Constructing chi-square tests for two-way price counts

Start from the sample mix

Two-outcome records such as rise versus decline can be scored with a chi-square test that applies a half-unit Yates adjustment to each observed-minus-expected gap. The chi-square-test turns those observed versus expected frequencies into one statistic used to judge whether a directional split is hard to attribute to chance.

Construction writes an expected-count for each outcome first. That baseline frequency is assigned from the overall sample mix, not from an automatic even-money split. Because the full sample rose on 52.1% of days and declined on 47.9%, weekday expected counts are taken from that mix rather than from an even split.

Apply the Yates-correction

The next step applies a Yates-correction: a half-unit reduction to each absolute observed-minus-expected gap before that gap is squared and scaled by the expected count.

The construction maps chi-square values above 3.84, 6.64, and 10.83 to 95%, 99%, and 99.9% confidence that the split is not only chance. That mapping is a conventional confidence-level: it relates the size of the statistic to how rarely an equally large departure would appear under the stated baseline.

A Monday illustration

A Monday illustration compares observed counts of 669 and 865 with expected counts of 799 and 735, a 130-count departure on each side. After the half-unit adjustment, that Monday illustration produces a chi-square of 43.8, which exceeds the 10.83 threshold tied to 99.9% confidence.

The even-money-simplification

When the baseline is treated as even money, the statistic simplifies to the square of one less than the absolute count difference, divided by the total number of trials. That even-money-simplification is a reduced two-outcome formula used only when the two results are treated as equally likely.