What is f test

The two-tailed version tests against the alternative that the variances are not equal. 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. The choice is determined by the problem. For example, if we are testing a new process, we may only be interested in knowing if the new process is less variable than the old process.

The F hypothesis test is defined as: H 0 :. The more this ratio deviates from 1, the stronger the evidence for unequal population variances. The details of conducting a one-way ANOVA fall into three categories: 1 writing hypotheses, 2 keeping the calculations organized, and 3 using the F-tables. The null hypothesis is that all of the population means are equal, and the alternative is that not all of the means are equal. Quite often, though two hypotheses are really needed for completeness, only H o is written:.

Keeping the calculations organized is important when you are finding the variance within. Remember that the variance within is found by squaring, and then summing, the distance between each observation and the mean of its sample. Though different people do the calculations differently, I find the best way to keep it all straight is to find the sample means, find the squared distances in each of the samples, and then add those together. It is also important to keep the calculations organized in the final computing of the F-score.

If you remember that the goal is to see if the variance between is large, then its easy to remember to divide variance between by variance within. Using the F-tables is the third detail. Though the null hypothesis is that all of the means are equal, you are testing that hypothesis by seeing if the variance between is less than or equal to the variance within.

The number of degrees of freedom is m -1, n — m , where m is the number of samples and n is the total size of all the samples together.

The young bank manager in Example 1 is still struggling with finding the best way to staff her branch. She knows that she needs to have more tellers on Fridays than on other days, but she is trying to find if the need for tellers is constant across the rest of the week. She collects data for the number of transactions each day for two months. Here are her data:. You can enter the number of transactions each day in the yellow cells in Figure 6. As you can then see in Figure 6.

Because her F-score is larger than the critical F-value, or alternatively since the p-value 0. She will want to adjust her staffing so that she has more tellers on some days than on others.

The F-distribution is the sampling distribution of the ratio of the variances of two samples drawn from a normal population. It is used directly to test to see if two samples come from populations with the same variance. Though you will occasionally see it used to test equality of variances, the more important use is in analysis of variance ANOVA. ANOVA, at least in its simplest form as presented in this chapter, is used to test to see if three or more samples come from populations with the same mean.

By testing to see if the variance of the observations comes more from the variation of each observation from the mean of its sample or from the variation of the means of the samples from the grand mean, ANOVA tests to see if the samples come from populations with equal means or not. ANOVA has more elegant forms that appear in later chapters. It forms the basis for regression analysis, a statistical technique that has many business applications; it is covered in later chapters.

The F-tables are also used in testing hypotheses about regression results. This is also the beginning of multivariate statistics. Notice that in the one-way ANOVA, each observation is for two variables: the x variable and the group of which the observation is a part.

In later chapters, observations will have two, three, or more variables. The F-test for equality of variances is sometimes used before using the t-test for equality of means because the t-test, at least in the form presented in this text, requires that the samples come from populations with equal variances. You will see it used along with t-tests when the stakes are high or the researcher is a little compulsive.

Tiemann is licensed under a Creative Commons Attribution 4. Skip to content Main Body. Previous: Chapter 5. The t-Test. Next: Chapter 7. Some Non-Parametric Tests. Share This Book Share on Twitter.

These group means are distributed around the overall mean for all 40 observations, which is 9. If the group means are clustered close to the overall mean, their variance is low. However, if the group means are spread out further from the overall mean, their variance is higher. Clearly, if we want to show that the group means are different, it helps if the means are further apart from each other. In other words, we want higher variability among the means. The graph below shows the spread of the means.

Each dot represents the mean of an entire group. The further the dots are spread out, the higher the value of the variability in the numerator of the F-statistic. What value do we use to measure the variance between sample means for the plastic strength example? Just keep in mind that the further apart the group means are, the larger this number becomes. We also need an estimate of the variability within each sample. To calculate this variance, we need to calculate how far each observation is from its group mean for all 40 observations.

Technically, it is the sum of the squared deviations of each observation from its group mean divided by the error DF. If the observations for each group are close to the group mean, the variance within the samples is low. However, if the observations for each group are further from the group mean, the variance within the samples is higher.

In the graph, the panel on the left shows low variation in the samples while the panel on the right shows high variation. The more spread out the observations are from their group mean, the higher the value in the denominator of the F-statistic. You can think of the within-group variance as the background noise that can obscure a difference between means. The F-statistic is the test statistic for F-tests.

In general, an F-statistic is a ratio of two quantities that are expected to be roughly equal under the null hypothesis, which produces an F-statistic of approximately 1. The F-statistic incorporates both measures of variability discussed above. Let's take a look at how these measures can work together to produce low and high F-values. Look at the graphs below and compare the width of the spread of the group means to the width of the spread within each group.

The low F-value graph shows a case where the group means are close together low variability relative to the variability within each group. The high F-value graph shows a case where the variability of group means is large relative to the within group variability. In order to reject the null hypothesis that the group means are equal, we need a high F-value.