# What is a good metric for comparing a single value to a distribution?

I'd like to compare single values from one distribution to another distribution, effectively transforming the second distribution in such way, that its values reflect both datasets simultaneously, i.e. the values of the second distribution correspond to some sort of normalized difference between the initial values and the first distribution. I thought about using z-score or one sample effect size (Cohen's D) for this task, but I'm unsure if these approaches are justified.

## 1 Answer

The z-score could help you understand how far away a data point in one distribution is from the mean of that distribution, in terms of standard deviations. This can be useful for comparing individual data points across distributions and assessing the significance of the difference.

Cohen's d, on the other hand, provides a standardized measure of the effect size between two distributions. It indicates the difference between the means of the distributions in terms of standard deviations. It's particularly useful for quantifying the magnitude of the difference between two distributions.

If your goal is to transform the second distribution to reflect a normalized difference between the initial values and the first distribution, using Cohen's d might be more appropriate. This is because Cohen's d specifically focuses on the difference between means, and it will give you an effect size that can help you understand the practical significance of the difference.Keep in mind that both z-score and Cohen's d have their limitations Before deciding which approach to use, consider the characteristics of your data.

• Yes, Cohen's D focuses on the difference between means, but if you're comparing a value to a distribution, then you only have mean value of the first distribution. What would be an assumption here - to treat a single value from the second distribution as mean of that distribution? Commented Aug 21, 2023 at 7:30
• When comparing a single value to a distribution, assuming that the single value represents the mean of the second distribution could be a reasonable approximation for the purpose of calculating Cohen's d. Imagine you have a single value and a group of values (a distribution). If you want to compare the single value to the group, you can sort of pretend that the single value is the "average" of that group. This can help you calculate Cohen's d, a measurement that tells you how different the single value is from the group. Commented Aug 21, 2023 at 12:10