01The question: what does equality of variances mean?

Comparing two populations requires distinguishing their location, dispersion, and complete probability distributions. A mean describes a center; a variance measures quadratic dispersion around that center. Equality of one characteristic does not automatically imply equality of the others.

Let X and Y be real random variables with finite second moments. The question is whether their population parameters satisfy σ_X² = σ_Y². With samples alone, we observe estimates of these parameters and must quantify their uncertainty. A mathematical identity, an asymptotic approximation, and an inferential conclusion answer different questions.

02Means, moments, and dispersion

An order-r raw moment is an expectation of powers. A central moment instead uses deviations from the mean. For ordinary positive integer moments, assume absolute integrability: E[|X|^r] < ∞.

Squaring prevents positive and negative deviations from canceling and gives greater weight to larger deviations. If a variable is measured in meters, its variance is measured in square meters; its standard deviation returns dispersion to the original unit.

We must also distinguish the parameter σ², the random estimator S², and its observed value s². A population moment is not, by definition, a test statistic.

03What implies what: distribution, location, and scale

A scale transformation changes variance itself: Var(a + bX) = b²Var(X). Preserving standardized shape therefore need not preserve dispersion in the original units.

The reverse implication also fails: a standard normal variable and a variable taking −1 and 1 with probability 1/2 both have mean zero and variance one, although one is continuous and the other discrete.

04Lʳ convergence and mean-square convergence

Convergence concerns a sequence of random variables and a limit, rather than an informal resemblance between two samples. For r ≥ 1, with the variables defined on the same probability space, L^r convergence means:

Convergence of E[X_n] to E[X] alone does not give L¹ convergence: the expectation of a signed difference can conceal cancellations. L¹ distance uses the absolute value. Likewise, a test of means does not measure the L² distance between variables.

In L², the norm ‖X‖₂ = (E[|X|²])^1/2 arises from the inner product E[XY]. This Hilbert-space structure explains the usefulness of quadratic tools; it does not turn every kind of convergence into L² convergence. Probability theory also studies other L^r spaces. [1]

05Why convergence in distribution is not enough

The chain L² ⇒ convergence in probability ⇒ convergence in distribution is valid; the reverse implications fail in general. Convergence in distribution controls expectations of bounded continuous functions, whereas x² is unbounded.

For each integer n ≥ 1, consider a variable whose mass increasingly concentrates at zero, while retaining rare and increasingly extreme events:

For any ε > 0 and sufficiently large n, P(|X_n| > ε) = 1/n → 0. Hence X_n → 0 in probability and in distribution. Nevertheless:

This example even converges to zero in L¹, since E[|X_n|] = 1/√n → 0, yet its variances still fail to converge to the variance of the limit.

A sufficient additional condition for convergence in distribution to yield convergence of the first two moments is uniform integrability of the squares:

For example, sup_nE[|X_n|^2+δ] < ∞ for some δ > 0 is sufficient for this control. Bounded second moments alone are insufficient, as the counterexample shows. [1]

06Existence of moments and determination of the distribution

Once a distribution is fixed, every existing moment has a definite value. A separate question is whether knowing all the moments uniquely identifies that distribution. This is the moment problem. Having finite moments of every order does not, by itself, solve it.

Distributions with finite support or support contained in a compact interval are determined by their moments. Countability of an infinite support alone is insufficient. General sufficient conditions include a moment-generating function finite on an open neighborhood of zero and Carleman’s criterion. These conditions also cover continuous distributions. [2]

07Characteristic functions and the Fourier transform

A characteristic function provides a representation identifying the complete law of a real random variable. Its general definition uses the distribution as a measure, so it does not presume that a density exists:

It exists for every real t because |e^itX| = 1. Two variables have the same law if and only if their characteristic functions agree for every t. This uniqueness does not require moments to exist. [3]

If E[X²] < ∞, derivatives at zero recover the relevant moments:

The exponent’s sign and the Fourier normalization factors depend on the chosen convention. Here we use e^itx. Inversion theorems, under their respective conditions, recover a measure or density; they do not introduce a free constant justifying an arbitrary difference between variances.

Characteristic functions do not automatically make the moments of a sequence converge either. Pointwise convergence to a limiting characteristic function gives convergence in distribution, but passing to unbounded moments still requires additional tail control.

08From populations to samples: consistency

Let X₁, …, X_n be independent and identically distributed observations from a population with finite second moment. The usual sample variance is:

The strong law of large numbers, applied to X and X², gives X̄_n → E[X] and n⁻¹∑X_i² → E[X²] almost surely. Thus S_n² → σ_X² almost surely. A finite fourth moment is unnecessary for this consistency, although certain results about its asymptotic distribution may require it.

For two samples, as both sizes grow, the estimators converge to their respective parameters:

If both samples are known to come from the same law, equality of the parameter is already part of that assumption. Their observed sample variances may still differ. If common origin is a conjecture, evidence must come from the data.

This asymptotic argument is not automatically an approach to the size of a finite population: sampling without replacement changes dependence and design. Nor does it justify ignoring temporal correlation, clusters, or paired measurements.

09The F test: assumptions, derivation, and interval

For two independent samples, each i.i.d. from a normal population with positive variance, define ρ = σ_X²/σ_Y². Equality means ρ = 1. The two-sided test contrasts H₀: ρ = 1 against H₁: ρ ≠ 1. [4]

Under H₀, the statistic F = S_X²/S_Y² follows this distribution without the factor ρ. Degrees of freedom identify the numerator and denominator order; they must not be inadvertently interchanged.

Let q_u be the lower-tail quantile defined by P(F_n−1,m−1 ≤ q_u) = u. At level α, reject when the observed ratio r lies outside [q_α/2, q_1−α/2]. This quantile convention agrees with R’s qf.

The ratio is dimensionless. An interval containing one is compatible with equality at the corresponding level, but does not prove it. If reduced variability is the target, the one-sided direction should be specified before examining results. Normality is a condition for this exact result; a nonsignificant normality test cannot certify it. [5]

10Robust alternatives and the scope of their hypotheses

The classical F test is sensitive to departures from normality. For skewed data or extreme observations, consider procedures that do not rely on the same normal-theory pivot. Robustness must always be related to the design and the target parameter.

Brown–Forsythe constructs Z_ij = |X_ij − median_i| and compares the means of Z across groups. Its calibration is approximate, unlike the exact normal-theory F test. When distributional shapes differ substantially, equal expected absolute deviations and equal variances are not generally equivalent. [6] [7]

For a common location–scale family, interpretation as a scale comparison is more direct. If variance does not exist, as for a Cauchy law, the target must change, for example to a quantile-based measure of dispersion, and that change must be stated.

If the substantive objective is to compare means, Welch’s test permits this without imposing equal variances. Mechanically choosing between Student and Welch after a preliminary variance test adds a selection step; the procedure should follow the design and inferential objective. [8]

11Non-rejection, power, and practical equivalence

A p-value greater than α means that the test supplies insufficient evidence to reject H₀ at the chosen level. It is not the probability that H₀ is true. An uninformative study may produce this result even when meaningful differences exist.

Power is defined against a specific alternative: π(ρ) = P_ρ(reject H₀). For the equal-tail two-sided F test, a positive alternative ratio ρ, and normal-theory assumptions:

These are prospective calculations for a specified scenario, rather than estimates derived from the example’s p-value. The relevant difference and the sample sizes should be planned together; equal means cannot replace that work.

To establish practical equivalence, prespecify limits L and U representing tolerable differences, with 0 < L < 1 < U. The question becomes whether L < ρ < U:

With two one-sided tests at level α, declare equivalence only when both reject. Under the normal model, this is equivalent to the two-sided 1 − 2α confidence interval for ρ lying entirely within (L,U). At α = 0.05 this uses a 90% interval, rather than a 95% interval.

As a conceptual illustration, a 90% interval [0.83,1.18] would lie inside (0.80,1.25). These reciprocal margins express multiplicative tolerance; they are neither a universal recommendation nor results from the example below. The substantive problem must justify them.

12A fully reproducible numerical example

We use two synthetic samples generated from independent normal models X ∼ N(10,1) and Y ∼ N(10,4). Their population means agree by construction; their population variances differ. The vectors are fixed to six decimal places in the code, so the results do not depend on R’s random-number generator.

At level 0.05, equality is rejected within the normal model. The interval lies below one: the evidence favors a smaller variance for X than for Y. The point estimate is about 20.5% of Y’s variance, but the interval shows substantial uncertainty. The population ratio used in simulation is 1/4; individual sample ratios need not match it exactly.

As sensitivity analyses, Brown–Forsythe yields F = 7.973726 and p = 0.009182; Fligner–Killeen yields χ² ≈ 6.840015 and p ≈ 0.008914. Agreement here does not make their hypotheses identical or establish universal robustness. Choose the primary method before searching for the most favorable result.

13R code: computation, sensitivity, and diagnostics

The code below uses only base R functions. It reproduces the test, checks its formulas, computes the two sensitivity analyses, and applies the equivalence criterion using illustrative margins. Q–Q plots and a boxplot support inspection without becoming an automatic assumptions test. Inferential validity still depends on the design and model.

# Igualdad de varianzas / Equality of variances
# Datos sinteticos fijos / Fixed synthetic data
# R base: stats + graphics; no external packages.
x <- c(9.867324, 10.005934, 9.230461, 10.707670,
       10.857477, 8.684811, 9.164998, 10.428310,
       9.504457, 10.836370, 11.206876, 9.653071)
y <- c(10.769160, 11.227355, 10.982203, 8.646258,
       7.303017, 7.526147, 11.072729, 7.522827,
       8.943126, 8.643210, 9.186375, 12.431695,
       7.238559, 7.421358, 10.896078)
stopifnot(all(is.finite(c(x, y))), var(x) > 0, var(y) > 0)

# Resumen / Summary
resumen <- data.frame(
  group = c("X", "Y"), n = c(length(x), length(y)),
  mean = c(mean(x), mean(y)), variance = c(var(x), var(y)))
print(resumen, digits = 8)

# F bilateral e IC / Two-sided F test and CI
alpha <- 0.05
nu1 <- length(x) - 1
nu2 <- length(y) - 1
r <- var(x) / var(y)
test_F <- var.test(x, y, ratio = 1, conf.level = 1 - alpha)
p_manual <- min(1, 2 * min(pf(r, nu1, nu2),
                         pf(r, nu1, nu2, lower.tail = FALSE)))
ci_manual <- c(r / qf(1 - alpha/2, nu1, nu2),
               r / qf(alpha/2, nu1, nu2))
stopifnot(abs(unname(test_F$statistic) - r) < 1e-10,
          abs(test_F$p.value - p_manual) < 1e-10,
          max(abs(as.numeric(test_F$conf.int) - ci_manual)) < 1e-10)
print(test_F)

# Brown-Forsythe: ANOVA de desviaciones absolutas / of absolute deviations
datos <- data.frame(
  value = c(x, y),
  group = factor(c(rep("X", length(x)), rep("Y", length(y)))))
datos$absdev <- abs(datos$value -
                   ave(datos$value, datos$group, FUN = median))
bf <- anova(lm(absdev ~ group, data = datos))
print(bf)
fk <- fligner.test(value ~ group, data = datos)
print(fk)

# Equivalencia / Equivalence: illustrative margins, not universal
L <- 0.80
U <- 1.25
ci90 <- c(r / qf(1 - alpha, nu1, nu2),
          r / qf(alpha, nu1, nu2))
p_lower <- pf(r / L, nu1, nu2, lower.tail = FALSE)
p_upper <- pf(r / U, nu1, nu2)
p_tost <- max(p_lower, p_upper)
equivalent <- all(ci90 > L & ci90 < U)
stopifnot(identical(equivalent, p_tost < alpha))
print(list(CI90 = ci90, margins = c(L, U),
           p_TOST = p_tost, equivalent = equivalent))

# Potencia prospectiva / Prospective power under independent normal samples
power_F <- function(n1, n2, rho, alpha = 0.05) {
  stopifnot(n1 >= 2, n2 >= 2, n1 == as.integer(n1),
            n2 == as.integer(n2), rho > 0, alpha > 0, alpha < 1)
  df1 <- n1 - 1
  df2 <- n2 - 1
  q <- qf(c(alpha/2, 1 - alpha/2), df1, df2)
  pf(q[1]/rho, df1, df2) +
    pf(q[2]/rho, df1, df2, lower.tail = FALSE)
}
sizes <- c(12, 30, 60, 120)
print(data.frame(n_per_group = sizes,
  power_rho_0.5 = vapply(sizes, function(n) power_F(n, n, 0.5), numeric(1))))
stopifnot(abs(power_F(12, 15, 1) - alpha) < 1e-10)

# Diagnosticos / Diagnostics; no automatic normality-test gate
op <- par(mfrow = c(1, 3))
boxplot(value ~ group, data = datos, col = "#fffdf7",
        border = "#8b4513", xlab = "Group", ylab = "Value")
qqnorm(x, main = "X: normal Q-Q"); qqline(x, col = "#8b4513")
qqnorm(y, main = "Y: normal Q-Q"); qqline(y, col = "#8b4513")
par(op)
sessionInfo()

The manual calculation and var.test should agree numerically. This example does not establish equivalence within (0.80,1.25). Brown–Forsythe is implemented through ANOVA of absolute deviations, and fligner.test follows R’s documented procedure. [5] [7]

14Homogeneity, the structure of the phenomenon, and abstraction

Statistical homogeneity must be specified: equal variances, equal distributions, and exchangeability are distinct properties. Their relevance depends on the unit of analysis, measurement scale, selection process, and relationships among observations.

A population mixing subgroups shows how the phenomenon’s structure enters total variability. If G identifies the subgroup and X has finite second moment, the law of total variance states:

Two populations can have identical within-subgroup variances yet different total variances when the separation of subgroup means or the mixture composition changes. For example, if each subgroup has variance one, their means are −a and a, and each has weight 1/2, total variance is 1 + a². Structural heterogeneity thus enters an explicit mathematical result.

From a materialist perspective, statistical abstraction selects relationships within the phenomenon and must justify that selection. A theorem establishes what follows from assumptions; research determines whether those assumptions adequately represent the material process. Larger samples can reduce uncertainty without correcting an ill-defined population, omitted dependence, or a misspecified mixture.

15Conclusions: conditions before shortcuts

Equal variances can follow from a known distributional identity or be investigated through appropriate estimation and testing. Equal means, standardized shape, and sample size cannot replace that justification.

L² convergence permits passage to the limiting variance; convergence in distribution needs additional conditions controlling second moments. Characteristic functions identify complete laws, whereas determination by moments requires its own conditions.

With finite data, the variance ratio and its confidence interval explain the size and uncertainty of dispersion differences. Power evaluates the design’s ability to detect differences; equivalence requires prior substantive margins. Each conclusion must match its target, model, and actual supporting evidence.

References and further reading

1. MIT OpenCourseWare. Fundamentals of Probability (2018), lectures 16–19. Read source.
2. Lin, G. D. (2017). Recent developments on the moment problem. Journal of Statistical Distributions and Applications, 4, 5. DOI: 10.1186/s40488-017-0059-2. Read source.
3. Duke University. The Probability Workbook: Characteristic functions (aka Fourier Transforms). Read source.
4. NIST/SEMATECH. e-Handbook of Statistical Methods, §1.3.5.9, F-Test for Equality of Two Variances. Read source.
5. R Core Team. stats: var.test — F Test to Compare Two Variances. Read source.
6. NIST/SEMATECH. e-Handbook of Statistical Methods, §1.3.5.10, Levene Test for Equality of Variances. Read source.
7. R Core Team. stats: fligner.test — Fligner-Killeen Test of Homogeneity of Variances. Read source.
8. R Core Team. stats: t.test — Student’s t-Test (Welch procedure when var.equal = FALSE). Read source.