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12.4: Facts About the F Distribution

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    Here are some facts about the \(\bf F\) distribution.

    1. The curve is not symmetrical but skewed to the right.
    2. There is a different curve for each set of degrees of freedom.
    3. The \(F\) statistic is greater than or equal to zero.
    4. 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.
    5. 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.
    This graph has an unmarked Y axis and then an X axis that ranges from 0.00 to 4.00. It has three plot lines. The plot line labelled F subscript 1, 5 starts near the top of the Y axis at the extreme left of the graph and drops quickly to near the bottom at 0.50, at which point is slowly decreases in a curved fashion to the 4.00 mark on the X axis. The plot line labelled F subscript 100, 100 remains at Y = 0 for much of its length, except for a distinct peak between 0.50 and 1.50. The peak is a smooth curve that reaches about half way up the Y axis at its peak. The plot line labelled F subscript 5, 10 increases slightly as it progresses from 0.00 to 0.50, after which it peaks and slowly decreases down the remainder of the X axis. The peak only reaches about one fifth up the height of the Y axis.
    Figure \(\PageIndex{7}\)

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