Methods and formulas for 2 Variances

Formula

The standard deviation equals the square root of the variance.

Notation

Term	Description
	mean of sample i
S2i	variance of sample i
Xij	jth measurement of the ith sample
ni	size of sample i

Formula for the test statistic

When n₁ = n₂ , the test statistic is Z². If the null hypothesis, ρ = ρ₀ is true, then Z² is distributed as a chi-square distribution with 1 degree of freedom. Z² is given by:

where se(ρ₀) is the standard error of the pooled kurtosis, which is given by:

where r_i = ( n_i - 3) / n_i and is the pooled kurtosis, which is given by:

se²(ρ₀) can also be expressed in terms of the kurtosis values of the individual samples, , as follows:

where:

Formula for p-value

Let z² be the value of Z² that is obtained from the data. Under the null hypothesis, H₀: ρ = ρ₀ , Z is distributed as the standard normal distribution. Therefore, the p-values for the alternative hypotheses (H₁) are given by the following.

Notation

Term	Description
Si	the standard deviation of sample i
ρ	the ratio of the population standard deviations
ρ0	the hypothesized ratio of the population standard deviations
α	the significance level for the test = 1 - (the confidence level / 100)
ni	the number of observations in sample i
	the kurtosis value for sample i
Xij	the jth observation in sample i
mi	the trimmed mean for sample i with trim proportions of

Formula

When n₁ ≠ n₂ , there is no test statistic. Rather, the p-value is calculated by inverting the confidence interval procedure. The p-value for the test is given by:

P = 2 min (α_L, α_U)

where α_L is the smallest value of α for which the following holds:

and α_U is the smallest value of α for which the following holds:

where c_α is an equalizer constant described below and se(ρ₀) is the standard error of the pooled kurtosis, which is given by:

where r_i = (n_i - 3) / n_i and is the pooled kurtosis which is given by:

se(ρ₀) can also be expressed in terms of the kurtosis values of the individual samples. For more information go to Test for Bonett's method with balanced designs.

Equalizer constant

The constant c_α is included as a small-sample adjustment to mitigate the effect of unequal tail error probabilities in unbalanced designs. The value of c_α is given by:

The constant vanishes when the designs are balanced, and its effect becomes negligible with increasing sample sizes.

Finding α_L and α_U

Finding α_L and α_U is equivalent to finding the roots of the functions L(z , n₁ , n₂ , S₁ , S₂ ) and L(z , n₂ , n₁ , S₂ , S₁ ), where L(z , n₁ , n₂ , S₁ , S₂) is given by:

Let:

To calculate α_U, apply the previous steps using the function, L(z, n₂, n₁, S₂, S₁), instead of the function, L(z, n₁, n₂, S₁, S₂).

Notation

Term	Description
Si	the standard deviation of sample i
ρ	the ratio of the population standard deviations
ρ0	the hypothesized ratio of the population standard deviations
α	the significance level for the test = 1 - (the confidence level / 100)
zα	the upper α percentile point of the standard normal distribution
ni	the number of observations in sample i
Xij	the jth observation in sample i
mi	the trimmed mean for sample i with trim proportions of

Formula

Confidence intervals are obtained by inverting the test procedure. More specifically, Minitab solves the following equation for ρ:

where c_α/2 is an equalizer constant (described below) and se(ρ) is the standard error of the pooled kurtosis (described below). Typically, this equation has two solutions, a solution, L < S₁ / S₂, and a solution U > S₁ / S₂. L is the lower confidence limit, and U is the upper confidence limit. For more information, go to Bonett's Method, which is a white paper that has simulations and other information about Bonett's Method.

The confidence limits for the ratio of the variances are obtained by squaring the confidence limits for the ratio of the standard deviations.

Equalizer constant

The constant c_α is included as a small-sample adjustment to mitigate the effect of unequal tail error probabilities in unbalanced designs. The value of c_α is given by:

The constant vanishes when the designs are balanced, and its effect becomes negligible with increasing sample sizes.

Standard error of the pooled kurtosis

se(ρ) is the standard error of the pooled kurtosis which is given by:

where r_i = (n_i - 3) / n_i and is the pooled kurtosis which is given by:

se(ρ) can also be expressed in terms of the kurtosis values of the individual samples. For more information, see the section on the Test for Bonett's method with balanced designs.

Notation

Term	Description
α	the significance level for the test = 1 - (the confidence level / 100)
Si	the standard deviation of sample i
ρ	the ratio of the population standard deviations
zα/2	the upper α/2 percentile point of the standard normal distribution
ni	the number of observations in sample i
Xij	the jth observation in sample i
mi	the trimmed mean for sample i with trim proportions of

Formula

Levene’s test is appropriate for continuous data. Levene’s test is not available for summarized data.

To test the null hypothesis that σ₁ / σ₂ = ρ with Levene’s test, Minitab performs a one-way ANOVA on the values Z_1j and ρZ_2j (where j = 1, …, n₁ or n₂).

The Levene's test statistic equals the value of the F-statistic in the resulting ANOVA table. The Levene's test p-value equals the p-value in this ANOVA table.

• H. Levene (1960). Contributions to Probability and Statistics. Stanford University Press, CA.
• M.B. Brown and A.B. Forsythe (1974). "Robust Tests for the Equality of Variance," Journal of the American Statistical Association, 69, 364–367.

Degrees of freedom

Under the null hypothesis, the test statistic follows an F-distribution with degrees of freedom DF1 and DF2.

DF1 = 1

DF2 = n₁ + n₂ – 2

Notation

Formula

For continuous data, Minitab calculates the confidence limits for the ratio (ρ) between the population standard deviations with the following formulas. To obtain limits for the ratio between population variances, square the values below.

Notation

Term	Description
t α	the α critical value of a t distribution with n1 + n2 – 2 degrees of freedom
ηi	the median of sample i
Zij	where j = 1, 2, ... , ni and i = 1, 2, and Xij are individual observations
Mi	the mean of Zij
Si2	the sample variance of Zij
vi
ρ	σ1 / σ2
n1	the size of the first sample
n2	the size of the second sample

The F-test is appropriate for normal data. To test the null hypothesis that σ₁ / σ₂ = ρ with the F-test, Minitab uses the following formulas.

Formula for the test statistic

Formula for degrees of freedom

Under the null hypothesis, the F-statistic follows an F-distribution with degrees of freedom DF1 and DF2.

DF1 = n₁ – 1

DF2 = n₂ – 1

Formula for p-value

Notation

When the data follow a normal distribution, Minitab calculates the confidence bounds for the ratio (ρ) between the population standard deviations with the following formulas. To obtain bounds for the ratio between population variances, square the values below.

Formula

When you specify a "not equal" alternative hypothesis, a 100(1 – α)% confidence interval for ρ is given by:

When you specify a "less than" alternative hypothesis, a 100(1 – α)% upper confidence bound for ρ is given by:

When you specify a "greater than" alternative hypothesis, a 100(1 – α)% lower confidence bound for ρ is given by:

Notation