Close Save changes. Help F1 or? Example Comparing Packing Machines Section In a packing plant, a machine packs cartons with jars. New Machine Old Machine Hypothesis Test Confidence Interval Minitab Do the data provide sufficient evidence to conclude that, on the average, the new machine packs faster? Are these independent samples? Yes, since the samples from the two machines are not related. Are these large samples or a normal population? Normal Probability Plot for Old Machine. Minitab: 2-Sample t-test - Pooled The following steps are used to conduct a 2-sample t-test for pooled variances in Minitab.
The following dialog boxes will then be displayed. When entering values into the samples in different columns input boxes, Minitab always subtracts the second value column entered second from the first value column entered first. The pooled variance is indicated by the horizontal reference line. It is the weighted average of the sample variances. If you think that all groups have the same variance, the pooled variance estimates that common variance.
In two-sample t tests and ANOVA tests, you often assume that the variance is constant across different groups in the data. The pooled variance is an estimate of the common variance. It is a weighted average of the sample variances for each group, where the larger groups are weighted more heavily than smaller groups.
You can download a SAS program that computes the pooled variance and creates the graphs in this article. Although the pooled variance provides an estimate for a common variance among groups, it is not always clear when the assumption of a common variance is valid. The visualization in this article can help, or you can perform a formal "homogeneity of variance" test as part of an ANOVA analysis.
His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis.
Hi, I am a graduate student in South Korea. Thank you for posting such a nice lecture. It was very helpful. I have a question, though. But when you get the Standard Error to get t-value using the pooled variance, how would you do it? But I do not understand why you do not consider all three samples. I mean, it seems like my friend only took into account 2 samples according to the above equation.
Rule of Thumb: If the ratio of the larger variance to the smaller variance is less than 4, then we can assume the variances are approximately equal and use the two sample t-test. For example, suppose sample 1 has a variance of The ratio of the larger sample variance to the smaller sample variance would be calculated as:.
Since this ratio is less than 4, we could assume that the variances between the two groups are approximately equal. Thus, we would use the two sample t-test which means we would calculate the pooled variance. Suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, we collect a random sample of turtles from each population with the following information:. Consider this parent sample:. Now, if you use "literature's formula" to compute the pooled variance, you will get 2.
Instead, if you use "my formula", you will get correct answer. Please understand, I use extreme examples here to show people that the formula indeed wrong. If I use "normal data" which doesn't have a lot of variations extreme cases , then the results from those two formulae will be very similar, and people could dismiss the difference due to rounding error, not because the formula itself is wrong.
This is explained, motivated, and analyzed in some detail in the Wikipedia entry for pooled variance. It does not estimate the variance of a new "meta-sample" formed by concatenating the two individual samples, like you supposed.
As you have already discovered, estimating that requires a completely different formula. Pooled variance is used to combine together variances from different samples by taking their weighted average, to get the "overall" variance.
The problem with your example is that it is a pathological case, since each of the sub-samples has variance equal to zero. Such pathological case has very little in common with the data we usually encounter, since there is always some variability and if there is no variability, we don't care about such variables since they carry no information. You need to notice that this is a very simple method and there are more complicated ways of estimating variance in hierarchical data structures that are not prone to such problems.
As about your example in the edit, it shows that it is important to clearly state your assumptions before starting the analysis. There are several scenarios possible, you can assume that all the points come from the same distribution for simplicity, let's assume normal distribution ,.
Depending on your assumptions, particular method may, or may not be adequate for analyzing the data. In the first case, you wouldn't be interested in estimating the within-group variances, since you would assume that they all are the same. Nonetheless, if you aggregated the global variance from the group variances, you would get the same result as by using pooled variance since the definition of variance is.
In the second case, means differ, but you have a common variance. This example is closest to your example in the edit. In this scenario, the pooled variance would correctly estimate the global variance, while if estimated variance on the whole dataset, you would obtain incorrect results, since you were not accounting for the fact that the groups have different means.
In the third case it doesn't make sense to estimate the "global" variance since you assume that each of the groups have its own variance.
You may be still interested in obtaining the estimate for the whole population, but in such case both a calculating the individual variances per group, and b calculating the global variance from the whole dataset, can give you misleading results. If you are dealing with this kind of data, you should think of using more complicated model that accounts for the hierarchical nature of the data. The fourth case is the most extreme and quite similar to the previous one.
In this scenario, if you wanted to estimate the global mean and variance, you would need a different model and different set of assumptions. In such case, you would assume that your data is of hierarchical structure, and besides the within-group means and variances, there is a higher-level common variance, for example assuming the following model.
In such case, you would use a hierarchical model that takes into consideration both the lower-level and upper-level variability. To read more about this kind of models, you can check the Bayesian Data Analysis book by Gelman et al. This is however much more complicated model then the simple pooled variance estimator. The problem is if you just concatenate the samples and estimate its variance you're assuming they're from the same distribution therefore have the same mean.
But we are in general interested in several samples with different mean.
0コメント