SOP: Testing a Sample Variance Against a Specified Value by the Chi-Square Test

GIFSQ — Global Institute for Food Safety & Quality

Principle

The chi-square (χ²) test is used for testing the equality of the variance of a given lot with a specified value. The sample variance is scaled by the degrees of freedom and divided by the specified variance; the resulting χ² value is compared with the critical values of the χ² distribution to determine whether the sample variance differs significantly from the specified value.

Materials

• The sample variance s² from n observations
• The specified (lot) variance σ₀²
• The book’s chi-square table for (n − 1) degrees of freedom

Procedure

Determine the variance s² of the sample from n observations

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State the specified variance σ₀² to be tested against

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Compute the χ² value using the expression below

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Compare the calculated χ² with the tabulated critical value at the chosen significance level for (n − 1) degrees of freedom; if the calculated value is greater, the difference is significant

Calculation

χ² = (n − 1)s² / σ₀²

Where,

• s² = variance of the sample
• σ₀² = specified (lot) variance
• n = size of the sample
• χ² has (n − 1) degrees of freedom

Worked example — variance of a new filling line

The specified variance σ₀² for the standard filling line of juice into bottles is 0.104. From a new line recommended, 15 tests were made and the variance was found to be 0.182.

χ² = 14 × 0.182 / 0.104 = 24.5

From the chi-square table, the critical value for 14 degrees of freedom is 23.69 at the 5% level. The calculated value is greater and hence is significant; consequently the new unit causes a significant variation.

Precautions

• The χ² test for variance requires no assumption about the distribution of the variable in the population.
• State the significance level (5% or 1%) before comparing with the tabulated value.

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.