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Critical values of F-Statistics
An on-line statistical calculator

This calculator will compute the critical values of F-statistics corresponding to nN (numerator) and nD (denominator) degrees of freedom, at the desired probability level. See for example Table 3-1 on page 44 of Stanton A. Glanz "Primer of Biostatistics", 3rd Edition, McGraw-Hill, New York, 1992. The numerator and denominator degrees of freedom must be whole numbers correponding to the sample sizes.

Degrees of freedom - numerator:
Degrees of freedom - denominator:
Probability level (alpha):



The following is Problem 3-2 from Glanz (1992). The effect of "environmental smoke" on lung disease was studied by measuring the mean forced mid-respiratory flow for five groups of people. Each group contained 200 subjects.

group respiratory flow standard deviation
nonsmokers working in clean environment 3.17 0.74
nonsmokers working in smoky environment 2.72 0.71
light smokers
2.63 0.73
moderate smokers
2.29 0.70
heavy smokers
2.12 0.72

Is there any evidence that the mean respiratory flow is any different among the five different experimental groups?


First we estimate the "within groups" variance as s2wit = ( 0.742 + 0.712 + 0.732 + 0.702 + 0.722 )/5 = 0.5186.

Next we compute the standard deviation of the sample means:

Number of data points5
Standard deviation:0.407958331205529
Standard error of the mean:0.182444512112587

From these results, the "between groups" estimate of variance is s2bet = 200 × (0.4082) = 33.29. The F-statistics is defined as F = s2bet / s2wit = 33.29 / 0.5186 = 64.19.

We are now ready to test the null hypothesis, which assumes that there is no difference between aerial from in the five different groups of test subjects. If the hypothesis is true, the critical value of F at (say) 95% confidence level (α = 0.05) should be larger than 64.19.

The numerator degrees of freedom are equal to the number of groups minus one: nN = 5 - 1 = 4. The denominator degrees of freedom are number of groups × (number of subjects minus one: nD = 4 × (200 - 1) = 796.

Type the "4" and "796" into the editable fields in the form above, select the confidence level (0.05) and clisk on the Calculate button. The results are:

Degrees of freedom: numerator4
Degrees of freedom: denominator796
Probability level:0.049999990430516
Critical value of F:2.383056640625


The critical value of F at 95% probability level is much lower (2.38) than the observed value of F (64.19), which means that the null hypothesis is false. The data does suggest that the differenes between aerial flow seen within different groups (smokers, nonsmokers) are significant.

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Sat Apr 20 18:22:32 2019