When you decide to use a weighted mean instead of a regular mean, it really depends on the situation and the data you have. Here are some important points I've learned:

When to Use Weighted Mean:

1. Importance of Different Data: If some pieces of data are more important than others, using a weighted mean will show a clearer overall picture. For example, in a school where homework and tests have different effects on your final grade, you’d want to use a weighted mean.

2. Different Group Sizes: If you're comparing groups that have different numbers of people, a weighted mean is the way to go. For instance, if you're figuring out an average score from groups that are not the same size, you should adjust those scores to reflect how big each group is.

3. Using Proportions: A weighted mean works well when your data has different amounts or value for each piece. For example, in a survey where each question counts for a different number of points.

How It Differs from Regular Mean:

• The regular mean (or average) is simple. You just add up all the values and then divide by how many there are. It works well when all the data is similar.

• The weighted mean is a bit different. You take each piece of data, multiply it by its importance (or weight), add those results together, and then divide by the total of all the weights:

$$\text{Weighted Mean} = \frac{\sum (x_i \cdot w_i)}{\sum w_i}$$

In this formula, $x_i$ are the data points, and $w_i$ are their weights.

So, to sum it up: if certain values are more important or if you're working with data from different-sized groups, go for the weighted mean!