?
Understanding Scatter Plots & Line of Best Fit
A best-fit line is a model, not a path that every data point must follow. It is placed so the points are balanced around it as much as possible. The vertical gap between a point and the line is called a residual.
A point above the line has a positive residual because the actual result was greater than the prediction. A point below it has a negative residual.
When residuals are generally small and scattered without a visible pattern, a straight-line model is more trustworthy. A curved arc or a changing spread of residuals signals that a line may hide an important feature.
When drawing a line by hand, use the middle of the overall cloud rather than trying to connect the first and last points. Choose two clear points that lie on the drawn line, even if neither is an original data point. Find the rise over the run to calculate slope.
Its units give the rate a prediction changes, such as extra dollars per hour or extra centimeters per year. The intercept can be useful only when a horizontal value of zero makes sense in the situation. A model for shoe size and height, for instance, should not be treated as a realistic claim about a person with zero height.
Outliers need careful attention. An outlier is far from the main pattern, possibly because of a recording mistake, unusual conditions, or genuine variation. One point far to the left or right can be especially influential because it can tilt the whole line.
Check the original data before removing any point. Keep it if it is real, but explain its effect. Most importantly, a pattern does not establish cause.
Ice cream sales and sunburn cases may rise together because warm weather affects both. A hidden factor like weather is called a confounding variable.
Predictions are safest within the values already measured. This is why teachers often expect interpolation rather than a far outside estimate. Suppose a graph includes practice times from one to ten hours.
Predicting a score for six hours uses nearby evidence. Predicting for fifty hours assumes the same pattern continues, even though scores have limits and the relationship may level off.
In real work, scientists use scatter plots to compare measurements, coaches track training data, and shops study sales records. State predictions with sensible rounding, include units, and describe the result as an estimate rather than a certainty.