10.3
One-Way ANOVA: Equal Sample Sizes
Consider performing a one-way ANOVA on two different datasets, each containing the heights of students from three samples.
Notice that in both datasets, all three samples have equal sample sizes.
Here, we can state the null hypothesis that the mean heights of all three samples are equal. The alternative hypothesis is that at least one of the means is different from the rest.
First, calculate the sample means and sample variances for both datasets. Observe that only the means of the first samples in both datasets differ substantially, but the sample variances are identical.
Next, calculate the F statistic for both datasets and find the P-values.
The different means of the first samples in both datasets cause a substantial change in the variance between samples. However, the variance within samples remains identical, as it doesn't require the sample mean during calculation.
The different values of variance between samples in both datasets affect the F statistic, leading to different results.
So, we can conclude that the F statistic is substantially affected by the sample mean.
One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
Different sample means can result in different values for the variance estimate: variance between samples. This is because the variance between samples is calculated as the product of the sample size and the variance between the sample means. Thus, two datasets with equal sample sizes can have two different values for variance between samples.
In contrast, it is possible for two different datasets with equal sample sizes to have equal sample variances but different sample means. Since the variance within samples, also called the pooled variance, is calculated as the mean of sample variances, the variance within samples can be equal for two datasets with equal sample sizes.
The computed F statistic value for the two datasets differs since the datasets show unequal values for variance between samples but equal values for variance within samples.
Consider performing a one-way ANOVA on two different datasets, each containing the heights of students from three samples.
Notice that in both datasets, all three samples have equal sample sizes.
Here, we can state the null hypothesis that the mean heights of all three samples are equal. The alternative hypothesis is that at least one of the means is different from the rest.
First, calculate the sample means and sample variances for both datasets. Observe that only the means of the first samples in both datasets differ substantially, but the sample variances are identical.
Next, calculate the F statistic for both datasets and find the P-values.
The different means of the first samples in both datasets cause a substantial change in the variance between samples. However, the variance within samples remains identical, as it doesn't require the sample mean during calculation.
The different values of variance between samples in both datasets affect the F statistic, leading to different results.
So, we can conclude that the F statistic is substantially affected by the sample mean.
From Chapter 10:
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10.3 : One-Way ANOVA: Equal Sample Sizes
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10.1 : What is an ANOVA?
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10.2 : One-Way ANOVA
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10.4 : One-Way ANOVA: Unequal Sample Sizes
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10.5 : Multiple Comparison Tests
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10.6 : Bonferroni Test
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10.7 : Two-Way ANOVA
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