Respuesta :
You can't simply consider the mean of the three means, because each mean is achieved by a different number of students.
Consider an extreme example, just to understand the concept: if 1 students gets a grade of 100 and one million students get a grade of 0, we can't say that the combined mean is 50!
What we need to do is called a weighted average, i.e. an average where we consider the weight of every entry - in this case the weight is given by how many students achieved a certain grade.
So, if each value [tex] x_i [/tex] comes with a weight [tex] a_i [/tex], the weighted average is computed as
[tex] \frac{a_1x_1+a_2x_2+\ldots+a_nx_n}{a_1+a_2+\ldots+a_n} [/tex]
Applying this formula to your case, we have
[tex] \frac{30\cdot 80+40\cdot 85+\ldots+30\cdot 80}{30+40+30} = \frac{2400+3400+2400}{100} = \frac{8200}{100} = 82 [/tex]
Total number of students = 30 + 40 + 30 = 100
Total grade for section 1 = 30 x 80 = 2400
Total grade for section 2 = 40 x 85 = 3400
Total grade for section 3 = 30 x 80 = 2400
Total grades for all the 100 students = 2400 + 3400 + 2400 = 8200
Mean grade for all the 100 students = 8200 ÷ 100 = 82
Answer: 82
