Suppose processes A and B both have mean yield 82. Most A values lie from 80 to 84, while B ranges broadly from 70 to 94. Their means look identical, but predictability of the next batch and the standard error of a mean difference are not the same at all.
This unit asks one question.
How does uncertainty in a mean comparison change when two groups have different spread?
Spread is background noise for a mean comparison
An observed difference between two means must be compared with natural variation within each group. Tightly clustered values can make a small mean difference clear; widely spread values can bury the same difference in sampling fluctuation. Assuming equal population variances is homoscedasticity; assuming different variances is heteroscedasticity.
Equal variance does not mean two sample SDs must match exactly. Samples from the same population variance have different SDs each time. Conversely, two accidentally similar SDs do not guarantee equal uncertainty in a mean comparison when distribution shapes or sample sizes differ.
First place dot plots, box plots, and SDs for each group side by side. Ask whether extra width reflects an overall scale change, one or two outliers, a right tail, or a mixture of subgroups. A variance test does not replace these plots.
Spread tests do not ask the same question in the same way
A common null is approximately H0: σ₁²=σ₂²=…=σk², but test statistics measure spread differently and have different distributional assumptions.
- A two-group F test compares the ratio of sample variances with an F distribution; it is sensitive to the normal model.
- The Bartlett test can be efficient across groups but is sensitive to non-normality and outliers.
- The Levene test compares absolute distances of observations from their group center as a new response.
- A Levene variant using the median rather than the mean center is often called Brown–Forsythe and can be more robust to tails and outliers.
Rather than memorize names, ask which center and distance each method uses and how sensitive it is to departures from normality.
A non-significant spread test means the current sample lacks evidence to detect a variance difference. Small n can miss even a large variance ratio. Do not use it as proof of equality or automatic permission for a pooled t-test.
Do not split the mean test by one preliminary test
An older common procedure first tested equal variances, then chose Welch t-test when significant and pooled t-test otherwise. This two-step automatic branch complicates final error rates and uncertainty, and is unstable in small samples.
Choose the analysis before looking at data, using design and scientific assumptions. For two independent group means, Welch's method separately reflects group variances and n, making it a useful robust option when equal variance is uncertain. It is not the sole answer for every design: paired, clustered, repeated-measures, or censored data need a structure-matched model.
In-Silico Lab: keep the mean fixed and change only width
In the Lab, both groups have the same true mean. Change only B's SD ratio, n, and seed.
- Raise the SD ratio from 1 to 3 and see that the sample-SD ratio is not exactly the set value every time.
- Repeat new samples at n=8 and n=40 and compare fluctuation in estimated spread.
- Find a sample where SD and median absolute deviation do not move in the same direction.
- Answer again whether equal means imply equal stability.
평균을 고정하고 산포만 바꾸어 보세요
같은 평균을 가진 두 집단에서 B의 SD 비율을 바꾸며 표본 SD와 robust distance가 어떻게 움직이는지 봅니다.
처음이라면: 무엇을 눌러야 하나요?
- 1. 질문을 먼저 읽기Lab 제목에서 이번에 비교할 한 가지를 확인합니다.
- 2. 조건 하나만 바꾸기처음에는 n, 효과, 산포 같은 입력 중 하나만 바꾸십시오.
- 3. 새 합성 표본 누르기새 합성 데이터가 만들어집니다. 같은 조건도 표본에 따라 달라질 수 있습니다.
- 4. 그림과 계산 결과 비교하기바꾸기 전후 무엇이 움직이고 무엇이 그대로인지 한 문장으로 적어보십시오.
막히면 초기화로 돌아가 기본 결과를 본 뒤 조건 하나만 바꾸십시오. 이 Lab은 정답 판정기가 아니라 패턴 관찰 도구입니다.
같은 설정의 합성 관측
계산 결과
표본 산포 차이는 자료가 만드는 증거입니다. p값 하나로 두 모집단 분산이 같다고 증명할 수 없습니다.
교육용 synthetic model · bjs-comparison-sequence-v1. 실제 연구 판단에는 실험단위, 결측, 분포, 다중성, 사전계획과 도메인 기준을 별도로 반영해야 합니다.
The Lab does not make spread-test p-values compete. It first shows which data-generating conditions change which spread summaries. With real data, also inspect raw-data plots, sample size, causes of outliers, measurement limits, and independence.
JMP output puts several sensitivities side by side
F·Levene·Brown-Forsythe류 결과는 같은 산포 질문을 서로 다른 민감도로 봅니다.
표본 SD의 크기와 불확실성을 그림으로 먼저 확인합니다.
p값 하나가 아니라 분포 모양·n·이상값과 함께 판단합니다.
The goal is not to follow a historical JMP menu path. Read how each result examines the same H0, and how much Std Dev and CIs show the actual scale. Do not pick the most favorable p-value among several tests.
Separate spread and mean questions in reporting
Poor statement:
Levene p=.31, so the two variances are equal and we used a pooled t-test.
Better statement:
Sample SDs for independent batches were 1.8 in A and 3.1 in B. B showed wider spread and a right tail. We estimated the mean difference using the pre-specified Welch method and presented the spread comparison as sensitivity information with raw-data plots and a median-based Levene result.
If variance difference itself is the research question, report a CI for a variance ratio or scale difference. If it is an assumption check for a mean comparison, the mean effect and CI are central and the spread test is supporting diagnosis.
Takeaways
- Equal variance is a population-model assumption, not matching sample SDs.
- Spread differences change standard error and predictability for mean differences.
- F and Bartlett are sensitive to departures from normality; Levene methods use center distances for more robustness.
- A non-significant spread test does not prove equal variances.
- Do not mechanically change analysis after a spread test.
- Raw-data plots and experimental units come before every test.
The next unit compares one group mean with an external target and joins effect, standard error, CI, and p-value in one result statement.
Official supplementary resources
The values, figures, and Lab in this article are educational synthetic material. They cannot be the sole basis for real research, clinical, quality, or regulatory decisions.