1. The Chi-Square Test for a Population Variance

Two packing lines can have the same average fill and different levels of consistency. We will test a claim about the variance itself. That calls for a different reference curve, with one tail that stretches much farther than the other.

2. When the business cares about consistency

Two machines can fill to the same average while one produces much more variable packs. The manager may want to test consistency. That is a claim about the variance, in squared units.

Compare the sample variance with the benchmark variance, scaled by the degrees of freedom. For independent normal weights, use the chi-square reference. A larger sample variance pushes the score to the right.

3. Equal tail areas do not mean equal distances

The variance score cannot be negative. The chi-square curve has a longer right tail. The two marked boundaries each leave two and a half percent outside, but they sit at different distances from the center of the curve.

Look to the right for evidence of greater variation and to the left for less variation. When either change matters, split the false-alarm budget between the two tails.

4. Check whether pack-weight variance has changed

Use variance in both numerator and denominator. The score is twenty-seven point three six, between the two cutoffs shown on the previous graph. The sample does not establish a change in consistency.

A question chosen in advance about greater variation reads only the right tail. For lower variation, read the left tail. The large left-tail probability makes sense because the observed sample variance is above its benchmark.
