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Analysis of Variance/ANOVA

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Source: Table created by Jonathan Osters

In this tutorial, you're going to learn about the process for analysis of variance. This is also called ANOVA. And this is a hypothesis test that allows us to test more than two population means.

So let's take a look at when we would use an ANOVA scenario. A factory supervisor wants to know whether it takes his workers different amounts of time to complete the task based on their proficiency level. So he has apprentices, novices, and masters. And he randomly selects 10 from each group and has them perform the task. The summary of the data, time in minutes to complete the task, is shown in this table here. And we want to know if these sample means are significantly different from each other. And this is where the analysis of variance is going to come in.

Comparing three or more means requires a new hypothesis test that we didn't know before, called analysis of variance. Typically it's called ANOVA. The AN is for "analysis", the O is for "of", and the VA is for "variance". We compare the means by analyzing the sample variances from the independently selected sample. So what you might notice is that independently selected samples is one of the conditions for the test.

The conditions are independent samples from the populations. Each population has to be normally distributed. And the variances, and therefore the standard deviations of all those normal distributions, are the same. In this scenario we're not going to actually check these. We're just going to assume that they're met, because we want to give you sort of the overall flavor for what an ANOVA test looks like.

So in an ANOVA test, the null hypothesis is that all the means are the same for the masters, the novices, and the apprentices. So the mean time required to complete the task is the same for each of the experience levels. The alternative hypothesis is that at least one of these is different from the others. We will pick an alpha level and we'll go ahead and assume the conditions are met.

When we do an ANOVA test, the statistic that we use is not going to be a z or t like it's been in the past. It's going to be what we call an F. And an F statistic is calculated by taking the quotient of the variability between the samples and the variability within each sample.

Now, this number, this F number, will be small when the null hypothesis is true. This arrow is pointing to the entire fraction. The value of F will be small when the null hypothesis is true. A large value of F indicates that there's more variability between the samples than there are within the samples. Which would be rare if the null hypothesis was true. So a large F provides evidence against the null hypothesis, versus a small F serves mainly to uphold the null hypothesis. We wouldn't reject it if F was small. So a small F, once again, is consistent with the null hypothesis, versus a large F statistic is evidence against the null hypothesis.

Now almost always, we're going to calculate the ANOVA F statistic and the p-value with technology. All but the most simple, straightforward problems will be calculated on technology. So in our scenario, the F statistic, which I calculated with technology, is 1.418. That is not a very large value of F. We can see that the p-value that got returned was 0.26. You should know by now that 0.26 is a very large p-value. And since the p-value is in fact very large, greater than our 0.05 significance level, we fail to reject the null hypothesis. As it turns out, there's no evidence that suggests that the time required to complete the task differs significantly with proficiency level.

And so to recap, ANOVA allows us to compare three or more means. We do this by comparing the variability within each sample to the variability between the samples. The null hypothesis is that all the means are the same, alternative hypothesis is that at least one of them is different. The F and the p-value are almost always calculated on technology. So we ran through, very briefly, an analysis of variance test. We calculated our F statistic. Good luck, and we'll see you next time.