Correlation patterns and honest claims

Task 3 continued · L6 p. 4.

4. Three datasets

Predictions: A positive, B negative, C zero (symmetric U-shape).

x = np.arange(-3, 4)
datasets = {"A": x, "B": -x, "C": x ** 2}
fig, axes = plt.subplots(1, 3, figsize=(9, 3))
for ax, (name, y) in zip(axes, datasets.items()):
    ax.scatter(x, y)
    ax.set(xlabel="x", ylabel="y", title=name)
    r = np.corrcoef(x, y)[0, 1]
    print(name, "r =", round(r, 3))
plt.tight_layout()
plt.show()

Output: A r = 1.0, B r = -1.0, C r = 0.0.

• A: y increases by exactly 1 whenever x increases by 1: perfect positive linear correlation.
• B: y decreases by exactly 1 whenever x increases by 1: perfect negative linear correlation.
• C: y falls from 9 to 0 as x goes from −3 to 0, then rises back to 9. The decreasing and increasing halves cancel, so r = 0, yet y is completely determined by x (x = −2 and x = 2 both give 4).

Zero correlation is not independence. Pearson r measures only straight-line association. C has a perfect curved relationship with r = 0, so "no correlation" must not be read as "unrelated". Always plot.

5. Same population for plot and statistic

1. One point represents one student (an entry score and final grade pair).
2. No. 0.48 was computed on all 1,200 DS100 students; the 50 plotted points are a selection and could have a different correlation.
3. The number of pairs tells the reader which observations were summarised and how much evidence supports the value. A statement should say, for example, "Among all 1,200 DS100 students, entry score and final grade have a positive linear association (r ≈ 0.48)."

6. Rewrite the causal claim

Unsupported: "The correlation is positive, so increasing entry score will cause a higher final grade."

Supported: "Among the 1,200 DS100 students, higher entry scores are associated with higher final grades (r ≈ 0.48). These observational records show how the two scores vary together; they do not show that raising a student's entry score would raise their final grade." A factor that could relate to both: prior mathematical preparation (or study habits, or time available for study).