The curve is not symmetrical but skewed to the right.There is a different curve for each set of degrees of freedom.The *F* statistic is greater than or equal to zero.As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal as can be seen in the two figures below. Figure (b) with more degrees of freedom is more closely approaching the normal distribution, but remember that the *F* cannot ever be less than zero so the distribution does not have a tail that goes to infinity on the left as the normal distribution does.Other uses for the *F* distribution include comparing two variances and two-way Analysis of Variance. Two-Way Analysis is beyond the scope of this chapter.

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### References

Data from a fourth grade classroom in 1994 in a private K – 12 school in San Jose, CA.

Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. *A Handbook of Small Datasets: Data for Fruitfly Fecundity.* London: Chapman & Hall, 1994.

Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. *A Handbook of Small Datasets.* London: Chapman & Hall, 1994, pg. 50.

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Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 118.

“MLB Standings – 2012.” Available online at http://espn.go.com/mlb/standings/_/year/2012.

Mackowiak, P. A., Wasserman, S. S., and Levine, M. M. (1992), “A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich,” *Journal of the American Medical Association*, 268, 1578-1580.

### Chapter Review

The graph of the *F* distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom. The *F* statistic is the ratio of a measure of the variation in the group means to a similar measure of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A small *F* statistic will result, and the area under the *F* curve to the right will be large, representing a large *p*-value. When the null hypothesis of equal group means is incorrect, then the numerator should be large compared to the denominator, giving a large *F* statistic and a small area (small *p*-value) to the right of the statistic under the *F* curve.

When the data have unequal group sizes (unbalanced data), then techniques from (Figure) need to be used for hand calculations. In the case of balanced data (the groups are the same size) however, simplified calculations based on group means and variances may be used. In practice, of course, software is usually employed in the analysis. As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look at your data!

What happens to the curves as the degrees of freedom for the numerator and the denominator get larger?

*Use the following information to answer the next seven exercise.*

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Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in (Figure).