Variances and Regression Residuals
Hartley's Fmax Table: Critical Values for Homogeneity of Variance
Critical values of the ratio of the largest to the smallest sample variance, a quick check for equal variances.
- k = 2 to 12 groups
- Exact integration
- Exact p-values
Hartley's Fmax Table
Jump to a significance level below. Rows: df per group (n − 1) · Columns: number of groups k · Cells: critical Fmax. Reject equal variances if Fmax > the cell.
α = 0.05
| Number of groups (k) → df (n − 1) ↓ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 39.00 | 87.49 | 142.5 | 202.4 | 266.2 | 333.2 | 403.1 | 475.4 | 549.8 | 626.2 | 704.4 |
| 3 | 15.44 | 27.76 | 39.51 | 50.88 | 61.98 | 72.83 | 83.48 | 93.94 | 104.2 | 114.4 | 124.4 |
| 4 | 9.605 | 15.46 | 20.56 | 25.21 | 29.54 | 33.63 | 37.52 | 41.24 | 44.81 | 48.27 | 51.61 |
| 5 | 7.146 | 10.75 | 13.72 | 16.34 | 18.70 | 20.88 | 22.91 | 24.83 | 26.65 | 28.38 | 30.03 |
| 6 | 5.820 | 8.363 | 10.38 | 12.11 | 13.64 | 15.04 | 16.32 | 17.51 | 18.64 | 19.70 | 20.70 |
| 7 | 4.995 | 6.940 | 8.440 | 9.701 | 10.80 | 11.80 | 12.70 | 13.54 | 14.31 | 15.05 | 15.74 |
| 8 | 4.433 | 6.002 | 7.185 | 8.166 | 9.015 | 9.771 | 10.46 | 11.08 | 11.67 | 12.21 | 12.72 |
| 9 | 4.026 | 5.338 | 6.312 | 7.109 | 7.793 | 8.398 | 8.942 | 9.440 | 9.899 | 10.33 | 10.73 |
| 10 | 3.717 | 4.845 | 5.670 | 6.339 | 6.909 | 7.410 | 7.860 | 8.268 | 8.644 | 8.993 | 9.319 |
| 12 | 3.277 | 4.160 | 4.791 | 5.295 | 5.720 | 6.091 | 6.420 | 6.718 | 6.991 | 7.242 | 7.477 |
| 15 | 2.862 | 3.532 | 3.999 | 4.367 | 4.673 | 4.938 | 5.171 | 5.381 | 5.572 | 5.747 | 5.909 |
| 20 | 2.464 | 2.949 | 3.278 | 3.533 | 3.743 | 3.922 | 4.079 | 4.219 | 4.345 | 4.461 | 4.567 |
| 30 | 2.074 | 2.396 | 2.609 | 2.770 | 2.901 | 3.012 | 3.108 | 3.193 | 3.269 | 3.338 | 3.402 |
| 60 | 1.667 | 1.843 | 1.955 | 2.039 | 2.105 | 2.161 | 2.208 | 2.250 | 2.287 | 2.320 | 2.351 |
| ∞ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
α = 0.01
| Number of groups (k) → df (n − 1) ↓ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 199.0 | 447.5 | 729.2 | 1035.7 | 1362.2 | 1705.3 | 2062.6 | 2432.3 | 2813.1 | 3203.8 | 3603.6 |
| 3 | 47.47 | 84.56 | 119.8 | 153.8 | 187.0 | 219.3 | 251.1 | 282.3 | 313.0 | 343.2 | 373.1 |
| 4 | 23.15 | 36.70 | 48.43 | 59.09 | 69.00 | 78.33 | 87.20 | 95.68 | 103.8 | 111.7 | 119.3 |
| 5 | 14.94 | 22.06 | 27.90 | 33.00 | 37.61 | 41.85 | 45.81 | 49.53 | 53.06 | 56.42 | 59.63 |
| 6 | 11.07 | 15.60 | 19.16 | 22.19 | 24.89 | 27.32 | 29.57 | 31.65 | 33.61 | 35.46 | 37.22 |
| 7 | 8.885 | 12.09 | 14.55 | 16.60 | 18.39 | 20.00 | 21.47 | 22.82 | 24.08 | 25.26 | 26.37 |
| 8 | 7.496 | 9.937 | 11.77 | 13.27 | 14.58 | 15.73 | 16.78 | 17.74 | 18.63 | 19.46 | 20.24 |
| 9 | 6.541 | 8.494 | 9.931 | 11.10 | 12.11 | 12.99 | 13.79 | 14.52 | 15.19 | 15.81 | 16.39 |
| 10 | 5.847 | 7.465 | 8.639 | 9.587 | 10.39 | 11.10 | 11.74 | 12.31 | 12.84 | 13.33 | 13.79 |
| 12 | 4.906 | 6.103 | 6.952 | 7.629 | 8.198 | 8.693 | 9.133 | 9.531 | 9.894 | 10.23 | 10.54 |
| 15 | 4.070 | 4.926 | 5.520 | 5.986 | 6.374 | 6.708 | 7.003 | 7.267 | 7.508 | 7.728 | 7.933 |
| 20 | 3.318 | 3.900 | 4.294 | 4.599 | 4.849 | 5.062 | 5.249 | 5.415 | 5.566 | 5.703 | 5.830 |
| 30 | 2.628 | 2.990 | 3.228 | 3.409 | 3.556 | 3.679 | 3.787 | 3.882 | 3.967 | 4.044 | 4.115 |
| 60 | 1.962 | 2.145 | 2.262 | 2.348 | 2.417 | 2.475 | 2.524 | 2.567 | 2.606 | 2.641 | 2.672 |
| ∞ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Tip: click any value to highlight its row and column.
How to read this table
- Compute the variance in each group; all groups should be the same size n.
- Fmax = largest variance / smallest variance.
- Find the row for df = n − 1 and the column for the number of groups k.
- Reject equal variances if Fmax exceeds the table value.
Worked example
Four treatment groups of 11 patients have variances 4.1, 3.6, 12.8 and 5.0, so Fmax = 12.8 / 3.6 = 3.56. Row df = 10, column k = 4 gives 5.670, so the variances are not significantly different at the 5% level.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
Values are computed exactly by numerical integration of the distribution of the ratio of the largest to the smallest of k independent chi-square variables. The widely reprinted Pearson and Hartley (1954) table was computed with approximations and can differ by up to about 1% for many groups.
Frequently asked questions
Is Fmax a good test?
It is simple but, like all variance ratio tests, very sensitive to non-normality. Levene's or the Brown-Forsythe test is more robust for real data.
What if my groups have different sizes?
Use the smaller df (from the smaller group) for a conservative answer, or use Levene's test instead.