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What Is The F Distribution? Discuss Its uses And Characteristics

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  • Course Code: MAN6712
  • Course Title: What Is The F Distribution? Discuss Its uses And Characteristics
  • University: Unversity of Michigan Law

F Distribution and F Statistics

The F-Distribution is a right skewed distribution that is commonly used in the application of statistical analysis of variance. In Probability theory and in stats the F-distribution is also known as Snedecor's F distribution, which reflects a continuous probability distribution that arises frequently as the null distribution of a test statistics, which as above stated, is also referred as analysis of variance.  

It is equally important to understand the F statistics in order to compute the F test Distributions. The equations that are commonly used for the computation of F Statistics are as follow:

  • f = [ s12/σ12 ] / [ s22/σ22 ]
  • f = [ s12 * σ22 ] / [ s22 * σ12 ]
  • f = [ Χ21 / v1 ] / [ Χ22 / v2 ]
  • f = [ Χ21 * v2 ] / [ Χ22 * v1 ]

 

The distribution of all possible values for that can be determined from the Statistics is called an F-Distribution where v1 = n1-1 and v2 = n2-1. The curve of the F-distribution is indeed dependent on the degree of freedom with the variables v1 and v2. In terms of explanation of F-Distribution it is to be noted that the F-distribution shows the number of degree of freedom that are associated with the standard deviation in the numerator of statistics is usually taken on a primary basis or stated firstly.

The key properties of F distribution are as follows:

  • The distribution mean is equal to v2 / (v2 - 2) for v2 > 2.
  • Variance is always equal with [ 2 * v22 * ( v1 + v1 - 2 ) ] / [ v1 * ( v2 - 2 )2 * ( v2 - 4 ) ] for v2 > 4.

Usage of F Distribution

One-Way Analysis of Variance

In terms of statistics, the application and usage of one-way variance, which is also called as one-way ANOVA, is a type of statistical tool that can be used for the purpose of comparing means of two or more samples with the help of a F-Distribution. The technique can be used for the purpose of numerical response data, the variable “Y”, that is almost one variable and numerically which is usually the categorical input data, the “X” is always one variable, hence it is called one-way.

Test of Two Variance

An F test is generally used for the purpose of testing that variances of the two populations are equal. The test that would be applied can be both in the way of two-tailed test or a one tailed test. The application of the F Test is done in order to check whether the variances between the two populations are equal or not. The one-tailed version only tests in one direction, that is, the variance from the first population is either greater than or less than (but not both) the second population variance. On the other hand the F-test distribution is also used for the purpose of checking the significance of the overall model and data whether the same can be statistically applied or not.   

Characteristics of F distribution

The characteristics of F distribution are listed below:

  • The distribution curve in F distribution is not symmetrical. It is skewed to the right, that is, the curve is positively skewed.
  • As the degrees of freedom v1 and v2 increase, the skewness of F distribution decreases. In other words, when the degrees of freedom for the denominator and the numerator become larger and the curve approximates normal distribution.
  • Value of F distribution is either positive or zero as the variances represent the square of the deviations. Thus, this distribution cannot assume negative values. The value of this distribution lies between 0 and infinity.
  • Each set of degrees of freedom has different distribution curve.
  • The shape of the F distribution is dependent on its parameters, v1 and v2 degrees of freedom.
  • In this distribution, the value of mean and variance are calculated in the following way:

 

  • Values of the area on the left hand side of the distribution curve can be obtained by taking the reciprocal of the F values corresponding to the right hand side and by interchanging the degrees of freedom in the denominator and the numerator. It is known as the reciprocal property of the F distribution. It is represented as:

 

This property is majorly used in situations when the lower tail F values needed to be determined in correspond to the upper tail F values.

Critical value of F distribution

The critical values for F distribution are calculated at p = 0.10 level of significance, p = 0.05 level of significance and p = 0.01 level of significance. The critical value is also known as the F statistic. To obtain the critical value, F ratio, which has (x, y) degrees of freedom associated with it. The intersection between x and y values represents the critical F ratio. If the obtained F ratio is equal or more than the critical F value, then the outcome is significant at that level of probability.

For example, from an F test, suppose, the obtained F ratio is 3.96 with (2, 24) degrees of freedom in case with ANOVA. In the table for the critical value of F for the 0.05 level of significance with ANOVA, the intersection point for 2 columns and 24 rows gives the value of 3.40. As the obtained F ratio is larger than 3.40, hence, it can be concluded that the obtained F ratio is likely to occur with a chance of p < 0.05, that is, the result is significant at 0.05 level of significance. Similar procedure is followed to obtain the critical value of F statistic for the 0.01 level of significance.  

Examples of the F distribution

As F distribution represents continuous probability distribution, hence, it can be defined for infinite number of different values. This distribution is used for various types of applications, such as, hypothesis testing regarding the equality of two population variances and evaluating the validity of multiple regression equations. Household income is an example of positively skewed distribution. For example, in a particular region, half of the total households have incomes below $50000 and the other half have income more than $50000. Hence, the median income is around $50000. $0 is the smallest possible income for households with less than $50000 income and for households with more than $50000 income, the income value can go up to any amount. Hence, there is a huge imbalance in the incomes between the two groups and it pushes the mean substantially higher than the median, say, $120000.

 

Figure 1: F distribution for household incomes

It shows F distribution curves for different degrees of freedom for the 0.05 level of significance. All the curves are positively skewed. With increase in the degrees of freedom, shape of the distribution curve shifts to the right. Null and alternate hypotheses are created to test the case study. The F statistic obtained from the hypothesis testing is compared with the critical value of F for the corresponding degrees of freedom at the 0.05 level of significance in the above case. If the obtained F value is higher than the critical value of F, then the outcome is considered to be significant at the 0.05 level of significance.

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