Question

1. Give an example of using a weighted mean. how might it be a better average than another measure of center like the mean?

2. Fill in the second column of the table
with any numbers you want.
This column will represent number of students
who had a certain score on a test.

Score	N (weight)
60
70
80
90

Answer 1

1. Weighted mean is a concept of calculating the mean of a data using their weight(importance, number of similar data points, etc.).

Example

In school there are various tests for a particular subject. Let the following be the data for the weightage of the different tests:

First class test - 20%

Second class test - 20%

Homework - 10%

Final exam - 50%

Let each of the above be marked out of 100. Now, if a particular student's score is (50, 60, 60, 100) in the same order as above then his weighted score in the subject can be calculated as:

Here, the final exam has more weightage than all other tests/hw. That's why despite scoring low on first and second class test as well as on the homework, the student was able to get a decent score of 78.

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Weighted mean allows the contribution of a more important data point in the sample to have a greater effect on the mean, thereby ensuring that the inequality among the data set is reflected in the mean.

A normal (arithmetic) mean assumes equal weightage of all the data points for calculating the centre of a data set. In the above example, if arithmetic mean would be (50+60+60+100)/4 = 67.5, which is much lower than the weighted mean of 78. Thus even if the student worked hard and did extremely well in the final exam (which matters the most), his score doesn't reflect any of that.

Alternately, suppose that another student scored (80,80,80,30) in the same subject. His weighted score will be

Thus even if both the students got equal total marks (270 ,which would give equal arithmetic mean),the weighted scores are much different. This ensures that the student with a proper understanding of all the concepts (implying better marks in the final exam) gets his proper due in the form of marks and grades.

2. The second question seems incomplete so it is being solved assuming that after filling the numbers, its arithmetic and weighted average has to be calculated.

Score	N (weight)
60	20
70	12
80	8
90	5

From the above table, the arithmetic mean of the scores would be (60+70+80+90)/4 = 75.

Weighted mean of score:

Total score/Total number of students = (60*20+70*12+80*8+90*5)/(20+12+8+5) = 69.56