Answer:
Sample variance plays a significant role in influencing the estimated standard error and measures of effect size such as r^2 (r squared) and Cohen's d.
Estimated Standard Error: The estimated standard error is inversely related to the sample variance. A larger sample variance indicates greater variability within the data, leading to a larger estimated standard error. Conversely, a smaller sample variance indicates less variability and a smaller estimated standard error.
Effect Size (r^2): The effect size measure r^2, also known as the coefficient of determination, represents the proportion of variance in the dependent variable that can be explained by the independent variable(s). A larger sample variance increases the total variance, which can potentially increase the numerator of r^2. Consequently, a larger sample variance might result in a larger effect size estimate.
Effect Size (Cohen's d): Cohen's d is a measure of the standardized difference between two means. Sample variance affects Cohen's d through its role in calculating the pooled standard deviation. The pooled standard deviation is the square root of the average of the variances from the two groups being compared. If the sample variance is larger, the pooled standard deviation will also be larger, leading to a smaller Cohen's d. Similarly, if the sample variance is smaller, the pooled standard deviation will be smaller, resulting in a larger Cohen's d.
In summary, sample variance influences the estimated standard error, r^2, and Cohen's d. A larger sample variance generally leads to a larger estimated standard error and potentially larger effect size estimates for r^2. However, for Cohen's d, a larger sample variance leads to a smaller effect size estimate.