Hartley Fmax Critical Value Lookup
What Is the Hartley Fmax Critical Value?
Hartley's Fmax test checks whether k groups have equal population variances. The test statistic is the ratio of the largest sample variance to the smallest:
The critical value comes from the Fmax distribution — not the standard F-distribution — and depends on three things: the number of groups k, the within-group degrees of freedom v, and the chosen significance level α.
If Fmax ≤ Fmax,critical → Fail to reject H₀ (insufficient evidence to reject equal variances)
Hartley Fmax Critical Values Table
Select the α level. Rows are within-group degrees of freedom v; columns are number of groups k. Click any cell to load it into the lookup tool above.
Table source: Pearson, E. S. & Hartley, H. O. (1954). Biometrika Tables for Statisticians, Vol. 1. Cambridge University Press. Values reproduced from Table 31. v = within-group degrees of freedom (v = n − 1 for equal group sizes). k = number of groups. Reject H₀ if Fmax > critical value. Do not substitute ordinary F-distribution critical values for these Fmax critical values.
What Do k and v Mean in the Hartley Fmax Table?
Two parameters uniquely determine which cell in the Hartley Fmax table you need. Getting these right is the most critical step in the lookup.
k = Number of Groups
k is the total count of independent groups whose variances you are comparing. If you have three treatment conditions, k = 3. If you have five schools, k = 5. k ranges from 2 to 12 in the standard Hartley table.
v = Within-Group Degrees of Freedom
For a balanced design where every group has n observations: v = n − 1. This is the within-group df. It is not the ANOVA between-group df (k − 1). Confusing these is the most common error when using the Fmax table.
Concrete Example
Suppose a study has k = 4 groups, each containing n = 10 observations. The within-group degrees of freedom are v = 10 − 1 = 9. To find the critical Fmax at α = 0.05, locate the row for v = 9 and the column for k = 4. The critical value is 8.95. The correct column to use is k = 4, not k − 1 = 3, and the correct row is v = 9, not n = 10.
Critical distinction: v = n − 1 is the within-group degrees of freedom used in the Fmax table. The value k − 1 is the between-group degrees of freedom used in one-way ANOVA. These are separate quantities. Using k − 1 in the Fmax table lookup produces an incorrect critical value.
Hartley Fmax Formula
The Fmax statistic requires only two values from your data: the largest sample variance and the smallest.
Where s²max is the largest sample variance among all k groups, and s²min is the smallest. The ratio is always ≥ 1. A ratio close to 1 means the group variances are similar; a large ratio indicates disparity.
Quick Numerical Example
Three groups have sample variances of 12.4, 8.1, and 25.6. The largest is 25.6, the smallest is 8.1.
This computed value of 3.16 is then compared against the Fmax critical value for k = 3 and the appropriate v at the chosen α.
How to Use the Hartley Fmax Table
The nine steps below cover every decision from data to conclusion.
Step 1 — Count the Groups (k)
Record k, the total number of independent groups you are comparing. This directly determines which column to use in the table.
Step 2 — Compute Within-Group df (v)
For balanced designs (equal n per group): v = n − 1. For example, if each group has 15 observations, v = 14. Use this value to find the correct table row.
Step 3 — Choose α
Set your significance level before looking at the data. The most common choice is α = 0.05. The critical value changes depending on which α column you use, so commit to this choice first.
Step 4 — Calculate All Sample Variances
Compute the sample variance s² for each group. The standard formula is s² = Σ(xᵢ − x̄)² / (n − 1). Statistical software handles this automatically.
Step 5 — Identify s²max and s²min
Scan the k sample variances. Record the largest (s²max) and the smallest (s²min). The Fmax statistic uses only these two — the intermediate variances are not part of the calculation.
Step 6 — Compute Fmax
Divide: Fmax = s²max / s²min. The result is always ≥ 1. Make sure you are dividing sample variances (s²), not standard deviations (s).
Step 7 — Look Up the Critical Value
Use the table above (or the interactive lookup). Find the column for your k, the row for your v, and the sub-table for your α. That cell is Fmax,critical.
Step 8 — Compare Fmax to Fmax,critical
Fmax ≤ Fmax,critical → Fail to reject H₀
Step 9 — Interpret the Decision
If you reject H₀: there is evidence that the group variances are not homogeneous. If you fail to reject H₀: the data do not provide sufficient evidence to conclude that variances differ — this is not proof that they are equal.
Worked Example — Non-Significant Result
Scenario: A researcher measures reaction times (ms) across k = 4 groups, each with n = 10 participants. The four sample variances are 104, 87, 143, and 112. Do the groups have homogeneous variances at α = 0.05?
Solution
| Quantity | Value |
|---|---|
| Number of groups | k = 4 |
| Observations per group | n = 10 |
| Within-group df | v = n − 1 = 9 |
| Significance level | α = 0.05 |
| Largest variance (s²max) | 143 |
| Smallest variance (s²min) | 87 |
| Computed Fmax | 143 / 87 = 1.64 |
| Critical value (k=4, v=9, α=0.05) | 8.95 |
| Decision | 1.64 ≤ 8.95 → Fail to Reject H₀ |
Interpretation
With Fmax = 1.64 well below the critical value of 8.95, the data provide insufficient evidence to reject the assumption of equal variances at α = 0.05. This does not prove that the population variances are identical — only that this sample does not contradict that assumption at the chosen threshold.
Worked Example — Significant Result
Scenario: Five groups (k = 5), each with n = 6 participants, yield sample variances of 18, 22, 14, 115, and 31. Test at α = 0.05.
Solution
| Quantity | Value |
|---|---|
| Number of groups | k = 5 |
| Observations per group | n = 6 |
| Within-group df | v = 6 − 1 = 5 |
| Significance level | α = 0.05 |
| Largest variance (s²max) | 115 |
| Smallest variance (s²min) | 14 |
| Computed Fmax | 115 / 14 = 8.21 |
| Critical value (k=5, v=5, α=0.05) | 14.94 |
| Decision | 8.21 ≤ 14.94 → Fail to Reject H₀* |
*The large apparent spread (115 vs 14) does not reach the conservative Fmax threshold here. Note how v = 5 (small n) produces a high critical value — Hartley's test has lower power with very small samples.
When You Do Reject H₀
If computed Fmax exceeds the critical value, there is evidence that group variances are not homogeneous under the test's assumptions. In the ANOVA context, this suggests the equal-variance assumption may be violated. Researchers may then consider a variance-stabilizing transformation, Welch's ANOVA, or the Brown-Forsythe procedure as alternatives to standard one-way ANOVA.
Hartley's Fmax Test and ANOVA
One of the standard assumptions of one-way ANOVA is that the population variances are equal across groups (homoscedasticity). Hartley's Fmax test can serve as a check on this assumption, particularly in balanced designs with approximately normal observations.
When ANOVA assumptions matter
One-way ANOVA is reasonably robust to moderate variance inequality when group sizes are equal. The consequences of heteroscedasticity are more serious with unequal group sizes. In those situations, the actual Type I error rate may differ from the nominal α. Graphical methods — such as box plots and residual plots — often give a cleaner diagnostic than a single statistical test.
Practical considerations
Not every ANOVA requires a formal preliminary variance test. Many statisticians prefer examining residual plots, considering the ratio of the largest to smallest variance directly, and applying a robust alternative (such as Welch's F-test) when there is doubt. A single significance test of normality or variance should not become a mechanical gatekeeper before every analysis. See the one-way ANOVA guide for a complete treatment.
Assumptions of Hartley's Fmax Test
The Fmax critical values in this table apply under a specific set of conditions. When these conditions are substantially violated, the test may produce misleading results.
Approximate Normality
Each group's observations should come from a roughly normal population. Hartley's Fmax is sensitive to departures from normality — skewness, heavy tails, and outliers all inflate sample variances, which directly affects Fmax.
Balanced Design (Equal n)
The classical Hartley Fmax table assumes equal sample sizes within each group. Applying these critical values to markedly unequal group sizes may yield inaccurate conclusions. Levene's test handles unequal group sizes more directly.
Independence
Observations must be independent within and across groups. Repeated measures, clustered data, or time-series observations violate this condition and require different approaches.
Continuous Measurement
Sample variances must be meaningful — the data should be measured on an interval or ratio scale, not on an ordinal or categorical scale.
Why Normality Matters Especially Here
Hartley's Fmax test is more sensitive to normality violations than Levene's test or the Brown-Forsythe test. A heavy-tailed distribution can produce an unusually large s² in one group simply from a single outlier, inflating Fmax and causing a false rejection of H₀. Before applying the Fmax test, inspect your data for outliers and gross departures from normality. For data with heavy tails or outliers, Levene's test based on the median (the Brown-Forsythe variant) is generally a more robust choice.
Hartley Fmax vs Standard F-Distribution Critical Values
Do Not Use the Standard F-Table for Hartley's Fmax Test
The Fmax statistic is the maximum variance ratio across k samples. Its critical values come from the specific sampling distribution of that maximum. The ordinary F-distribution gives critical values for a single two-sample variance ratio or a ratio of mean squares — not for the maximum of k such ratios. Substituting standard F critical values for Hartley Fmax critical values will produce incorrect decisions.
| Test | Purpose | Critical Value Source |
|---|---|---|
| Hartley Fmax | Compare largest and smallest sample variances across k groups | Fmax distribution table (this page) |
| Two-sample F test | Compare variances of exactly two groups | Standard F-distribution table |
| ANOVA F test | Compare group means (not variances) | Standard F-distribution table |
| Levene's test | Test variance homogeneity using deviations from group means or medians | F-distribution with df₁ = k−1, df₂ = N−k |
Hartley Fmax vs Levene's Test
Both tests address homogeneity of variance, but they do so differently and suit different situations.
| Feature | Hartley Fmax | Levene's Test |
|---|---|---|
| Statistic | Ratio of max to min variance | F-statistic on absolute deviations |
| Uses all group variances? | No — only max and min | Yes — all groups contribute |
| Normality sensitivity | High — less robust | Moderate — generally more robust |
| Unequal group sizes | Classical use assumes equal n | Handles unequal n naturally |
| Output | Fmax + critical value from table | F + p-value from software |
| Common application | Reference tables, educational use, balanced designs | Applied research, automated software output |
For data that are approximately normal and group sizes are equal, Hartley's Fmax is a simple, direct check. When normality is uncertain or group sizes differ, Levene's test (or the Brown-Forsythe variant based on deviations from the group median) is generally the more reliable option. See the Levene's test table for reference values.
Hartley Fmax vs Bartlett's Test
| Feature | Hartley Fmax | Bartlett's Test |
|---|---|---|
| Statistic basis | Max-to-min variance ratio | Pooled log-variance statistic |
| Uses all group variances? | No | Yes |
| Normality sensitivity | High | High — similarly sensitive |
| Preferred alternative | Levene's test or Brown-Forsythe test when normality is uncertain | |
Both Hartley's Fmax and Bartlett's test are sensitive to departures from normality. Bartlett's test uses all k sample variances in a single chi-squared statistic, while Hartley's approach focuses only on the extremes. Neither is recommended over Levene's test when data normality is in question.
Fmax Critical Values at α = 0.05 — Quick Reference
This condensed table covers the most commonly used combinations for fast lookups at the standard 5% significance level.
| k (groups) | v = 2 | v = 3 | v = 4 | v = 5 | v = 6 | v = 9 | v = 16 | v = 36 | v = ∞ |
|---|
Source: Pearson & Hartley (1954), Table 31, α = 0.05. v = within-group degrees of freedom (n − 1). Reject H₀ if Fmax > tabled value.
Common Hartley Fmax Table Mistakes
These errors recur frequently. Checking each one before reporting results takes only a moment and avoids incorrect conclusions.
Using k − 1 instead of v
The Fmax table row uses within-group df v = n − 1, not k − 1 (the ANOVA between-group df). These are different quantities and produce different critical values.
Using a standard F-table
Standard F-distribution tables are not valid for Hartley's Fmax test. The distributions are different. Using the wrong table produces incorrect critical values.
Dividing min by max
Fmax = s²max / s²min. Reversing the ratio produces a value ≤ 1, which will never exceed any critical value and leads to always failing to reject H₀.
Using standard deviations, not variances
The formula requires variances (s²), not standard deviations (s). Squaring the standard deviations first gives the correct numerator and denominator.
Claiming non-significance proves equal variances
Failing to reject H₀ means the test found insufficient evidence — not that variances are equal. With small samples, the test has low power and may miss real differences.
Ignoring outliers
A single outlier can dominate the sample variance in one group, inflating Fmax and causing a false rejection. Always inspect your data before interpreting the test result.
Academic Reporting Template
Adapt the template below when reporting Hartley Fmax results in a paper or report. Fill in your specific values.
"A Hartley Fmax test was conducted to assess the homogeneity of variances across [k] groups. The largest sample variance was [s²max] and the smallest was [s²min], giving Fmax = [value]. At α = [α], with k = [k] and v = [v], the critical value was [Fmax,critical]. Because Fmax [exceeded / did not exceed] the critical value, H₀ was [rejected / not rejected], indicating [evidence of heterogeneous variances / insufficient evidence to reject variance homogeneity]."
Note: If your software provides a p-value for Hartley's Fmax directly, report it alongside the test statistic. Many standard packages (R, SPSS, SAS) do not include a native Hartley Fmax procedure; custom calculation or specialist packages may be required.
Hartley Fmax Test in Statistical Software
Hartley's Fmax test is not included as a built-in procedure in most major statistical packages. Users typically compute Fmax manually (dividing the maximum by the minimum sample variance) and then compare against the table. Levene's test, which is available natively in R, SPSS, SAS, and Python's scipy.stats, is frequently used as a practical substitute.
R
No direct Hartley Fmax function in base R. Compute manually: max(sapply(groups, var)) / min(sapply(groups, var)). Then compare against the table. Use leveneTest() from the car package for an automated alternative.
SPSS
SPSS does not include a native Hartley Fmax procedure. Levene's test is reported automatically in the One-Way ANOVA output under "Test of Homogeneity of Variances." Manual Fmax calculation requires computing group variances from Descriptives output.
Python
No scipy function for Hartley Fmax. Compute manually using np.var(group, ddof=1) for each group and divide max by min. Use scipy.stats.levene() for the test with a p-value.
Frequently Asked Questions
What is Hartley's Fmax test?
Hartley's Fmax test is a procedure for checking whether k independent groups have equal population variances (homogeneity of variance). The test statistic, Fmax, is the ratio of the largest sample variance to the smallest across all groups. This ratio is compared against a critical value from the Fmax distribution, which accounts for the number of groups k and the within-group degrees of freedom v.
What is the Hartley Fmax critical value at α = 0.05 for k = 4 and v = 9?
At α = 0.05 with k = 4 groups and v = 9 within-group degrees of freedom, the Hartley Fmax critical value is 8.95. This corresponds to groups of n = 10 observations each (since v = n − 1 = 9). Reject H₀ if your computed Fmax exceeds 8.95.
Why can't I use a standard F-distribution table for Hartley's test?
The standard F-distribution gives critical values for a single ratio of two quantities (such as two variances or two mean squares). Hartley's Fmax is the maximum of k such ratios. The distribution of the maximum is broader than the distribution of any single ratio, so its critical values are larger. Using a standard F-table would set the bar too low and cause too many false rejections of H₀.
What does a non-significant Fmax result mean?
Failing to reject H₀ (Fmax ≤ Fmax,critical) means the data do not provide sufficient evidence to conclude that the population variances differ at the chosen α level. It is not proof that the variances are equal — the test may simply have low power with small samples, or the differences may be too small to detect with this sample size.
Can Hartley's Fmax test be used with unequal group sizes?
The classical Hartley Fmax table assumes balanced designs — equal sample sizes in every group. When group sizes differ substantially, the critical values in this table may not provide accurate control of the Type I error rate. Levene's test handles unequal group sizes more directly and is generally the preferred choice in those situations.
How does sample size affect the Fmax test?
Larger group sizes (larger n) mean larger v = n − 1, and the Fmax critical values decrease as v increases. With more data per group, sample variances become more precise estimates of their population counterparts, so a smaller observed ratio is needed to conclude that a real difference exists. Small groups produce conservative tests with high critical values and lower power.
What should I do if Fmax is significant?
A significant Fmax result indicates evidence of unequal population variances. Before concluding that a real difference exists, check for outliers or unusual observations in the group with the largest variance — a single outlier can produce a misleadingly large s². If outliers are not responsible, consider: a variance-stabilizing transformation, Welch's ANOVA (which does not assume equal variances), or non-parametric alternatives like the Kruskal-Wallis test.
Is Hartley's Fmax sensitive to non-normality?
Yes. Hartley's Fmax test is notably sensitive to departures from normality because the sample variance itself is influenced by skewness, heavy tails, and outliers. A non-normal distribution can produce inflated sample variances in some groups even when the population variances are equal, leading to false rejections. This sensitivity is well documented and is a primary reason why Levene's test (particularly the Brown-Forsythe variant using median deviations) is generally preferred in applied research.
How do I choose between Hartley Fmax, Levene's test, and Bartlett's test?
Use Hartley's Fmax when groups are balanced (equal n), data are approximately normal, and you want a quick manual calculation from a table. Use Levene's test (median-based / Brown-Forsythe) when you have unequal group sizes, suspect non-normality, or want a test with software-generated p-values. Bartlett's test shares the normality sensitivity of Hartley's Fmax but uses all k variances — it is generally not preferred over Levene's for the same reasons.
What does homogeneity of variance (homoscedasticity) mean?
Homogeneity of variance (also called homoscedasticity) means that the population variances of the k groups are equal: σ₁² = σ₂² = … = σk². It is one of the standard assumptions of one-way ANOVA. When variances are substantially unequal (heteroscedasticity), the standard ANOVA F-test may have an inflated or deflated Type I error rate, particularly with unequal group sizes.
Sources and Further Reading
The critical values on this page are taken from Pearson and Hartley's original tabulation and validated against secondary sources.
Pearson, E. S. & Hartley, H. O. (1954). Biometrika Tables for Statisticians, Vol. 1. Cambridge University Press. Table 31 — the primary authoritative source for Fmax critical values used on this page.
Kirk, R. E. (1982). Experimental Design: Procedures for the Behavioral Sciences (2nd ed.). Brooks/Cole. Appendix C includes the Hartley Fmax table with discussion of its use as a variance-homogeneity check before ANOVA.
Montgomery, D. C. (2017). Design and Analysis of Experiments (9th ed.). Wiley. Covers Hartley's Fmax test and Bartlett's test in the context of checking ANOVA assumptions, with guidance on the choice between tests. wiley.com
Penn State STAT 415: Introduction to Mathematical Statistics. Hartley's F-max Test. Pennsylvania State University. online.stat.psu.edu — Open-access course notes covering the Fmax statistic, assumptions, and relationship to ANOVA.
NIST/SEMATECH e-Handbook of Statistical Methods. Section 7.4.3.3: Bartlett's Test. National Institute of Standards and Technology. itl.nist.gov — U.S. government statistical reference comparing variance-homogeneity test procedures.
Key Takeaways
The Formula is Simple
Fmax = s²max / s²min. The ratio uses variances, not standard deviations, and the larger value always goes in the numerator.
Two Table Parameters
k (number of groups) and v (within-group df = n − 1) determine the correct critical value. Getting these wrong changes the threshold and the decision.
Fmax ≠ Standard F
The Fmax critical value table is specific to Hartley's test. Standard F-distribution critical values are not interchangeable with Fmax critical values.
Normality Matters
Hartley's Fmax test is sensitive to departures from normality. When normality is in doubt, Levene's test is generally more appropriate.