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Grade 8 · Data and regression

Sports practice data: Separate trend from causal claims

Sports practice data investigation: separate trend from causal claims. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Sports practice data: Separate trend from causal claims. Points revealed: 1 of 5. Full-data least-squares slope: 2. Full-data mean output: 7. Correlation: 0.992Points, least-squares line, and residuals-8-4048-40040xRead numeric axes; drawing scales differ.
Data are constructed, not observations. The residual pattern is fixed and c≥0. Axes use x∈[−8,8], y∈[−40,40]. Full-data correlation is undefined when every output is equal; no causal or population inference is made.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Points revealed
1 of 5
Full-data least-squares slope
2
Full-data mean output
7
Correlation
0.992

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A scatter plot separates a trend from the deviations around it. These five constructed points have residuals whose sum and linear trend both vanish, so y=ax+b is their exact least-squares line. Correlation measures linear association and cannot, by itself, establish cause. This investigation begins with trend slope = 2; trend intercept = 1; residual scale = 0.25. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

yᵢ = axᵢ+b+c·eᵢ; e=(1,−2,0,2,−1), x=(1,2,3,4,5)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Read each plotted input-output pair; the five inputs are fixed at one through five.

  2. STEP 2

    Follow the changing model

    Playback reveals points and their vertical residual segments. The reference line is fitted to the complete five-point set.

  3. STEP 3

    Check and explain the relationship

    Change the residual scale and compare the scatter with the same fitted slope. A zero slope can coexist with variation.

Your turn to explain

Make a prediction. Test your reasoning.

Sports practice data: For the five-point residual pattern in this investigation, what are the least-squares slope and mean output?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Slope 2; mean output 7. The residuals sum to zero and have zero covariance with the five inputs.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 8 · Statistics

Read a scatter plot without claiming causation

A scatter plot can show direction, form, and unusual points, but an association alone does not identify a cause.