Scatter Plots and Lines of Best Fit
Plot bivariate data on a scatter plot, describe the association (direction, form, strength), and draw or use a line of best fit to make predictions. Interpret the slope and y-intercept of the best-fit line in context.
A lesson of Algebra 1, in Linear Functions, about 30 minutes.
By the end of this lesson, you will be able to describe the association in a scatter plot, fit a line to data by hand, use it to predict, and say clearly why a strong correlation is not evidence of a cause.
Real Data Never Sits on a Line
Every line so far went exactly through its points. Measure anything real and it does not.
Height against shoe size, hours studied against exam scores, temperature against ice creams sold: the points scatter. There is no line through all of them, so the question changes.
Not "which line passes through these points" but "which line comes closest to all of them".
That change is bigger than it sounds. The answer stops being the line and becomes a line, chosen because it describes the data well, and it arrives with a question attached: how well?
Describing What You See
Three things, in this order, every time.
Direction. As $x$ increases, does $y$ tend to increase (positive), decrease (negative), or neither (none)?
Strength. How tightly do the points hug a line? Strong means a narrow band; weak means a cloud with a lean.
Form. Straight, or curved? A line of best fit is only honest if the pattern is straight in the first place.
$$\text{"a strong negative linear association"}$$
That phrase is a complete answer, and each word is worth a point. Watch for outliers too: single points far from the pattern, which drag a fitted line toward themselves and deserve a comment rather than silent deletion.
Sort It: Describe the Association
Fitting a Line by Hand
You are not computing a regression by hand. You are drawing a sensible line and then finding its equation.
- Draw a line following the trend, with roughly as many points above it as below. It does not have to touch any of them
- Pick two points ON YOUR LINE, not two data points, and read their coordinates
- Work out the slope between those two
- Use point-slope and tidy
Step 2 is the one that goes wrong. Using two actual data points fits a line through those two and ignores the rest, which is exactly what a line of best fit is not. Read your drawn line, at places where it crosses corners cleanly.
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