Answer:
- average: 71.0625
- standard deviation:
Explanation:
Let
a = average score of 325 students = 72
b = average score of 75 students = 67
S = total score of 325 students
T = total score of 75 students
u = standard deviation of the score of 325 students = 15
v = standard deviation of the score of 75 students = 12
C = average of the squares of the scores of 325 students
D = average of the squares of the scores of 75 students
The total score of 325 students is calculated as:
S = 325a
= 325(72)
S = 23,400
While the total score of 75 students is calculated in similar manner:
T = 75b
= 75(67)
T = 5,025
To get the total score of 400 students, we just add the total score of 325 students and the total score of 75 students. So, the total score of 400 students is given by
S + T = 23,400 + 5,025
S + T = 28,425
To get the average score of 400 students, we just divide the total score of 400 students by the number of students, which is 400. In terms of equation,
[tex]\text{average score of 400 students} = \frac{S + T }{400}
\\ = \frac{28,425}{400}
\\ \boxed{\text{average score of 400 students} = 71.0625}[/tex]
To calculate the standard deviation of the scores of 400 students, we use the following formula:
[tex]\sigma_{400} = \sqrt{\left( C + D \right) - \left( \frac{S + T}{400} \right)^2}[/tex]
where
[tex]\sigma_{400}[/tex] = standard deviation of the scores of 400 students
[tex] C + D [/tex] = average of the squares of the scores of 400 students
[tex]\frac{S + T}{400}[/tex] = average of the scores of 400 students = 71.0625
Because
C = average of the squares of the scores of 325 students
D = average of the squares of the scores of 75 students
u = standard deviation of the score of 325 students = 15
v = standard deviation of the score of 75 students = 12
a = average score of 325 students = 72
b = average score of 75 students = 67
We have the following
[tex]u^2 = C - a^2
\\ \Rightarrow \boxed{C = a^2 + u^2}
[/tex] (1)
[tex]v^2 = D - b^2 \\ \Rightarrow \boxed{D = v^2 + b^2}[/tex] (2)
So, using equations (1) and (2),
[tex]C + D = (a^2 + u^2) + (v^2 + b^2)
\\ = (72^2 + 15^2) + (12^2 + 67^2)
\\ \boxed{C + D = 10,042}[/tex]
Hence, the standard deviation of the scores of 400 students is calculated as
[tex]\sigma_{400} = \sqrt{\left( C + D \right) - \left( \frac{S + T}{400} \right)^2}
\\ = \sqrt{\left( 10,042 \right) - \left(71.0625 \right)^2}
\\ \boxed{\sigma_{400} \approx 70.65 }[/tex]
