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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

If all the weights are equal, then the weighted mean is the same as the simple arithmetic mean.

## Definition

Formally, the weighted mean of a non-empty set of data $\{x_1, x_2, \dots , x_n\},$ with non-negative weights is

$\bar{x} = \frac{ \sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}$.

Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed).

The simple average $\frac {1}{n}\sum\limits_{i=1}^n {x_i}$ is a special case of the weighted mean where all data points have equal weights.

## Example

Players searching for certain cards usually open several Pro Kit Boxes, and mostly also different types of boxes. Let's assume a player is looking for engines and obtains 200 cards from opening two kinds of boxes:

• 180 cards from (engine drop rate: 45.73 %) and
• 20 cards from (engine drop rate: 4.24 %).

The simple average for engine cards from these boxes would be

$\dfrac{ 45.73 % + 4.24 %}2 =\; $$\mathbf{ 24.99 %} . However, this does not take into account that the majority of cards comes from Finish Line Boxes, so the drop rate of engine cards should be much closer to the 45.73 % of the Finish Line Box than just in the middle. Therefore, the weighted average is used: \dfrac{180 \cdot 45.73 % + 20 \cdot 4.24 %}{200} =\;$$ \mathbf{ 41.58 %}$

As drop rates are expected values, the player can expect that around 83 cards (41.58 % of 200) will be engines.