Can zero correlation hide a predictable pattern?

A feature screening script rejects any column with correlation near zero. Could it discard a feature that predicts a target perfectly?

Use the nine inputs $x=-4,-3,\ldots,4$. We will keep them fixed and change only the relationship that produces $y$. These are exact constructed examples, not sampled evidence about a real system.

Compare a line with a curve

Use $y=\text{scale}\times x$ and scale 1 as the reference. Predict the scatter shape and Pearson correlation, then record the run.

Change only Relationship to $y=\text{scale}\times x^2$. Keep scale 1 and all nine inputs fixed. Predict the sign and magnitude of the correlation before running. Can you still predict $y$ from $x$?

Nine fixed inputs x = −4 through 4. Construct y using the selected relationship and scale. Pearson correlation measures linear association. It is undefined if either variable is constant. These constructed relationships are not causal evidence.

Relationshipy = scale × xy = scale × x²y = scaleScale-2-1012

Prediction and reason

Record runReference settings

0/16 runs. Each run keeps its own inputs and results. Work stays in this tab across reloads; download a report before closing it.

Write a prediction to record your reference.

Observations: cite runs and valuesExplanation: what accounts for the change?Your next test, prediction, and a result that would challenge it

Download report

Describe the scatter pattern and compare it with the reported number before opening the explanation.

Why the curve has zero linear correlation

The line has Pearson correlation 1. The parabola has correlation 0 even though $y=x^2$ gives an exact prediction for every input.

The symmetric inputs have mean zero. In the correlation numerator, each positive $x$ contributes the opposite of its matching negative $x$, because their $y$ values are equal. The contributions cancel. Both columns vary, so the denominator is positive: the correlation is genuinely zero, not undefined.

A measure of linear association

Pearson correlation is

$$r=\frac{\sum_i(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum_i(x_i-\bar x)^2\sum_i(y_i-\bar y)^2}}.$$

The centered products measure whether larger-than-average values occur together. The denominator accounts for the scales of both columns. A constant column makes that denominator zero, so its correlation is undefined.

Use a scatter plot to inspect structure that a single linear summary can miss. Neither a strong correlation nor a visible curve establishes cause and effect. NIST explains the role and limits of scatter plots.

Design a comparison

Choose a relationship and an unused scale. Predict the scatter plot and correlation before recording a run. Keep one control fixed in each comparison. Seek a pair with different numeric ranges but the same correlation, then a pair whose correlations have opposite signs.

As a boundary case, keep the line relationship and choose scale 0. Predict whether the result should be zero or undefined. Download your report, including why a feature-screening decision needs more than one statistic.

Check your boundary case

Positive rescaling preserves a nonconstant line’s correlation of 1; negative rescaling reverses it to −1. Scale 0 makes every output zero, so the denominator vanishes. The viewer shows a dash rather than inventing a correlation. The parabolic example also demonstrates that zero correlation does not imply independence.

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