Weighted Arithmetic Mean
When calculating the arithmetic mean, the importance of all the items are considered to be equal. However, there may be situations in which all the items under considerations are not of equal importance. For example, when we want to find the average number of marks per students in different subjects like mathematics, statistics, physics and biology. These subjects do not have equal importance. Thus, the arithmetic mean computed by considering the relative importance of each item is called the weighted arithmetic mean. To give due importance to each item under consideration, we assign a number called a weight to each item in proportion to its relative importance.
The weighted arithmetic mean is computed by using the following formula:
$${\overline X _w} = \frac{{\sum wx}}{{\sum w}}$$
Here:
$${\overline X _w}$$ stands for weighted arithmetic mean
$$x$$ stands for values of the items and
$$w$$ stands for the weight of the item
Example:
A student obtained the marks 40, 50, 60, 80, and 45 in math, statistics, physics, chemistry and biology respectively. Assuming weights 5, 2, 4, 3, and 1 respectively for the above mentioned subjects, find the weighted arithmetic mean per subject.
Solution:
Subject 
Mark Obtained
$$x$$ 
Weight
$$w$$ 
$$wx$$

Math

$$40$$

$$5$$

$$200$$

Statistics

$$50$$

$$2$$

$$100$$

Physics

$$60$$

$$4$$

$$240$$

Chemistry

$$80$$

$$3$$

$$240$$

Biology

$$45$$

$$1$$

$$45$$

Total

$$\sum w = 15$$

$$\sum wx = 825$$

Now we will find the weighted arithmetic mean as:
$${\overline X _w} = \frac{{\sum wx}}{{\sum w}} = \frac{{825}}{{15}} = 55$$ marks/subject.